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34
doc/04.tex
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@ -24,25 +24,35 @@ In the concrete syntax for the constructs we add the closing keywords ``\llang{i
\] \]
\begin{figure}[t] \begin{figure}[t]
\arraycolsep=10pt
\[\trans{c}{\llang{skip}}{c}\ruleno{Skip$_{bs}$}\] %\def\arraystretch{2.9}
\[\trans{c}{\llang{skip}}{c}\ruleno{Skip$_{bs}$}\]\vskip2mm
\[ \[
\crule{\trans{c}{\mbox{$S_1$}}{c^\prime}} \trule{\begin{array}{cc}
\sembr{e}\;s\ne 0 & \trans{c}{\mbox{$S_1$}}{c^\prime}
\end{array}}
{\trans{c=\inbr{s, \_, \_}}{\llang{if $\;e\;$ then $\;S_1\;$ else $\;S_2\;$}}{c^\prime}} {\trans{c=\inbr{s, \_, \_}}{\llang{if $\;e\;$ then $\;S_1\;$ else $\;S_2\;$}}{c^\prime}}
{\sembr{e}\;s\ne 0}\ruleno{If-True$_{bs}$} \ruleno{If-True$_{bs}$}
\] \]\vskip2mm
\[ \[
\crule{\trans{c}{\mbox{$S_2$}}{c^\prime}} \trule{\begin{array}{cc}
\sembr{e}\;s=0 & \trans{c}{\mbox{$S_1$}}{c^\prime}
\end{array}}
{\trans{c=\inbr{s, \_, \_}}{\llang{if $\;e\;$ then $\;S_1\;$ else $\;S_2\;$}}{c^\prime}} {\trans{c=\inbr{s, \_, \_}}{\llang{if $\;e\;$ then $\;S_1\;$ else $\;S_2\;$}}{c^\prime}}
{\sembr{e}\;s=0}\ruleno{If-False$_{bs}$} \ruleno{If-False$_{bs}$}
\] \]\vskip3mm
\[ \[
\crule{\trans{c}{\llang{$S$}}{c^\prime},\;\trans{c^\prime}{\llang{while $\;e\;$ do $\;S\;$}}{c^{\prime\prime}}} \trule{\begin{array}{ccc}
{\sembr{e}\;s\ne 0} & \trans{c}{\llang{$S$}}{c^\prime} & \trans{c^\prime}{\llang{while $\;e\;$ do $\;S\;$}}{c^{\prime\prime}}\\
\end{array}
}
{\trans{c=\inbr{s, \_, \_}}{\llang{while $\;e\;$ do $\;S\;$}}{c^{\prime\prime}}} {\trans{c=\inbr{s, \_, \_}}{\llang{while $\;e\;$ do $\;S\;$}}{c^{\prime\prime}}}
{\sembr{e}\;s\ne 0}\ruleno{While-True$_{bs}$} \ruleno{While-True$_{bs}$}
\] \]\vskip3mm
\[ \[
\ctrans{c=\inbr{s, \_, \_}}{\llang{while $\;e\;$ do $\;S\;$}}{c}{\sembr{e}\;s=0}\ruleno{While-False$_{bs}$} \trule{\sembr{e}\;s=0}
{\trans{c=\inbr{s, \_, \_}}{\llang{while $\;e\;$ do $\;S\;$}}{c}}
\ruleno{While-False$_{bs}$}
\] \]
\caption{Big-step operational semantics for control flow statements} \caption{Big-step operational semantics for control flow statements}
\label{bs_stmt_cf} \label{bs_stmt_cf}

12
doc/05.tex
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@ -94,8 +94,8 @@ Finally, we need two transformations for states:
\[ \[
\begin{array}{rcl} \begin{array}{rcl}
\mbox{\textbf{enter}}\,\inbr{\sigma_g,\,\_,\,\_}\,S & = & \inbr{\sigma_g,\,S,\,\bot}\\ \primi{enter}{\inbr{\sigma_g,\,\_,\,\_}\,S} & = & \inbr{\sigma_g,\,S,\,\bot}\\
\mbox{\textbf{leave}}\,\inbr{\sigma_g,\,\_,\,\_}\,\inbr{\_,\,S,\,\sigma_l}& = & \inbr{\sigma_g,\,S,\,\sigma_l} \primi{leave}{\inbr{\sigma_g,\,\_,\,\_}\,\inbr{\_,\,S,\,\sigma_l}}& = & \inbr{\sigma_g,\,S,\,\sigma_l}
\end{array} \end{array}
