From 6150761c9f4dc88f320143a3a4bbd12e6294135b Mon Sep 17 00:00:00 2001 From: Jacob Neumann Date: Wed, 7 Jul 2021 23:59:16 -0600 Subject: [PATCH] Begin documentation of Language --- Language.sml | 19 +-- documentation/language.pdf | Bin 0 -> 136689 bytes tex/language.tex | 266 +++++++++++++++++++++++++++++++++++++ tex/spec.sty | 1 + 4 files changed, 269 insertions(+), 17 deletions(-) create mode 100644 documentation/language.pdf create mode 100644 tex/language.tex diff --git a/Language.sml b/Language.sml index d854c33..55581b3 100644 --- a/Language.sml +++ b/Language.sml @@ -6,6 +6,7 @@ sig val everything : 'S language val nothing : 'S language + val singleton : ''S list -> ''S language val just : ''S list list -> ''S language val Or @@ -21,9 +22,6 @@ sig val lengthLess : int -> 'S language val lengthGreater : int -> 'S language - val Sub : ''S list -> ''S language - val Sup : ''S list -> ''S language - val str : char language -> string -> bool end @@ -35,6 +33,7 @@ struct fun everything (x:'S list) = true fun nothing (x : 'S list) = false + val singleton = Fn.equal fun just ([] : ''S list list) s = false | just (x::xs) s = (s=x) orelse just xs s @@ -48,20 +47,6 @@ struct fun lengthLess n s = (List.length s)n - - fun isPrefix [] _ = true - | isPrefix _ [] = false - | isPrefix (c::cs) (c'::cs') = - (c=c') andalso (isPrefix cs cs') - fun isSubList [] _ = true - | isSubList _ [] = false - | isSubList sub (c::cs) = - (isPrefix sub (c::cs)) - orelse (isSubList sub cs) - - fun Sub s0 s = isSubList s s0 - fun Sup s0 s = isSubList s0 s - fun str L = L o String.explode end diff --git a/documentation/language.pdf b/documentation/language.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aaa551ba50500f98d9e5498c5199953e01666ca4 GIT binary patch literal 136689 zcma(2Q*bU!w1$hu&Wvr__Ka=Yw(ZQ=wr$(CZQK6B^Pk%LZmp`l`lkD~t48^UEXaFwAU(jD+?^RxrH0Fbp!LcIGY?gzTIwg#XWiVGy&laWQowWDv74 zbTJh%HMTb~h2iIiadvSsHME8C*sM{Nwclcd>-khaKgTVWTz+%R1CF%QSliyj%U+g+ 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+\begin{example} + \begin{itemize} + \item $\Vals(\code{bool}) = \set{\code{true}, \code{false}}$ + \item $\Vals(\code{bool -> bool}) = \set{\code{not}, \code{Fn.id}, \code{(fn _ => true)}, \code{(fn _ => false)}}$ + \end{itemize} +\end{example} +\begin{definition} + A \keyword{decision problem} of type \code{t} is a subset + \[ L \sub \Vals(\code t) \] +\end{definition} +\begin{definition}\label{defn:computes} + A decision problem $L\sub\Vals(\code t)$ is said to be \keyword{decidable} if there is a \textbf{total} value \code{D : t -> bool} such that + \[ \code{D(v)}\stepsTo\code{true} \qtq{iff} \code{v}\in L \] + for all values \code{v:t}. In this case, we say that \code{D} \keyword{decides} (or \keyword{computes}) $L$ +\end{definition} +\begin{example} + The \textit{subset sum problem} is a family of decision problems of type \code{int list}, one for each value \code{n : int} + \[ \textsf{SUBSETSUM}_{\code n} = \compSet{\code{l}\in\Vals(\code{int list})}{\code{foldr op+ 0 l}\eeq \code{n}} \] + For each \code{n}, the problem $\textsf{SUBSETSUM}_{\code n}$ is decidable: we can define a total function + \[ \code{subsetSum n : int list -> bool} \] + such that $\code{subsetSum n l}\stepsTo\code{true}$ iff $\code{foldr op+ 0 l}\eeq\code{n}$. +\end{example} + +\begin{definition}\label{defn:language} + Given a total function \code{D : t -> bool}, define the \keyword{language} of \code{D} to be the set + \[ \mc L(\code{D}) = \compSet{\code v\in\Vals(\code t)}{\code{D(v)}\stepsTo\code{true}}. \] +\end{definition} + + +\begin{definition} + An \keyword{equality type} is any type \code{t} such that + \[ \code{ (op =) : t * t -> bool } \] + is well-typed, i.e. any type whose values we can compare with the \code{=} and \code{<>} operators. +\end{definition} + +\begin{definition} + Given an equality type \code{Sigma}, a \keyword{decision problem over the alphabet \code{Sigma}} is a subset + \[ L \sub \Vals(\code{Sigma list}). \] +\end{definition} + +\section{Specification} + +\begin{note} + Throughout, \code{Sigma} will denote some equality type, the type of our ``alphabet''. Though some of our results will hold when \code{Sigma} is allowed to be a general polymorphic type (e.g. \autoref{lemma:boolAlg}), we are mainly concerned with situations where \code{Sigma} is an equality type (and the values \code{singleton} and \code{just} demand that \code{Sigma} be an equality type). + + A paradigm example is to take + \begin{codeblock} + type Sigma = char + \end{codeblock} + + +\end{note} + +\begin{lemma} + For any total function \code{D : Sigma list -> bool} and any subset $L\sub\Vals(\code{Sigma list})$, the following are equivalent: + \begin{enumerate} + \item[(1)] \code{D} computes $L$ (\autoref{defn:computes}) + \item[(2)] For \textit{all} values \code{v : Sigma list}, + \[ \code{D(v)} \quad\stepsTo\quad \begin{cases} + \code{true} &\text{ if }\code{v}\in L\\ + \code{false} &\text{ if }\code v\not\in L + \end{cases} + \] + \item $\mc L(\code D) = L$ (\autoref{defn:language}) + \end{enumerate} +\end{lemma} +%\begin{note} +% A total function \code{f : t -> bool} decides $L$ iff $\mc L(\code f)=L$. +%\end{note} +\TypeSpec{'S language = 'S list -> bool}{}{} + +\spec{everything}{'S language}{}{\code{everything : Sigma language} is a total function computing the decision problem $\Vals(\code{Sigma list})$ over the alphabet \code{Sigma}.} +\spec{nothing}{'S language}{}{\code{nothing : Sigma language} is a total function computing the decision problem $\emptyset$.} +\spec{singleton}{''S list -> ''S language}{}{\code{(singleton v): Sigma language} is a total function computing the decision problem $\set{\code v}$} +\spec{just}{''S list list -> ''S language}{}{$\code{(just [v}_1,\code{v}_2,\ldots,\code{v}_n\code{]): Sigma language}$ is a total function computing the decision problem $\set{\code v_1,\code v_2,\ldots,\code v_n}$} + +\spec{Or}{'S language * 'S language -> 'S language}{\code{L1} and \code{L2} are total}{\code{Or(L1,L2)} is a total function such that + \[ \mc L(\code{Or(L1,L2)}) = \mc L(\code{L1}) \cup \mc L(\code{L2}) \] +} +\spec{And}{'S language * 'S language -> 'S language}{\code{L1} and \code{L2} are total}{\code{And(L1,L2)} is a total function such that + \[ \mc L(\code{And(L1,L2)}) = \mc L(\code{L1}) \cap \mc L(\code{L2}) \] +} +\spec{Not}{'S language -> 'S language}{\code{L} is total}{For any \code{L : Sigma language}, \code{Not(L)} is a total function such that + \[ \mc L(\code{Not(L)}) = \Vals(\code{Sigma list}) \setminus \mc L(\code{L}) \] +} +\spec{Xor}{'S language * 'S language -> 'S language}{\code{L1} and \code{L2} are total}{\code{Xor(L1,L2)} is a total function such that + \[ \mc L(\code{Xor(L1,L2)}) = \mc L(\code{Or(L1,L2)}) \setminus \mc L(\code{And(L1,L2)}) \] +} + +\spec{lengthEqual}{int -> 'S language}{$\code{n}\geq\code0$}{\code{lengthEqual n} is a total