Some Math about Minesweeper
Desires
A complete algorithm for solving minesweeper. By ‘complete’ I mean one which can solve any game which is in principle solvable without guessing.
An implementation of a minesweeper app that guarantees only presenting solvable games, as proven by the above algorithm
The game, traditionally
Traditionally, the game Minesweeper consists of a finite-size grid of cells, some of which are said to contain a bomb. At the start of the game, the player has no knowledge of which cells contain a bomb. The game proceeds in a sequence of turns; on each turn, the player must choose one cell to reveal. Revealing a cell has one of two effects. If the cell contains a bomb, then the game ends immediately and is considered a loss. If the cell does not contain a bomb, then that fact is revealed to the user, and additionally the user is granted knowledge of how many bombs are present in the 3x3 neighorhood around the revealed cell. (Usually this information is presented by displaying a number in the place of the revealed cell.) Typically boms are placed randomly around the game board. This means that at the start of the game, from the player’s persective, all cells are equally likely to contain a bomb, and hence whether the first move is a loss or not is pure luck. For this reason, some Minesweeper implementations choose to defer the placement of bombs until after the first reveal in order to guarantee that the first reveal is a non-bomb. Others go further and guarantee that the first cell revealed and all surrounding cells in the 3x3 neighborhood around that cell are non-bombs. This gives a rough characterization of “the” traditional Minesweeper game. Implementations may vary on details, such as:
Whether or not cells on opposing sides of the game board are considered neighbors; ie, whether the game board is a rectangle or a torus
Behaviour on first reveal, as discussed above.
Size of the game board, and number of bombs
Placement of bombs
Behaviour on reveal of a cell with no bomb neighbors (a “zero-valued” cell): some implementations, upon such an action, will reveal all contiguous zero-valued cells
Note that in this characterization we do away with the concept of flagging a cell, which most Minsweeper implementations include. The ability to flag a cell can be seen as pure user convenience and does not affect the game, but removing it makes analysis simpler.
The game, for us
A game state consists of the following data:
A finite set of cells, typically denoted Cell %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Cell}
A valuation function Σ⊆P(Cell)×N %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma \subseteq \cl P(\text{Cell}) \times \bb N taking a set of cells (often called a neighborhood) to the number of bombs we know to be present between them. Note that the domain of Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma is not necessarily all of P(Cell) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cl P(\text{Cell}), since we won’t know the valuation of all sets of cells (unless the game is complete). We could equivalently consider this function to have signature Σ:P(Cell)→{?}∪N %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma : \cl P(\text{Cell}) \to \{ ? \} \cup \bb N, where ?∉N %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } ? \notin \bb N denotes that the valuation of a neighborhood is unkown.
A bomb function B:Cell→{0,1} %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B : \text{Cell} \to \{ 0, 1 \} giving whether or not each cell is actually a bomb or not. The value 1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1 denotes that it is a bomb, and 0 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 0 denotes that it is not
A cell c %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c is considered to be revealed if {c}∈dom(Σ) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \{c\} \in \text{dom}(\Sigma). If this is the case, then one also knows B(c) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B(c). A turn consists of choosing a cell c %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c to reveal. The result of revealing a cell is that the valuation function Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma is updated to at least include the singleton set {c} %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \{c\} in its domain. If Σ({c})=1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma(\{c\}) = 1, then we know B(c)=1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B(c) = 1 and the game is an immediate loss. If Σ({c})=0 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma(\{c\}) = 0, the game continues. Note that {c} %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \{c\} need not be the only neighborhood included; other neighborhoods may also be revaled. For instance, in the case of traditional minesweeper, the valuation function is also updated with the 3x3 neighborhood around the revealed cell. Note that this abstraction makes no assumptions about the shape of the game board (besides finiteness).
Completeness: the goal
One goal is to present a ‘complete’ algorithm for solving a game of minesweeper, where ‘complete’ is meant in the sense that the algorithm will beat the game in all cases where no guessing is required. To make that concrete, we introduce the following notions. A bomb function B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B and valuation function Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma are said to be compatible if they do not disagree; that is, if for every neighborhood C∈dom(Σ) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C \in \text{dom}(\Sigma) we have that Σ(C) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma(C) is indeed the number of bombs assigned by B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B. In other words, (∀C∈dom(Σ))(Σ(C)=∑c∈CB(c)) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Bigg( \forall C \in \t{dom}(\Sigma) \Bigg)\left( \Sigma(C) = \sum_{c \in C} B(c) \right) This can also be interpreted as saying that B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B ‘respects’ or ‘realizes’ Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma Say that a cell c %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c is determined relative to a valuation function Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma if every bomb function B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B compatible with Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma agrees on c %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c; that is, if (∃b0∈{0,1})(∀bomb-function B)(B compatible with Σ  ⟹  B(c)=b0) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } ( \exists b_0 \in \{0,1\} )( \forall \t{bomb-function } B )( B \text{ compatible with } \Sigma \implies B(c) = b_0 ) If a turn is spent revealing a determined cell, call that a progression. Say that a game is progressable if there exists a progression. Call a game solvable if there exists a sequence of progressions resulting in a win.