\] \]
@ -110,8 +110,8 @@ propagate this environment, adding $\withenv{\Gamma}{...}$ for each transition `
need now is to describe the rule for procedure calls: need now is to describe the rule for procedure calls:
\[ \[
\trule{\withenv{\Gamma}{\trans{\inbr{\mbox{\textbf{enter}}\;\sigma\,(\bar{a}@\bar{l})[\overline{a\gets\sembr{e}\sigma}],\,i,\,o}}{S}{\inbr{\sigma^\prime,\,i^\prime,\,o^\prime}}}} \trule{\withenv{\Gamma}{\trans{\inbr{\primi{enter}{\sigma\,(\bar{a}@\bar{l})[\overline{a\gets\sembr{e}\sigma}]},\,i,\,o}}{S}{\inbr{\sigma^\prime,\,i^\prime,\,o^\prime}}}}
{\withenv{\Gamma}{\trans{\inbr{\sigma,\,i,\,o}}{f (\bar{e})}{\inbr{\mbox{\textbf{leave}}\;\sigma^\prime\,\sigma,\,i^\prime,o^\prime}}}} {\withenv{\Gamma}{\trans{\inbr{\sigma,\,i,\,o}}{f (\bar{e})}{\inbr{\primi{leave}{\sigma^\prime\,\sigma},\,i^\prime,o^\prime}}}}
\ruleno{Call$_{bs}$} \ruleno{Call$_{bs}$}
\] \]
@ -146,7 +146,7 @@ affect the control stack, it gets threaded through all rules of operational sema
Now we specify additional rules for the new instructions: Now we specify additional rules for the new instructions:
\[ \[
\trule{\withenv{P}{\trans{\inbr{cs,\,st,\,\inbr{\mbox{\textbf{enter}}\;\sigma\,(\bar{a}@\bar{l})\overline{[a\gets z]},\,i,\,o}}}{p}{c^\prime}}} \trule{\withenv{P}{\trans{\inbr{cs,\,st,\,\inbr{\primi{enter}{\sigma\,(\bar{a}@\bar{l})\overline{[a\gets z]}},\,i,\,o}}}{p}{c^\prime}}}
{\withenv{P}{\trans{\inbr{cs,\,\bar{z}@st,\,\inbr{\sigma,\,i,\,o}}}{(\llang{BEGIN $\;\bar{a}\,\bar{l}$})p}{c^\prime}}} {\withenv{P}{\trans{\inbr{cs,\,\bar{z}@st,\,\inbr{\sigma,\,i,\,o}}}{(\llang{BEGIN $\;\bar{a}\,\bar{l}$})p}{c^\prime}}}
\ruleno{Begin$_{SM}$} \ruleno{Begin$_{SM}$}
\] \]
@ -158,7 +158,7 @@ Now we specify additional rules for the new instructions:
\] \]
\[ \[
\trule{\withenv{P}{\trans{\inbr{cs,\,st,\,\inbr{\mbox{\textbf{leave}}\;\sigma\,\sigma^\prime,\,i,\,o}}}{p^\prime}{c^\prime}}} \trule{\withenv{P}{\trans{\inbr{cs,\,st,\,\inbr{\primi{leave}{\sigma\,\sigma^\prime},\,i,\,o}}}{p^\prime}{c^\prime}}}
{\withenv{P}{\trans{\inbr{(p^\prime,\,\sigma^\prime)::cs,\,st,\,\inbr{\sigma,\,i,\,o}}}{\llang{END}p}{c^\prime}}} {\withenv{P}{\trans{\inbr{(p^\prime,\,\sigma^\prime)::cs,\,st,\,\inbr{\sigma,\,i,\,o}}}{\llang{END}p}{c^\prime}}}
\ruleno{EndRet$_{SM}$} \ruleno{EndRet$_{SM}$}
\] \]

87
doc/06.tex
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@ -30,29 +30,41 @@ We extend the configuration, used in the semantic description for statements, wi
This component will correspond to an optional return value for function/procedure calls (either integer value $n$ or ``$\hbox{---}$'', nothing). This component will correspond to an optional return value for function/procedure calls (either integer value $n$ or ``$\hbox{---}$'', nothing).