function such that + \begin{align*} + & \mc L(\code{lengthEqual n}) \\ + & = \compSet{\code{s}\in\Vals(\code{Sigma list})}{\code{((List.length s)=n)}\stepsTo\code{true}} + \end{align*} +} +\spec{lengthLess}{int -> 'S language}{$\code{n}\geq\code0$}{\code{lengthLess n} is a total function such that + \begin{align*} + & \mc L(\code{lengthLess n}) \\ + &= \compSet{\code{s}\in\Vals(\code{Sigma list})}{\code{((List.length s) 'S language}{$\code{n}\geq\code0$}{\code{lengthGreater n} is a total function such that + \begin{align*} + &\mc L(\code{lengthGreater n}) \\ + &= \compSet{\code{s}\in\Vals(\code{Sigma list})}{\code{((List.length s)>n)}\stepsTo\code{true}} + \end{align*} +} + +\section{Lemmas} +Throughout, \code{L,L1,L2,L3 : Sigma language} are total. + +\begin{proposition}[Totality] \label{prop:totality} + All values of the \code{Sigma language} type produced using the \code{Language} module methods are total: + \begin{itemize} + \item \code{everything} is total + \item \code{nothing} is total + \item For any value \code{v : Sigma list}, \code{(singleton v)} is total + \item For any value \code{l : Sigma list list}, \code{(just l)} is total + \item All the curried higher-order functions (\code{singleton}, \code{just}, \code{Or}, \code{And}, \code{Not}, \code{Xor}, \code{lengthEqual}, \code{lengthLess}, \code{lengthGreater}, and \code{str}) are all \textit{total} in the trivial sense: upon being supplied one argument, they evaluate to a value (a function expecting the next curried argument) + \item If \code{L1,L2 : Sigma language} are total, + \begin{itemize} + \item \code{Or(L1,L2)} is total + \item \code{And(L1,L2)} is total + \item \code{Not(L1)} is total + \item \code{Xor(L1,L2)} is total + \end{itemize} + \item For any $\code{n}\geq\code 0$, + \begin{itemize} + \item \code{lengthEqual n} is total + \item \code{lengthLess n} is total + \item \code{lengthGreater n} is total + \end{itemize} + \item If \code{L : char language} is total, so too is \code{str L} + \end{itemize} +\end{proposition} +\begin{proof} + We prove several paradigmatic cases, and the others can be done similarly. + + Recall \code{just} is implemented as + \auxLibIndent{Language.sml}{37}{38}{-20pt} + So we can prove the totality of \code{(just l)} by structural induction on \code{l : Sigma list list}. The base case is immediate: for any value \code{s : Sigma list}, $\code{just [] s}\stepsTo\code{false}$, a value. Inductively assuming \code{just xs s} valuable, then we can see that \code{(just (x::xs) s)} is also valuable, since \code{(s=x)} is valuable and, if \code{(s=x)} evaluates to \code{false}, then + \[ \code{just (x::xs) s} \quad\stepsTo\quad \code{just xs s} \] + which our IH tells us is valuable, completing the proof. + + + Recall \code{And} is implemented as + \auxLibIndent{Language.sml}{41}{41}{-20pt} + so if \code{L1} and \code{L2} are total, then \code{(L1 s)} and \code{(L2 s)} are valuable, hence \code{(L1 s) andalso (L2 s)} is valuable, proving \code{And(L1,L2)} total. + + Taking for granted that \code{List.length} is total, it follows that the expression \code{(List.length s)=n} is valuable. Since this is the body of \code{lengthEqual n s}, we get that \code{lengthEqual n} is total. + + Proving the totality of \code{str L} from the totality of \code{L} is a straightforward consequence of the totality of \code{String.explode}. +\end{proof} +\begin{lemma} + \[ \code{just} \qeq \code{(foldr Or