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doc/02.tex
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doc/02.tex
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\documentclass{article}
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\usepackage{amssymb, amsmath}
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\usepackage{amsmath, amsthm, amssymb}
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\usepackage{mathtools}
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\definecolor{darkgray}{gray}{0.30}
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\def\transarrow{\xrightarrow}
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\newcommand{\setarrow}[1]{\def\transarrow{#1}}
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\newcommand{\trans}[3]{{#1}\transarrow{\padding#2\padding}{#3}}
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\newcommand{\ctrans}[4]{{#1}\transarrow{\padding#2\padding}{#3},\;{#4}}
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\newcommand{\llang}[1]{\mbox{\lstinline[mathescape]|#1|}}
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\newcommand{\pair}[2]{\inbr{{#1}\mid{#2}}}
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\newcommand{\inbr}[1]{\left<{#1}\right>}
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\newcommand{\highlight}[1]{\color{red}{#1}}
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\newcommand{\ruleno}[1]{\eqno[\scriptsize\textsc{#1}]}
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\newcommand{\rulename}[1]{\textsc{#1}}
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\newcommand{\lfp}[1]{fix_{#1}}
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\newcommand{\gfp}[1]{Fix_{#1}}
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\newcommand{\vsep}{\vspace{-2mm}}
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\newcommand{\supp}[1]{\scriptsize{#1}}
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\newcommand{\sembr}[1]{\llbracket{#1}\rrbracket}
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\newcommand{\cd}[1]{\texttt{#1}}
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\newcommand{\free}[1]{\boxed{#1}}
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\newcommand{\binds}{\;\mapsto\;}
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\newcommand{\sv}[1]{\mbox{\textbf{#1}}}
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\newcommand{\bnd}[2]{{#1}\mkern-9mu\binds\mkern-9mu{#2}}
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\definecolor{light-gray}{gray}{0.90}
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\newcommand{\graybox}[1]{\colorbox{light-gray}{#1}}
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\lstdefinelanguage{ocaml}{
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keywords={let, begin, end, in, match, type, and, fun,
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function, try, with, class, object, method, of, rec, repeat, until,
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while, not, do, done, as, val, inherit, module, sig, @type, struct,
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if, then, else, open, virtual, new, fresh},
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sensitive=true,
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%basicstyle=\small,
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commentstyle=\scriptsize\rmfamily,
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keywordstyle=\ttfamily\bfseries,
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identifierstyle=\ttfamily,
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basewidth={0.5em,0.5em},
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columns=fixed,
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fontadjust=true,
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literate={fun}{{$\lambda$}}1 {->}{{$\to$}}3 {===}{{$\equiv$}}1 {=/=}{{$\not\equiv$}}1 {|>}{{$\triangleright$}}3 {\&\&\&}{{$\wedge$}}2 {|||}{{$\vee$}}2 {^}{{$\uparrow$}}1,
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morecomment=[s]{(*}{*)}
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}
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\lstset{
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mathescape=true,
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%basicstyle=\small,
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identifierstyle=\ttfamily,
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keywordstyle=\bfseries,
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commentstyle=\scriptsize\rmfamily,
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basewidth={0.5em,0.5em},
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fontadjust=true,
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escapechar=!,
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language=ocaml
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}
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\sloppy
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\newcommand{\ocaml}{\texttt{OCaml}\xspace}
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\theoremstyle{definition}
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\title{Statements, Stack Machine, Stack Machine Compiler\\
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(the first draft)
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}
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\author{Dmitry Boulytchev}
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\begin{document}
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\maketitle
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\section{Statements}
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More interesting language~--- a language of simple statements:
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$$
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\begin{array}{rcl}
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\mathscr S & = & \mathscr X \mbox{\lstinline|:=|} \;\mathscr E \\
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& & \mbox{\lstinline|read (|} \mathscr X \mbox{\lstinline|)|} \\
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& & \mbox{\lstinline|write (|} \mathscr E \mbox{\lstinline|)|} \\
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& & \mathscr S \mbox{\lstinline|;|} \mathscr S
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\end{array}
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$$
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Here $\mathscr E, \mathscr X$ stand for the sets of expressions and variables, as in the previous lecture.