The rules inself are summarized in the Fig.~\ref{bs_expr}. Note the use of double environment for evaluating the body of function in the rule We introduce two primitives to make the semantic description shorter:
\[
\begin{array}{c}
\primi{val}{\inbr{s,\,i,\,o,\,v}}=v\\
\primi{ret}{\inbr{s,\,i,\,o,\,\_}\,v}=\inbr{s,\,i,\,o,\,v}
\end{array}
\]
The rules themselves are summarized in the Fig.~\ref{bs_expr}. Note the use of double environment for evaluating the body of a function in the rule
\rulename{Call}; note also, that now semantics of expressions and statements are mutually recursive. \rulename{Call}; note also, that now semantics of expressions and statements are mutually recursive.
\setarrow{\xRightarrow} \setarrow{\xRightarrow}
\setsubarrow{_{\mathscr E}} \setsubarrow{_{\mathscr E}}
\begin{figure} \begin{figure}
\[\withenv{\Phi}{\trans{\inbr{\sigma, i, o, \hbox{---}}}{n}{\inbr{\sigma, i, o, n}}} \ruleno{Const} \] \arraycolsep=10pt
\[\withenv{\Phi}{\trans{\inbr{\sigma, i, o, \hbox{---}}}{x}{\inbr{\sigma, i, o, \sigma\;x}}} \ruleno{Var} \] \[\withenv{\Phi}{\trans{\inbr{\sigma,\, i,\, o,\, \_}}{n}{\inbr{\sigma,\, i,\, o,\, n}}} \ruleno{Const} \]\vskip2mm
\[\trule{\withenv{\Phi}{\trans{c}{A}{c^\prime=\inbr{\_,\, \_,\, \_,\, a}}},\; \[\withenv{\Phi}{\trans{\inbr{\sigma,\, i,\, o,\, \_}}{x}{\inbr{\sigma,\, i,\, o,\, \sigma\;x}}} \ruleno{Var} \]\vskip2mm
\withenv{\Phi}{\trans{c^\prime}{B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, b}}} \[\trule{\begin{array}{cc}
} \withenv{\Phi}{\trans{c}{A}{c^\prime}} & \withenv{\Phi}{\trans{c^\prime}{B}{c^{\prime\prime}}}
{\withenv{\Phi}{\trans{c}{A\otimes B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, a\oplus b}}}}
\ruleno{Binop}
\]
\[\trule{\begin{array}{c}
\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}},\\
\Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}},\\
{\setsubarrow{}
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\mbox{\textbf{enter}}\;\sigma_k\; (\overline{a}@\overline{l})\; [\overline{a_j\gets v_j}],i_k,o_k,\hbox{---}}}{s}{\inbr{\sigma^\prime, i^\prime, o^\prime, n}}}}
\end{array} \end{array}
} }
{\withenv{\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\mbox{\textbf{leave}}\;\sigma^\prime\;\sigma_0, i^\prime, o^\prime, n}}}} {\withenv{\Phi}{\trans{c}{A\otimes B}{\primi{ret}{(\primi{val}{c^\prime}\oplus\primi{val}{c^{\prime\prime}})}}}}
\ruleno{Binop}
\]\vskip2mm
\[\trule{\begin{array}{c}
\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}}\\
\Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}}\\
{\setsubarrow{}
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\primi{enter}{\sigma_k\; (\overline{a}@\overline{l})\; [\overline{a_j\gets v_j}]},\,i_k,o_k,\hbox{---}}}{s}{\inbr{\sigma^\prime, i^\prime, o^\prime, n}}}}
\end{array}
}
{\withenv{\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\primi{leave}{\sigma^\prime\;\sigma_0}, i^\prime, o^\prime, n}}}}
\ruleno{Call} \ruleno{Call}
\] \]
\caption{Big-step Operational Semantics for Expressions} \caption{Big-step Operational Semantics for Expressions}
@ -96,14 +108,16 @@ The rules themselves are shown on Fig.~\ref{bs_cps}. Note, the rule for the call
{\withenv{K,\,\Phi}{\trans{c}{\llang{skip}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{skip}}{c^\prime}}}
\ruleno{Skip} \ruleno{Skip}
\] \]
