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Lectures
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Dmitry Boulytchev
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5 changed files with 140 additions and 103 deletions
87
doc/06.tex
87
doc/06.tex
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@ -30,29 +30,41 @@ We extend the configuration, used in the semantic description for statements, wi
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This component will correspond to an optional return value for function/procedure calls (either integer value $n$ or ``$\hbox{---}$'', nothing).
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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
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We introduce two primitives to make the semantic description shorter:
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\[
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\begin{array}{c}
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\primi{val}{\inbr{s,\,i,\,o,\,v}}=v\\
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\primi{ret}{\inbr{s,\,i,\,o,\,\_}\,v}=\inbr{s,\,i,\,o,\,v}
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\end{array}
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\]
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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
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\rulename{Call}; note also, that now semantics of expressions and statements are mutually recursive.
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\setarrow{\xRightarrow}
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\setsubarrow{_{\mathscr E}}
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\begin{figure}
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\[\withenv{\Phi}{\trans{\inbr{\sigma, i, o, \hbox{---}}}{n}{\inbr{\sigma, i, o, n}}} \ruleno{Const} \]
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\[\withenv{\Phi}{\trans{\inbr{\sigma, i, o, \hbox{---}}}{x}{\inbr{\sigma, i, o, \sigma\;x}}} \ruleno{Var} \]
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\[\trule{\withenv{\Phi}{\trans{c}{A}{c^\prime=\inbr{\_,\, \_,\, \_,\, a}}},\;
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\withenv{\Phi}{\trans{c^\prime}{B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, b}}}
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}
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{\withenv{\Phi}{\trans{c}{A\otimes B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, a\oplus b}}}}
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\ruleno{Binop}
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\]
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\[\trule{\begin{array}{c}
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\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}},\\
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\Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}},\\
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{\setsubarrow{}
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\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}}}}
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\arraycolsep=10pt
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\[\withenv{\Phi}{\trans{\inbr{\sigma,\, i,\, o,\, \_}}{n}{\inbr{\sigma,\, i,\, o,\, n}}} \ruleno{Const} \]\vskip2mm
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\[\withenv{\Phi}{\trans{\inbr{\sigma,\, i,\, o,\, \_}}{x}{\inbr{\sigma,\, i,\, o,\, \sigma\;x}}} \ruleno{Var} \]\vskip2mm
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\[\trule{\begin{array}{cc}
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\withenv{\Phi}{\trans{c}{A}{c^\prime}} & \withenv{\Phi}{\trans{c^\prime}{B}{c^{\prime\prime}}}
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\end{array}
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}
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{\withenv{\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\mbox{\textbf{leave}}\;\sigma^\prime\;\sigma_0, i^\prime, o^\prime, n}}}}
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{\withenv{\Phi}{\trans{c}{A\otimes B}{\primi{ret}{(\primi{val}{c^\prime}\oplus\primi{val}{c^{\prime\prime}})}}}}
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\ruleno{Binop}
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\]\vskip2mm
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\[\trule{\begin{array}{c}
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\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}}\\
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\Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}}\\
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{\setsubarrow{}
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\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\primi{enter}{\sigma_k\; (\overline{a}@\overline{l})\; [\overline{a_j\gets v_j}]},\,i_k,o_k,\hbox{---}}}{s}{\inbr{\sigma^\prime, i^\prime, o^\prime, n}}}}
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\end{array}
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}
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{\withenv{\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\primi{leave}{\sigma^\prime\;\sigma_0}, i^\prime, o^\prime, n}}}}
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\ruleno{Call}
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\]
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\caption{Big-step Operational Semantics for Expressions}
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@ -96,14 +108,16 @@ The rules themselves are shown on Fig.~\ref{bs_cps}. Note, the rule for the call
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{\withenv{K,\,\Phi}{\trans{c}{\llang{skip}}{c^\prime}}}
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\ruleno{Skip}
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\]
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\[\trule{{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\;
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\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma[x\gets n],\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}
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}
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\[\trule{
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\begin{array}{cc}
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{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} & \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma[x\gets n],\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}