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Again, we define the semantics for this language
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$$
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\sembr{\bullet}_{\mathscr S} : \mathscr S \mapsto \mathbb Z^* \to \mathbb Z^*
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$$
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with the semantic domain of partial functions from integer strings to integer strings. This time we will
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use \emph{big-step operational semantics}: we define a ternary relation ``$\Rightarrow$''
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$$
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\Rightarrow \subseteq \mathscr C \times \mathscr S \times \mathscr C
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$$
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where $\mathscr C = \Sigma \times \mathbb Z^* \times \mathbb Z^*$~--- a set of all configurations during a
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program execution. We will write $c_1\xRightarrow{S}c_2$ instead of $(c_1, S, c_2)\in\Rightarrow$ and informally
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interpret the former as ``the execution of a statement $S$ in a configuration $c_1$ completes with the configuration
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$c_2$''. The components of a configuration are state, which binds (some) variables to their values, and input and
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output streams, represented as (finite) strings of integers.
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The relation ``$\Rightarrow$'' is defined by the following deductive system (see Fig.~\ref{bs_stmt}). The first
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three rules are \emph{axioms} as they do not have any premises. Note, according to these rules sometimes a program
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cannot do a step in a given configuration: a value of an expression can be undefined in a given state in rules
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$\rulename{Assign}$ and $\rulename{Write}$, and there can be no input value in rule $\rulename{Read}$. This style of
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a semantics description is called big-step operational semantics, since the results of a computation are
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immediately observable at the right hand side of ``$\Rightarrow$'' and, thus, the computation is performed in
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a single ``big'' step. And, again, this style of a semantic description can be used to easily implement a
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reference interpreter.
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With the relation ``$\Rightarrow$'' defined we can abbreviate the ``surface'' semantics for the language of statements:
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\setarrow{\xRightarrow}
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\[
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\forall S\in\mathscr S,\,\forall i\in\mathbb Z^*\;:\;\sembr{S}_{\mathscr S} i = o \Leftrightarrow \trans{\inbr{\bot, i, \epsilon}}{S}{\inbr{\_, \_, o}}
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\]
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\begin{figure}[t]
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\[\trans{\inbr{s, i, o}}{\llang{x := $\;\;e$}}{\inbr{s[x\gets\sembr{e}_{\mathscr E}\;s], i, o}}\ruleno{Assign}\]
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\[\trans{\inbr{s, z\llang{::}i, o}}{\llang{read ($x$)}}{\inbr{s[x\gets z], i, o}}\ruleno{Read}\]
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\[\trans{\inbr{s, i, o}}{\llang{write ($e$)}}{\inbr{s, i, o \llang{@} \sembr{e}_{\mathscr E}\;s}}\ruleno{Write}\]
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\[\trule{\trans{c_1}{S_1}{c^\prime},\;\trans{c^\prime}{S_2}{c_2}}{\trans{c_1}{S_1\llang{;}S_2}{c_2}}\ruleno{Seq}\]
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\caption{Big-step operational semantics for statements}
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\label{bs_stmt}
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\end{figure}
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\section{Stack Machine}
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Stack machine is a simple abstract computational device, which can be used as a convenient model to constructively describe
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the compilation process.
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In short, stack machine operates on the same configurations, as the language of statements, plus a stack of integers. The
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computation, performed by the stack machine, is controlled by a program, which is described as follows:
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\[
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\begin{array}{rcl}
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\mathscr I & = & \llang{BINOP $\;\otimes$} \\
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& & \llang{CONST $\;\mathbb N$} \\
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& & \llang{READ} \\
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& & \llang{WRITE} \\
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& & \llang{LD $\;\mathscr X$} \\
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& & \llang{ST $\;\mathscr X$} \\
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\mathscr P & = & \epsilon \\
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& & \mathscr I\mathscr P
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\end{array}
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\]
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Here the syntax category $\mathscr I$ stands for \emph{instructions}, $\mathscr P$~--- for \emph{programs}; thus, a program is a finite
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string of instructions.
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The semantics of stack machine program can be described, again, in the form of big-step operational semantics. This time the set of
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stack machine configurations is
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\[
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\mathscr C_{SM} = \mathbb Z^* \times \mathscr C
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\]
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where the first component is a stack, and the second~--- a configuration as in the semantics of statement language. The rules are shown on Fig.~\ref{bs_sm}; note,
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now we have one axiom and six inference rules (one per instruction).