\[\trule{{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\; \[\trule{
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma[x\gets n],\,i,\,o,\,\hbox{---}}}{K}{c^\prime}} \begin{array}{cc}
} {\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} & \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma[x\gets n],\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}
\end{array}}
{\withenv{K,\,\Phi}{\trans{c}{\llang{$x\,$:=$\,e$}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{$x\,$:=$\,e$}}{c^\prime}}}
\ruleno{Assign} \ruleno{Assign}
\] \]
\[\trule{{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\; \[\trule{\begin{array}{cc}
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o@[n],\,\hbox{---}}}{K}{c^\prime}} {\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} & \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o@[n],\,\hbox{---}}}{K}{c^\prime}}
\end{array}
} }
{\withenv{K,\,\Phi}{\trans{c}{\llang{write ($e$)}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{write ($e$)}}{c^\prime}}}
\ruleno{Write} \ruleno{Write}
@ -120,41 +134,44 @@ The rules themselves are shown on Fig.~\ref{bs_cps}. Note, the rule for the call
\ruleno{Seq} \ruleno{Seq}
\] \]
\[\crule{{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\; \[\trule{\begin{array}{cc}
\withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_1$}}{c^\prime}}} {\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} & \withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_1$}}{c^\prime}}\\
n\ne 0 &
\end{array}
}
{\withenv{K,\,\Phi}{\trans{c}{\llang{if $\;e\;$ then $\;s_1\;$ else $\;s_2$}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{if $\;e\;$ then $\;s_1\;$ else $\;s_2$}}{c^\prime}}}
{n\ne 0}
\ruleno{IfTrue} \ruleno{IfTrue}
\] \]
\[\crule{{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\; \[\trule{\begin{array}{cc}
\withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_2$}}{c^\prime}}} {\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,0}}}} & \withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_2$}}{c^\prime}}
\end{array}
}
{\withenv{K,\,\Phi}{\trans{c}{\llang{if $\;e\;$ then $\;s_1\;$ else $\;s_2$}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{if $\;e\;$ then $\;s_1\;$ else $\;s_2$}}{c^\prime}}}
{n= 0}
\ruleno{IfFalse} \ruleno{IfFalse}
\] \]
\[\crule{\begin{array}{c} \[\trule{\begin{array}{cc}
{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}}\\ {\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} &
\withenv{\llang{while $\;e\;$ do $\;s$}\diamond K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s$}}{c^\prime}} \withenv{\llang{while $\;e\;$ do $\;s$}\diamond K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s$}}{c^\prime}}\\
n \ne 0 &
\end{array}} \end{array}}
{\withenv{K,\,\Phi}{\trans{c}{\llang{while $\;e\;$ do $\;s$}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{while $\;e\;$ do $\;s$}}{c^\prime}}}
{n\ne 0}
\ruleno{WhileTrue} \ruleno{WhileTrue}
\] \]
\[\crule{{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\; \[\trule{\begin{array}{cc}
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}} {\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,0}}}} & \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}
\end{array}}
{\withenv{K,\,\Phi}{\trans{c}{\llang{while $\;e\;$ do $\;s$}}{c^\prime}}} {\withenv{K,\,\Phi}{\trans{c}{\llang{while $\;e\;$ do $\;s$}}{c^\prime}}}