Prompts
A prompt consists of a set Cell %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Cell} and a valuation function Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma. A prompt is the same as a game state but without the bomb function. The name prompt comes from thinking of it as a ‘puzzle to be solved’. Given two prompts P1=(C1,Σ1) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_1 = (C_1, \Sigma_1) and P2=(C2,Σ2) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_2 = (C_2, \Sigma_2), denote by P1∪P2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_1 \cup P_2 their union, formed by unioning their cells and valuation functions. This is only well-defined if the valuation functions agree on C1∩C2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C_1 \cap C_2. Given two prompts P1=(C1,Σ1) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_1 = (C_1, \Sigma_1) and P2=(C2,Σ2) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_2 = (C_2, \Sigma_2), denote by P1+P2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_1 + P_2 their coproduct, consisting of the cell set C1+C2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C_1 + C_2 and the valuation function constructed from Σ1,Σ2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma_1, \Sigma_2 in the natural way. Given a prompt P=(C,Σ) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P = (C, \Sigma) and cells x,y∈C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x, y \in C, denote by P/x=y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P / x = y the prompt (C∪{z},Σ∪{({x,z},1),({y,z},1)}) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (C \cup \{z\}, \Sigma \cup \{ (\{x, z\}, 1), (\{y, z\}, 1) \}) for some z∉C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } z \notin C. This adds a new cell z %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } z to P %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P along with assertions that Σ(x,z)=Σ(y,z)=1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma(x, z) = \Sigma(y, z) = 1, which states that x↔¬z↔y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x \leftrightarrow \neg z \leftrightarrow y and therefore x↔y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x \leftrightarrow y; ie, x %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x and y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } y evaluate equivalently. Relative to a prompt (C,Σ) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (C, \Sigma), for a vector c‾ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \ol c of cells denote by span(c‾) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{span}(\ol c) the set of valuations for those cells compatible with Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma. For instance, if C={a,b} %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C = \{a,b\} and Σ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma demands that a+b=1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a + b = 1 and nothing else, then we get span(⟨a,b⟩)={⟨0,1⟩,⟨1,0⟩} %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{span}(\tup{a,b}) = \{ \tup{0,1}, \tup{1,0} \}. Let P=P1+P2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P = P_1 + P_2 and take a vector of variables v‾∈C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \ol v \in C. Assume that this vector decomposes as the concatenation v‾=v‾1v‾2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \ol v = \ol v_1 \ol v_2, where v‾1∈C1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \ol v_1 \in C_1 and v‾2∈C2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \ol v_2 \in C_2. Then spanP(v‾)=spanP1(v‾1)×spanP2(v‾2) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{span}_P(\ol v) = \text{span}_{P_1}(\ol v_1) \times \text{span}_{P_2}(\ol v_2) (where × %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \times is not plain cartesian product but a “zipping” cartesian product which combines pairs by concatenation) Now need case for spanP(v‾) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{span}_P(\ol v) in P=P⋆/x=y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P = P^\star / x=y Conjecture: any prompt can be decomposed into a sequence of coproducts followed by quotients; the span of cell-vectors is the same in the original prompt as in its decomposition. The goal here is to first show some property of the span of cell-vectors, such as that (for instance) all spans are of odd cardinality. Then we use that to show that conjunction cannot be encoded as a prompt, for instance because if we had cells a,b,x %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a, b, x such that x↔a∧b %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x \leftrightarrow a \land b then we’d have that span(⟨a,b,x⟩)=4 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{span}(\tup{a, b, x}) = 4 is even, which contradicts that all spans are odd.
Completeness: an attempt
One attempt at a complete set of rules for solving a game of minesweeper are the following: Σ(A)=∣A∣; a∈AB(a)=1(bomb-intro)B(a)=1Σ({a})=1(bomb-elim)Σ(A)=0; a∈AB(a)=0(safe-intro)B(a)=0Σ({a})=0(safe-elim)Σ(U)=n; Σ(A)=k; A⊆UΣ(U∖A)=n−k(split) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{gather*} \frac{\Sigma(A) = \norm A;\ a \in A}{B(a) = 1} \hspace{10pt} \t{(bomb-intro)} \\ \\ \frac{B(a) = 1}{\Sigma(\{a\}) = 1} \hspace{10pt} \t{(bomb-elim)} \\ \\ \frac{\Sigma(A) = 0;\ a \in A}{B(a) = 0} \hspace{10pt} \t{(safe-intro)} \\ \\ \frac{B(a) = 0}{\Sigma(\{a\}) = 0} \hspace{10pt} \t{(safe-elim)} \\ \\ \frac{ \Sigma(U) = n;\ \Sigma(A) = k;\ A \subseteq U }{ \Sigma(U\setminus A) = n - k } \hspace{10pt} \t{(split)} \end{gather*} Unfortunately, this rule set is not complete. One counter-example is as follows: – – – – – 1 2 – a b c d – – – – Here a number denotes the valuation of a revealed cell, and letters and hyphens denote un-revealed cells. This valuation determines that B(d)=1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B(d) = 1. However, none of the proposed rules are applicable. Hence, the ruleset is incomplete.
Relation to Boolean Satisfiability
One tempting re-interpretation of the game is as a phrasing for the boolean satisfiability problem. Given a game state, we may consider each cell to be a variable. Then a valuation Σ(C)=n %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma(C) = n corresponds to the statement that exactly n %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n of the variables in C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C are true, which is expressible as a finite sentence. For instance, Σ({a,b,c})=2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma(\{a, b, c\}) = 2 becomes (a∧b∧¬c)∨(a∧¬b∧c)∨(¬a∧b∧c) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (a \land b \land \neg c) \lor (a \land \neg b \land c) \lor (\neg a \land b \land c) It is unclear to me at this moment whether the problem of solving minesweeper is equivalent to boolean satisfiability. While each minesweeper game can be interpreted as a boolean expression, I am unsure if the reverse is also true. Are there boolean sentences which cannot be encoded as minesweeper games? Any sentence which conists only of conjunctions and negations (under the conjunctons), and with no repeated variables, is easy to express. Let C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C be the set of variables, and let T⊆C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } T \subseteq C be the set of non-negated variables. Let Σ={(T,∣T∣)} %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Sigma = \{ (T, \norm T) \}. Then by split and bomb-intro we can deduce that B(c)=1  ⟺  c∈T %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B(c) = 1 \iff c \in T.