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As for the statement, with the aid of the relation ``$\Rightarrow$'' we can define the surface semantics of stack machine:
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\[
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\forall p\in\mathscr P,\,\forall i\in\mathbb Z^*\;:\;\sembr{p}_{SM}\;i=o\Leftrightarrow\trans{\inbr{\epsilon, \inbr{\bot, i, \epsilon}}}{p}{\inbr{\_, \inbr{\_, \_, o}}}
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\]
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\begin{figure}[t]
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\[\trans{c}{\epsilon}{c}\ruleno{Stop$_{SM}$}\]
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\[\trule{\trans{\inbr{(x\oplus y)\llang{::}st, c}}{p}{c^\prime}}{\trans{\inbr{y\llang{::}x\llang{::}st, c}}{(\llang{BINOP $\;\otimes$})p}{c^\prime}}\ruleno{Binop$_{SM}$}\]
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\[\trule{\trans{\inbr{z\llang{::}st, c}}{p}{c^\prime}}{\trans{\inbr{st, c}}{(\llang{CONST $\;z$})p}{c^\prime}}\ruleno{Const$_{SM}$}\]
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\[\trule{\trans{\inbr{z\llang{::}st, \inbr{s, i, o}}}{p}{c^\prime}}{\trans{\inbr{st, \inbr{s, z\llang{::}i, o}}}{(\llang{READ})p}{c^\prime}}\ruleno{Read$_{SM}$}\]
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\[\trule{\trans{\inbr{st, \inbr{s, i, o\llang{@}z}}}{p}{c^\prime}}{\trans{\inbr{z\llang{::}st, \inbr{s, i, o}}}{(\llang{WRITE})p}{c^\prime}}\ruleno{Write$_{SM}$}\]
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\[\trule{\trans{\inbr{(s\;x)\llang{::}st, \inbr{s, i, o}}}{p}{c^\prime}}{\trans{\inbr{st, \inbr{s, i, o}}}{(\llang{LD $\;x$})p}{c^\prime}}\ruleno{LD$_{SM}$}\]
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\[\trule{\trans{\inbr{st, \inbr{s[x\gets z], i, o}}}{p}{c^\prime}}{\trans{\inbr{z\llang{::}st, \inbr{s, i, o}}}{(\llang{ST $\;x$})p}{c^\prime}}\ruleno{ST$_{SM}$}\]
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\caption{Big-step operational semantics for stack machine}
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\label{bs_sm}
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\end{figure}
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\section{A Compiler for the Stack Machine}
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A compiler of the statement language into the stack machine is a total mapping
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\[
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\sembr{\bullet}_{comp} : \mathscr S \mapsto \mathscr P
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\]
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We can describe the compiler in the form of denotational semantics for the source language. In fact, we can treat the compiler as a \emph{static} semantics, which
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maps each program into its stack machine equivalent.
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As the source language consists of two syntactic categories (expressions and statments), the compiler has to be ``bootstrapped'' from the compiler for expressions
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$\sembr{\bullet}^{\mathscr E}_{comp}$:
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\[
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\begin{array}{rcl}
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\sembr{x}^{\mathscr E}_{comp}&=&\llang{[LD $\;x$]}\\
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\sembr{n}^{\mathscr E}_{comp}&=&\llang{[CONST $\;n$]}\\
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\sembr{A\otimes B}^{\mathscr E}_{comp}&=&\sembr{A}^{\mathscr E}_{comp}\llang{@}\sembr{B}^{\mathscr E}_{comp}\llang{@}(\llang{BINOP $\;\otimes$})
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\end{array}
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\]
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And now the main dish:
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\[
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\begin{array}{rcl}
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\sembr{\llang{$x$ := $e$}}_{comp}&=&\sembr{e}^{\mathscr E}_{comp}\llang{@}\llang{[ST $\;x$]}\\
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\sembr{\llang{read ($x$)}}_{comp}&=&\llang{[READ; ST $\;x$]}\\
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\sembr{\llang{write ($e$)}}_{comp}&=&\sembr{e}^{\mathscr E}_{comp}\llang{@}\llang{[WRITE]}\\
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\sembr{\llang{$S_1$;$\;S_2$}}_{comp}&=&\sembr{S_1}_{comp}\llang{@}\sembr{S_2}_{comp}
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\end{array}
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\]
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\end{document}
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