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\end{array}}
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{\withenv{K,\,\Phi}{\trans{c}{\llang{$x\,$:=$\,e$}}{c^\prime}}}
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\ruleno{Assign}
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\]
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\[\trule{{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\;
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\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o@[n],\,\hbox{---}}}{K}{c^\prime}}
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\[\trule{\begin{array}{cc}
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{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} & \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o@[n],\,\hbox{---}}}{K}{c^\prime}}
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\end{array}
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}
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{\withenv{K,\,\Phi}{\trans{c}{\llang{write ($e$)}}{c^\prime}}}
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\ruleno{Write}
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@ -120,41 +134,44 @@ The rules themselves are shown on Fig.~\ref{bs_cps}. Note, the rule for the call
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\ruleno{Seq}
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\]
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\[\crule{{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\;
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\withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_1$}}{c^\prime}}}
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\[\trule{\begin{array}{cc}
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{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} & \withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_1$}}{c^\prime}}\\
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n\ne 0 &
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\end{array}
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}
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{\withenv{K,\,\Phi}{\trans{c}{\llang{if $\;e\;$ then $\;s_1\;$ else $\;s_2$}}{c^\prime}}}
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{n\ne 0}
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\ruleno{IfTrue}
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\]
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\[\crule{{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\;
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\withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_2$}}{c^\prime}}}
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\[\trule{\begin{array}{cc}
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{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,0}}}} & \withenv{K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s_2$}}{c^\prime}}
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\end{array}
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}
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{\withenv{K,\,\Phi}{\trans{c}{\llang{if $\;e\;$ then $\;s_1\;$ else $\;s_2$}}{c^\prime}}}
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{n= 0}
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\ruleno{IfFalse}
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\]
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\[\crule{\begin{array}{c}
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{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}}\\
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\withenv{\llang{while $\;e\;$ do $\;s$}\diamond K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s$}}{c^\prime}}
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\[\trule{\begin{array}{cc}
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{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}} &
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\withenv{\llang{while $\;e\;$ do $\;s$}\diamond K,\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{\llang{$s$}}{c^\prime}}\\
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n \ne 0 &
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\end{array}}
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{\withenv{K,\,\Phi}{\trans{c}{\llang{while $\;e\;$ do $\;s$}}{c^\prime}}}
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{n\ne 0}
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\ruleno{WhileTrue}
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\]
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\[\crule{{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,n}}}};\;
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\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}}
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\[\trule{\begin{array}{cc}
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{\setsubarrow{_{\mathscr E}} \withenv{\Phi}{\trans{c}{e}{\inbr{\sigma,\,i,\,o,\,0}}}} & \withenv{\llang{skip},\,\Phi}{\trans{\inbr{\sigma,\,i,\,o,\,\hbox{---}}}{K}{c^\prime}}
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\end{array}}
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{\withenv{K,\,\Phi}{\trans{c}{\llang{while $\;e\;$ do $\;s$}}{c^\prime}}}
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{n = 0}
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\ruleno{WhileFalse}
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\]
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\[\trule{\begin{array}{c}
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{\setsubarrow{_{\mathscr E}}\withenv{\Phi}{\trans{c_{j-1}}{e_j}{c_j=\inbr{\sigma_j,\,i_j,\,o_j,\,v_j}}}}\\
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\Phi\;f = \llang{fun $\;f\;$ ($\overline{a}$) local $\;\overline{l}\;$ \{$s$\}} \\
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\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{---}}}}\\
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\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}}}
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\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{---}}}}\\
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\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}}}
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\end{array}}
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{\withenv{K,\,\Phi}{\trans{c_0=\inbr{\sigma_0,\,\_,\,\_,\,\_}}{f (\overline{e_k})}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}}}
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\ruleno{Call}
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