{n = 0}
\ruleno{WhileFalse} \ruleno{WhileFalse}
\] \]
\[\trule{\begin{array}{c} \[\trule{\begin{array}{c}
{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}}}\\ {\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}}}\\
\Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}} \\ \Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}} \\
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\mbox{\textbf{enter}}\;\sigma_k\; (\overline{a}@\overline{l})\; [\overline{a_j\gets v_j}],i_k,o_k,\hbox{---}}}{s}{\inbr{\sigma^\prime, i^\prime, o^\prime, \hbox{---}}}}\\ \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\primi{enter}{\sigma_k\; (\overline{a}@\overline{l})\; [\overline{a_j\gets v_j}]},i_k,o_k,\hbox{---}}}{s}{\inbr{\sigma^\prime, i^\prime, o^\prime, \hbox{---}}}}\\
\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\mbox{\textbf{leave}}\;\sigma^\prime\;\sigma_0, i^\prime, o^\prime, \hbox{---}}}{K}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}} \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\primi{leave}{\sigma^\prime\;\sigma_0}, i^\prime, o^\prime, \hbox{---}}}{K}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}}
\end{array}} \end{array}}
{\withenv{K,\,\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}}} {\withenv{K,\,\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}}}
\ruleno{Call} \ruleno{Call}

107
doc/07.tex
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@ -11,7 +11,7 @@ For an array $(n, f)$ we assume $\dom{f}=[0\,..\,n-1]$. An element selection fun
\[ \[
\begin{array}{c} \begin{array}{c}
[\bullet] : \mathscr A (\mathscr E) \to \mathbb N \to \mathscr E\\[2mm] \bullet[\bullet] : \mathscr A (\mathscr E) \to \mathbb N \to \mathscr E\\[2mm]
(n, f) [i] = \left\{ (n, f) [i] = \left\{
\begin{array}{rcl} \begin{array}{rcl}
f\;i &, & i < n\\ f\;i &, & i < n\\
@ -21,22 +21,6 @@ For an array $(n, f)$ we assume $\dom{f}=[0\,..\,n-1]$. An element selection fun
\end{array} \end{array}
\] \]
\begin{comment}
A set of (ASCII) characters:
\[
\mathscr C = [0\,..\,255]
\]
A string is an array of characters:
\[
\mathscr S = \mathscr A (\mathscr C)
\]
Adding strings and arrays to the language changes the set of values the programs operate on:
\end{comment}
We represent arrays by \emph{references}. Thus, we introduce a (linearly) ordered set of locations We represent arrays by \emph{references}. Thus, we introduce a (linearly) ordered set of locations
\[ \[
@ -57,12 +41,16 @@ unambiguously discriminate between the shapes of each value. To access arrays, w
\mathscr M = \mathscr L \to \mathscr A\,(\mathscr V) \mathscr M = \mathscr L \to \mathscr A\,(\mathscr V)
\] \]
We now add two more components to the states: a memory function $\mu$ and the first free memory location $l_m$. We now add two more components to the configurations: a memory function $\mu$ and the first free memory location $l_m$, and
define the following primitive:
%Enriching the value set extends the definitions of state (which now has to map variable names to values) and \[
%stack (similarly), but does not affect the definition of input/output streams, read and write primitives, etc. \primi{mem}{\inbr{s,\,\mu,\,l_m,\,i,\,o,\,v}}=\mu
\]
\subsection{Adding arrays/strings on expression level} which gives a memory function from a configuration.
