Commit 83979416 authored by Ralf Jung's avatar Ralf Jung

move \text... to their respective sections; get rid of \fork

parent b45812dd
......@@ -2,14 +2,14 @@
\section{Parameters to the logic}
\begin{itemize}
\item A set \textdom{Exp} of \emph{expressions} (metavariable $\expr$) with a
subset \textdom{Val} of values ($\val$). We assume that if $\expr$ is an
expression then so is $\fork{\expr}$. We moreover assume a value
\textsf{fRet} (giving the intended return value of a fork), and we assume that
\begin{align*}
\fork{\expr} &\notin \textdom{Val} \\
\fork{\expr_1} = \fork{\expr_2} &\implies \expr_1 = \expr_2
\end{align*}
% \item A set \textdom{Exp} of \emph{expressions} (metavariable $\expr$) with a
% subset \textdom{Val} of values ($\val$). We assume that if $\expr$ is an
% expression then so is $\fork{\expr}$. We moreover assume a value
% \textsf{fRet} (giving the intended return value of a fork), and we assume that
% \begin{align*}
% \fork{\expr} &\notin \textdom{Val} \\
% \fork{\expr_1} = \fork{\expr_2} &\implies \expr_1 = \expr_2
% \end{align*}
\item A set $\textdom{Ectx}$ of \emph{evaluation contexts} ($\ectx$) that includes the empty context $[\; ]$,
a plugging operation $\ectx[\expr]$ that produces an expression, and context composition $\circ$
satisfying the following axioms:
......@@ -20,7 +20,7 @@
\ectx[\expr_1] = \ectx[\expr_2] &\implies \expr_1 = \expr_2 \\
\ectx_1 \circ \ectx_2 = [\; ] &\implies \ectx_1 = \ectx_2 = [\; ] \\
\ectx[\expr] \in \textdom{Val} &\implies \ectx = [\;] \\
\ectx[\expr] = \fork{\expr'} &\implies \ectx = [\;]
% \ectx[\expr] = \fork{\expr'} &\implies \ectx = [\;]
\end{align*}
\item A set \textdom{State} of shared machine states (\eg heaps), metavariable $\state$.
......@@ -34,14 +34,14 @@ and notions of an expression to be \emph{reducible} or \emph{stuck}, such that
\lnot \textlog{reducible}(\expr')
\end{align*}
and the following hold
\begin{align*}
&\textlog{stuck}(\fork{\expr})& \\
&\textlog{stuck}(\val)&\\
&\ectx[\expr] = \ectx'[\expr'] \implies \textlog{reducible}(\expr') \implies
\expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(step-by-value)} \\
&\ectx[\expr] = \ectx'[\fork{\expr'}] \implies
\expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(fork-by-value)} \\
\end{align*}
% \begin{align*}
% &\textlog{stuck}(\fork{\expr})& \\
% &\textlog{stuck}(\val)&\\
% &\ectx[\expr] = \ectx'[\expr'] \implies \textlog{reducible}(\expr') \implies
% \expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(step-by-value)} \\
% &\ectx[\expr] = \ectx'[\fork{\expr'}] \implies
% \expr \notin \textdom{Val} \implies \Exists \ectx''. \ectx' = \ectx \circ \ectx'' &\mbox{(fork-by-value)} \\
% \end{align*}
\item A predicate \textlog{atomic} on expressions satisfying
\begin{align*}
......@@ -78,11 +78,11 @@ Let $\mcarp{M} \eqdef |\monoid| \setminus \{\mzero\}$.
{\cfg{\state}{\expr} \step \cfg{\state'}{\expr'}}
{\cfg{\state}{\tpool [i \mapsto \ectx[\expr]]} \step
\cfg{\state'}{\tpool [i \mapsto \ectx[\expr']]}}
\and
\infer
{}
{\cfg{\state}{\tpool [i \mapsto \ectx[\fork{\expr}]]} \step
\cfg{\state}{\tpool [i \mapsto \ectx[\textsf{fRet}]] [j \mapsto \expr]}}
% \and
% \infer
% {}
% {\cfg{\state}{\tpool [i \mapsto \ectx[\fork{\expr}]]} \step
% \cfg{\state}{\tpool [i \mapsto \ectx[\textsf{fRet}]] [j \mapsto \expr]}}
\end{mathpar}
\section{Syntax}
......@@ -567,10 +567,10 @@ We write $\provesalways$ to denote judgments that can only be extended with a bo
{\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \and \text{$\expr$ not a value}
}
{\hoare{\prop * \later\propC}{\expr}{\Ret\val. \propB * \propC}[\mask \uplus \mask']}
\and
\inferH{Fork}
{\hoare{\prop}{\expr}{\Ret\any. \TRUE}[\top]}
{\hoare{\later\prop * \later\propB}{\fork{\expr}}{\Ret\val. \val = \textsf{fRet} \land \propB}[\mask]}
% \and
% \inferH{Fork}
% {\hoare{\prop}{\expr}{\Ret\any. \TRUE}[\top]}
% {\hoare{\later\prop * \later\propB}{\fork{\expr}}{\Ret\val. \val = \textsf{fRet} \land \propB}[\mask]}
\and
\inferH{ACsq}
{\prop \vs[\mask \uplus \mask'][\mask] \prop' \\
......