nothing) o (map singleton)} \] +\end{lemma} +\begin{proof} + It suffices to show that for all values \code{l : Sigma list list}, + \[ \code{just l} \qeq \code{foldr Or nothing (map singleton l)} \] + which we do by structural induction on \code{l}. + + \bcBox \code{l = []}. Pick arbitrary \code{s : Sigma list}. Then + \begin{align*} + &\code{just [] s}\\ + &\qeq \code{false} \tag{Defn. \code{just}}\\ + &\qeq \code{nothing s} \tag{Defn. \code{nothing}}\\ + &\qeq \code{(foldr Or nothing []) s} \tag{Defn. \code{foldr}}\\ + &\qeq \code{(foldr Or nothing (map singleton [])) s} \tag{Defn. \code{map}} + \end{align*} + Establishing that $\code{just []}\eeq\code{foldr Or nothing (map singleton [])}$. + + \ihBox Suppose for some \code{xs : Sigma list list} that + \[ \code{just xs} \qeq \code{foldr Or nothing (map singleton xs)} \] + Now pick some \code{x : Sigma list}. We'll show that + \[ \code{just (x::xs)} \qeq \code{foldr Or nothing (map singleton (x::xs))} \] + Pick arbitrary \code{s : Sigma list}. + \begin{align*} + &\code{just (x::xs) s}\\ + &\eeq \code{(s=x) orelse just xs s} \tag{Defn. \code{just}}\\ + &\eeq \code{(x=s) orelse just xs s} \tag{Symmetry of \code=}\\ + &\eeq \code{(Fn.equal x s) orelse just xs s} \tag{Defn. \code{Fn.equal}}\\ + &\eeq \code{(singleton x s) orelse just xs s} \tag{Defn. \code{singleton}}\\ + &\eeq \code{(singleton x s) orelse (foldr Or nothing (map singleton xs)) s} \tag*{\refIH}\\ + &\eeq \code{Or(singleton x, (foldr Or nothing (map singleton xs))) s} \tag{Defn. \code{Or}, \autoref{prop:totality}, higher-order totality of \code{map} and \code{foldr}}\\ + &\eeq \code{(foldr Or nothing ((singleton x)::(map singleton xs))) s} \tag{Defn. \code{foldr}, \autoref{prop:totality}, higher-order totality of \code{map}}\\ + &\eeq \code{(foldr Or nothing (map singleton (x::xs))) s} \tag{Defn. \code{map}} + \end{align*} + so we're done. +\end{proof} +\begin{corollary} + \[ \code{nothing} \qeq \code{just []} \] +\end{corollary} + +\begin{lemma} \label{lemma:boolAlg} +For any total \code{L,L1,L2,L3 : Sigma language}, the following equivalences hold + \begin{itemize} + \item $\code{And(L1,And(L2,L3))} \qeq \code{And(And(L1,L2),L3)}$ + \item $\code{And(L1,L2)} \qeq \code{And(L2,L1)}$ + \item $\code{Or(L1,Or(L2,L3))} \qeq \code{Or(Or(L1,L2),L3)}$ + \item $\code{Or(L1,L2)} \qeq \code{Or(L2,L1)}$ + \item $\code{And(L,everything)} \qeq \code{L} \qeq \code{And(everything,L)}$ + \item $\code{Or(L,nothing)} \qeq \code{L} \qeq \code{Or(nothing,L)}$ + \item $\code{Not(Not(L))} \qeq \code{L}$ + \item $\code{Or(L1,And(L1,L2))} \qeq \code{L1} \qeq \code{And(L1,Or(L1,L2))}$ + \item $\code{Or(L1,And(L2,L3))} \qeq \code{And(Or(L1,L2),Or(L1,L3)))}$\\ + $\code{And(L1,Or(L2,L3))} \qeq \code{Or(And(L1,L2),And(L1,L3)))}$ + \item $\code{And(L,Not L)} \qeq \code{nothing}$\\ + $\code{Or(L,Not L)} \qeq \code{everything}$ + \end{itemize} +\end{lemma} + +\begin{lemma} + For all values $\code{n}\geq\code 0$, + \[ \code{Not(lengthGreater n)} \qeq \code{Or(lengthEqual n,lengthLess n)} \] +\end{lemma} +\begin{proof} + Pick arbitrary \code{s : Sigma list}, and let \code{m} denote the value of \code{List.length s}. + \begin{align*} + &\code{(Not(lengthGreater n)) s}\\ + &\eeq \code{not(lengthGreater n s)} \tag{Defn. \code{Not}, \autoref{prop:totality}}\\ + &\eeq \code{not((List.length s)>n} \tag{Defn. \code{lengthGreater}}\\ + &\eeq \code{not (m>n)} \tag{Defn. \code{m}}\\ + &\eeq \code{m=n orelse m