\subsection{Adding arrays on expression level}
On expression level, abstractly/concretely: On expression level, abstractly/concretely:
@ -74,19 +62,18 @@ On expression level, abstractly/concretely:
\end{array} \end{array}
\] \]
In addition, in a concrete syntax we supply two special forms for strings: \llang{'$x$'} as a denotation for integer code of ASCII symbol %In addition, in a concrete syntax we supply two special forms for strings: \llang{'$x$'} as a denotation for integer code of ASCII symbol
$x$, and \llang{"..."}~--- a string constant. %$x$, and \llang{"..."}~--- a string constant.
The semantics of enriched expressions is modified as follows. First, we add two additional premises to the rule for binary operators: The semantics of enriched expressions is modified as follows. First, we add two additional premises to the rule for binary operators:
\setsubarrow{_{\mathscr E}} \setsubarrow{_{\mathscr E}}
\[\trule{\begin{array}{c} \[\trule{\begin{array}{ccc}
\withenv{\Phi}{\trans{c}{A}{c^\prime=\inbr{\_,\, \_,\, \_,\, a}}}\\ \withenv{\Phi}{\trans{c}{A}{c^\prime}} &\phantom{XXXXX}& \withenv{\Phi}{\trans{c^\prime}{B}{c^{\prime\prime}}}\\
\withenv{\Phi}{\trans{c^\prime}{B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, b}}}\\ \primi{val}{c^\prime}\in\mathbb Z & & \primi{val}{c^{\prime\prime}}\in\mathbb Z
a\in\mathbb Z,\,b\in\mathbb Z
\end{array} \end{array}
} }
{\withenv{\Phi}{\trans{c}{A\otimes B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, a\oplus b}}}} {\withenv{\Phi}{\trans{c}{A\otimes B}{\primi{ret}{c^{\prime\prime}\;(\primi{val}{c^\prime}\oplus \primi{val}{c^{\prime\prime}})}}}}
\ruleno{Binop} \ruleno{Binop}
\] \]
@ -94,31 +81,40 @@ These two premises ensure that both operand expressions are evaluated into integ
kinds of expressions (see Figure~\ref{array_expressions}). kinds of expressions (see Figure~\ref{array_expressions}).
\begin{figure} \begin{figure}
\[\trule{\begin{array}{c} \[\trule{\begin{array}{ccc}
\withenv{\Phi}{\trans{c}{E}{c^\prime=\inbr{\_, \_, \_, l}}}\\ \withenv{\Phi}{\trans{c}{e}{c^\prime}} &\phantom{XXXX} &\withenv{\Phi}{\trans{c^\prime}{j}{c^{\prime\prime}}}\\
\withenv{\Phi}{\trans{c^\prime}{J}{\inbr{s^{\prime\prime}=\inbr{\_,\, \_,\, \_,\, \mu,\,\_},\, i^{\prime\prime},\, o^{\prime\prime},\, j}}}\\ l=\primi{val}{c^\prime} & &j=\primi{val}{c^{\prime\prime}}\\
l\in \mathscr L,\,(n,\,f)=\mu\,\;l\,j\in\mathbb N,\,j<n l\in\mathscr L & &j\in\mathbb N\\
(n,\,f)=\primi{mem}{l} & &j<n
\end{array}} \end{array}}
{\withenv{\Phi}{\trans{c}{E\,\mathtt{[}J\mathtt{]}}{\inbr{s^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, f\,j}}}} {\withenv{\Phi}{\trans{c}{e\,\mathtt{[}j\mathtt{]}}{\primi{ret}{c^{\prime\prime}(f\;j)}}}}