......@@ -389,28 +389,28 @@ $\rho\nequiv{n} \rho' \iff n=0 \lor \bigl(\dom(\rho)=\dom(\rho') \land
\typedsection{Weakest precondition}{\mathit{wp} : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \to \textdom{Prop}) \to \textdom{Prop} \in {\cal U}}
\begin{align*}
\mathit{wp}_\mask(\expr, q) &\eqdef \Lam W.
\begin{aligned}[t]
\{\, (n, \rs) &\mid \All W_F \geq W; k \leq n; \rs_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\fullSat{\state}{\mask \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \implies{}\\
&\qquad
(\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \rs'. \\
&\qquad\qquad
k \in (\fullSat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(\expr)(W'))~\land \\
&\qquad
(\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \rs'. \\
&\qquad\qquad
k - 1 \in (\fullSat{\state'}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k-1, \rs') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\
&\qquad
(\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \rs', \rs_1', \rs_2'. \\
&\qquad\qquad
k - 1 \in (\fullSat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land \rs' = \rs_1' \rtimes \rs_2'~\land \\
&\qquad\qquad
(k-1, \rs_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land
(k-1, \rs_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W'))
\,\}
\end{aligned}
\end{align*}
% \begin{align*}
% \mathit{wp}_\mask(\expr, q) &\eqdef \Lam W.
% \begin{aligned}[t]
% \{\, (n, \rs) &\mid \All W_F \geq W; k \leq n; \rs_F; \state; \mask_F \sep \mask. k > 0 \land k \in (\fullSat{\state}{\mask \cup \mask_F}{\rs \rtimes \rs_F}{W_F}) \implies{}\\
% &\qquad
% (\expr \in \textdom{Val} \implies \Exists W' \geq W_F. \Exists \rs'. \\
% &\qquad\qquad
% k \in (\fullSat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k, \rs') \in q(\expr)(W'))~\land \\
% &\qquad
% (\All\ectx,\expr_0,\expr'_0,\state'. \expr = \ectx[\expr_0] \land \cfg{\state}{\expr_0} \step \cfg{\state'}{\expr'_0} \implies \Exists W' \geq W_F. \Exists \rs'. \\
% &\qquad\qquad
% k - 1 \in (\fullSat{\state'}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land (k-1, \rs') \in wp_\mask(\ectx[\expr_0'], q)(W'))~\land \\
% &\qquad
% (\All\ectx,\expr'. \expr = \ectx[\fork{\expr'}] \implies \Exists W' \geq W_F. \Exists \rs', \rs_1', \rs_2'. \\
% &\qquad\qquad
% k - 1 \in (\fullSat{\state}{\mask \cup \mask_F}{\rs' \rtimes \rs_F}{W'}) \land \rs' = \rs_1' \rtimes \rs_2'~\land \\
% &\qquad\qquad
% (k-1, \rs_1') \in \mathit{wp}_\mask(\ectx[\textsf{fRet}], q)(W') \land
% (k-1, \rs_2') \in \mathit{wp}_\top(\expr', \Lam\any. \top)(W'))
% \,\}
% \end{aligned}
% \end{align*}
\begin{lem}
$\mathit{wp}$ is well-defined: $\mathit{wp}_{\mask}(\expr, q)$ is a valid proposition, and $\mathit{wp}$ is a non-expansive map. Besides, the dependency on the recursive occurrence is contractive, so $\mathit{wp}$ has a fixed-point.
\end{lem}
......
......@@ -34,6 +34,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% SETUP
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SetSymbolFont{stmry}{bold}{U}{stmry}{m}{n} % this fixes warnings when \boldsymbol is used with stmaryrd included
\extrarowheight=\jot % else, arrays are scrunched compared to, say, aligned
\newcolumntype{.}{@{}}
......@@ -84,17 +85,6 @@
\newtheorem{thm}{Theorem}
\newtheorem{exercise}{Exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% FONTS & FORMATTING
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SetSymbolFont{stmry}{bold}{U}{stmry}{m}{n} % this fixes warnings when \boldsymbol is used with stmaryrd included
\newcommand{\textdom}[1]{\textit{#1}}
\newcommand{\textlog}[1]{\textsf{#1}}
\newcommand{\textsort}[1]{\textlog{#1}}
\newcommand{\textlang}[1]{\texttt{#1}}
\newcommand{\textmon}[1]{\textsc{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% GENERIC MACROS
......@@ -217,6 +207,8 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CMRA (RESOURCE ALGEBRA) SYMBOLS & NOTATION & IDENTIFIERS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\textmon}[1]{\textsc{#1}}
\newcommand{\monoid}{M}
\newcommand{\melt}{a}
......@@ -238,6 +230,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MODEL-SPECIFIC SYMBOLS & NOTATION & IDENTIFIERS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\textdom}[1]{\textit{#1}}
\newcommand{\wIso}{\xi}
......@@ -245,7 +238,7 @@
\newcommand{\rsB}{s}
\newcommand{\pres}{\pi}
\newcommand{\wld}{w}
\newcommand{\ghostRes}{g}
%% Various pieces of syntax
......@@ -276,6 +269,8 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% LOGIC SYMBOLS & NOTATION & IDENTIFIERS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\textlog}[1]{\textsf{#1}}
\newcommand{\textsort}[1]{\textlog{#1}}
\newcommand{\Sig}{\mathcal{S}}
\newcommand{\SigType}{\mathcal{T}}
......@@ -421,7 +416,7 @@
\newcommand{\FALSE}{\textlog{False}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LANGUAGE-LEVEL SYNTAX AND SEMANTICS
% LANGUAGE SYNTAX AND SEMANTICS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\expr}{e}
\newcommand{\val}{v}
......@@ -433,7 +428,6 @@
\newcommand{\tpool}{T}
\newcommand{\cfg}[2]{{#1};{#2}}
\newcommand{\fork}[1]{\textlang{fork}\;{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DERIVED CONSTRUCTIONS
......
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