\ruleno{ArrayElement} \ruleno{ArrayElement}
\] \]
\[\trule{\withenv{\Phi}{\trans{c_j}{e_j}{c_{j+1}=\inbr{\inbr{\sigma_g^{j+1},\,S^{j+1},\,\sigma_l^{j+1},\,\mu^{j+1},\,l_{m_{j+1}}}, i_{j+1}, o_{j+1}, a_j}}},\,j\in [0..k]} \vskip5mm
{\withenv{\Phi}{\trans{c_0}{\mathtt{[}e_0, e_1,...,e_k\mathtt{]}}{\inbr{\inbr{\sigma_g^{j+1},\,S^{j+1},\,\sigma_l^{j+1},\,\mu^{j+1}[l_{m_{k+1}}\gets (k,\,\lambda i\,.\,a_i)],\, l_{m_{k+1}+1}}, i_{k+1}, o_{k+1}, l_{m_{k+1}}}}}}
\ruleno{ArrayConstructor}
\]
\[\trule{\begin{array}{c} \[\trule{\begin{array}{c}
\withenv{\Phi}{\trans{c}{E}{\inbr{s^{\prime\prime}=\inbr{\_,\,\_,\,\_,\,\mu,\,\_}, i^{\prime\prime}, o^{\prime\prime}, l}}}\\ \withenv{\Phi}{\trans{c_j}{e_j}{c_{j+1}}},\,j\in [0..k]\\
l\in\mathscr L,\,(n,\,f)=\mu\;l \inbr{s,\,\mu,\,l_m,\,i,\,o,\,\_}=c_{k+1}
\end{array}
}
{\withenv{\Phi}{\trans{c_0}{\mathtt{[}e_0, e_1,...,e_k\mathtt{]}}{\inbr{s,\,\mu\,[l_m\gets(k+1,\,\lambda n.\primi{val}{c_n})],\,l_{m+1},\,i,\,o,\,l_m}}}}
\ruleno{Array}
\]
\vskip5mm
\[\trule{\begin{array}{c}
\withenv{\Phi}{\trans{c}{e}{c^\prime}}\\
l=\primi{val}{c^\prime}\\
l\in\mathscr L\\
(n,\,f)=(\primi{mem}{c^\prime})\;l
\end{array} \end{array}
} }
{\withenv{\Phi}{\trans{c}{E\mathtt{.length}}{\inbr{s^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}}} {\withenv{\Phi}{\trans{c}{e\mathtt{.length}}{\primi{return}{c^\prime\;n}}}}
\ruleno{ArrayLength} \ruleno{ArrayLength}
\] \]
\caption{Big-step Operational Semantics for Array Expressions} \caption{Big-step Operational Semantics for Array Expressions}
\label{array_expressions} \label{array_expressions}
\end{figure} \end{figure}
\subsection{Adding arrays/strings on statement level} \subsection{Adding arrays on statement level}
On statement level, we add the single construct: On statement level, we add the single construct:
@ -127,17 +123,30 @@ On statement level, we add the single construct:
\] \]
This construct is interpreted as an assignment to an element of an array. The semantics of this construct is described by the following rule: This construct is interpreted as an assignment to an element of an array. The semantics of this construct is described by the following rule:
\[ \[
\trule{\setsubarrow{_{\mathscr E}} \trule{\setsubarrow{_{\mathscr E}}
\begin{array}{c} \begin{array}{ccccc}
\withenv{\Phi}{\trans{c}{E}{c^\prime=\inbr{\_,\_,\_, (n,\,f)}}}\\ \withenv{\Phi}{\trans{c}{e}{c^\prime}} & \phantom{XXX} & \withenv{\Phi}{\trans{c^\prime}{j}{c^{\prime\prime}}} & \phantom{XXX} & \withenv{\Phi}{\trans{c^{\prime\prime}}{g}{\inbr{s,\mu,l_m,i,o,v}}}\\
\withenv{\Phi}{\trans{c^\prime}{J}{c^{\prime\prime}=\inbr{\_,\_,\_,j}}}\\ l=\primi{val}{c^\prime} & & i=\primi{val}{c^{\prime\prime}} & & \\
\withenv{\Phi}{\trans{c^{\prime\prime}}{F}{c^{\prime\prime\prime}=\inbr{s^{\prime\prime},i^{\prime\prime},o^{\prime\prime},y}}}\\ l\in\mathscr L & & i\in\mathbb N & & \\[3mm]
j\in\mathbb N,\,j<n\\ \multicolumn{5}{c}{(n,\,f)=\mu\;l}\\
\setsubarrow{}\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\mathtt{update}(s^{\prime\prime}, E, (n,\,f [j\gets y])),i^{\prime\prime},o^{\prime\prime},\hbox{---}}}{K}{\widetilde{c}}} \multicolumn{5}{c}{i<n}\\
\multicolumn{5}{c}{\setsubarrow{}\withenv{\llang{skip},\,\Phi}{\trans{\inbr{s,\,\mu\,[l\gets (n,\,f\,[i\gets x])],\,l_m,\,i,\,o,\,\hbox{---}}}{K}{\widetilde{c}}}}
\end{array} \end{array}
} }
{\setsubarrow{}\withenv{K,\,\Phi}{\trans{c}{E\mathtt{[}J\mathtt{]}\llang{:=}F}{\widetilde{c}}}} {\setsubarrow{}\withenv{K,\,\Phi}{\trans{c}{e\mathtt{[}j\mathtt{]}\llang{:=}g}{\widetilde{c}}}}
\ruleno{ArrayAssign} \ruleno{ArrayAssign}
\] \]
\subsection{Strings}
With arrays in our hands, we can easily add strings as arrays of characters. In fact, on the source language the strings can be
introduced as a syntactic extension:
\begin{enumerate}
\item we add a character constants \llang{'c'} as a shortcut for their integer codes;
\item we add a string literals \llang{"abcd..."} as a shortcut for arrays \llang{['a', 'b', 'c', 'd', ...]}.
\end{enumerate}
Nothing else has to be done~--- now we have mutable reference-representable strings.

3
doc/lectures.tex
View file

@ -45,7 +45,7 @@
\newcommand{\trule}[2]{\frac{#1}{#2}} \newcommand{\trule}[2]{\frac{#1}{#2}}
\newcommand{\crule}[3]{\frac{#1}{#2},\;{#3}} \newcommand{\crule}[3]{\frac{#1}{#2},\;{#3}}
\newcommand{\withenv}[2]{{#1}\vdash{#2}} \newcommand{\withenv}[2]{{#1}\vdash{#2}}
\newcommand{\trans}[3]{{#1}\transarrow{\padding#2\padding}\subarrow{#3}} \newcommand{\trans}[3]{{#1}\transarrow{\padding{\textstyle #2}\padding}\subarrow{#3}}
\newcommand{\ctrans}[4]{{#1}\transarrow{\padding#2\padding}\subarrow{#3},\;{#4}} \newcommand{\ctrans}[4]{{#1}\transarrow{\padding#2\padding}\subarrow{#3},\;{#4}}
\newcommand{\llang}[1]{\mbox{\lstinline[mathescape]|#1|}} \newcommand{\llang}[1]{\mbox{\lstinline[mathescape]|#1|}}
\newcommand{\pair}[2]{\inbr{{#1}\mid{#2}}} \newcommand{\pair}[2]{\inbr{{#1}\mid{#2}}}
@ -70,6 +70,7 @@
\newcommand{\meta}[1]{{\mathcal{#1}}} \newcommand{\meta}[1]{{\mathcal{#1}}}
\renewcommand{\emptyset}{\varnothing} \renewcommand{\emptyset}{\varnothing}
\newcommand{\dom}[1]{\mathtt{dom}\;{#1}} \newcommand{\dom}[1]{\mathtt{dom}\;{#1}}
\newcommand{\primi}[2]{\mathbf{#1}\;{#2}}
\definecolor{light-gray}{gray}{0.90} \definecolor{light-gray}{gray}{0.90}
\newcommand{\graybox}[1]{\colorbox{light-gray}{#1}} \newcommand{\graybox}[1]{\colorbox{light-gray}{#1}}