m2-pas-dm/dm.tex

\documentclass[11pt]{article}
\usepackage{fullpage}

%% Corrige quelques erreurs de LaTeX2ε
\usepackage{fixltx2e}
\usepackage{xspace}
\usepackage{microtype}

%% Pour ne pas commettre les erreurs fréquentes décrites dans l2tabu
\usepackage[l2tabu,abort]{nag}

%% Saisie en UTF-8
\usepackage[utf8]{inputenc}
%% Fonte : cf. http://www.tug.dk/FontCatalogue/
\usepackage[T1]{fontenc}
\usepackage{lmodern}

%% Pour écrire du français
% \usepackage[frenchb]{babel}

%% Pour composer des mathématiques
\usepackage{mathtools}

\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}

\usepackage{stmaryrd}
\usepackage{semantic}

%% Commandes du paquet semantic : langage While
\reservestyle{\command}{\textbf}
\command{skip,while,do,if,then,else,not,and,or,true,false,int}
%% -> Utiliser \<begin> Pour styliser le mot-clé

\newcommand{\reduce}{\rightarrow^{*}}

\begin{document}

\author{Timoth\'ee Haudebourg \and Thibaut Marty}
\title{Homework PAS/SDL}
\date{November 18, 2016}

\maketitle

\section{Exercise 1}

\subsection{Domains}

The $State$ domain of evaluation is defined as:
\[State = (Var \rightharpoonup Value) \cup \bot\]
Where $Var$ is the set of variables, and $Value$ the set of possible values.
We use $\bot$ to represent the error state.
In our model, a record is represented as a partial application $Field \rightharpoonup Value$.
This means that a record field can contains a record itself.
To do so, we define the set of value as:
\begin{align*}
	Value &= \bigcup_{i\in\mathbb{N}} Value_i \\
	Value_n &= \mathbb{N} \cup (Field \rightharpoonup Value_{n+1}) \\
\end{align*}

\subsection{Denotational Semantic for expressions}

We define $\mathcal{A}$ the denotational semantics of arithmetic expressions as:

\[ \mathcal{A}|[ \bullet |] : AExpr \rightarrow State \rightarrow (Value \cup \bot) \]
where
\begin{align*}
  \mathcal{A} |[ n |] \sigma &= \mathcal{N} |[n|] \\
  \mathcal{A} |[ x |] \sigma &= \sigma x \\
  \mathcal{A} |[ e_1\ o\ e_2 |] \sigma &= \mathcal{A} |[ e_1 |] \sigma\ \bar{o}\ \mathcal{A} |[ e_2 |] \sigma \\
  \mathcal{A} |[ e.f |] \sigma &= \mathcal{A}|[e|] \sigma f\\
  \mathcal{A} |[ e_1{f \mapsto e_2} |] \sigma f &= \mathcal{A}|[ e_1 |] \sigma [f \mapsto \mathcal{A}|[ e_2 |] \sigma ]\\
  \mathcal{A} |[ \{f_1 = e_1 ; \dots ; f_n = e_n\} |] \sigma f_i &= \mathcal{A}|[ e_i |] \sigma \\
\end{align*}

For the undefined cases, for example when we try to add two records, we also note $\mathcal{A}|[e|] \sigma = \bot$.
In the rest of this homwork, we suppose $\mathcal{B}|[ \bullet |]$ the denotational semantic for booleans expressions already defined.

% TODO fin de la question : explications

\subsection{Structural Operational Semantic for commands}

We define $->$ the structural operational semantics for commands as:

\[ -> : (Statement \times State) \times ((Statement \times State) \cup State) \]

\subsubsection{Regular execution}
For all $\sigma \neq \bot$:
\[
  \inference{}{(skip, \sigma) -> \sigma}
\]
\[
  \inference{}{(x := a, \sigma) -> \sigma[x \mapsto \mathcal{A}|[a|] \sigma]}{\text{if }\mathcal{A}|[a|] \sigma \neq \bot}
\]
\[
  \inference{(S_1, \sigma) -> (S_1', \sigma')}{(S_1 ; S_2, \sigma) -> (S_1' ; S_2, \sigma')}
\]
\[
  \inference{(S_1, \sigma) -> \sigma'}{(S_1 ; S_2, \sigma) -> (S_2, \sigma')}
\]
\[
  \inference{}{(\<while>\ b\ \<do>\ S, \sigma) -> (S ; \<while>\ b\ \<do>\ S, \sigma)}{\text{if } \mathcal{B}|[b|] \sigma = \text{tt}}
\]
\[
  \inference{}{(\<while>\ b\ \<do>\ S, \sigma) -> \sigma)}{\text{if } \mathcal{B}|[b|] \sigma = \text{ff}}
\]
\[
  \inference{}{(\<if>\ b\ \<then>\ S_1\ \<else>\ S_2, \sigma) -> (S_1, \sigma)}{\text{if } \mathcal{B}|[b|] \sigma = \text{tt}}
\]
\[
  \inference{}{(\<if>\ b\ \<then>\ S_1\ \<else>\ S_2, \sigma) -> (S_2, \sigma)}{\text{if } \mathcal{B}|[b|] \sigma = \text{ff}}
\]

\subsubsection{Error handling}
An error can occurs when one tries to add (or compare) records.
In that case, according to our definition of arithmetic expressions, $\mathcal{A}|[e|] \sigma$ (or $\mathcal{B}|[b|]$) will returns $\bot$.
From this point, the execution of the program makes no sense, and we will return the error state.

\[
  \inference{}{(x := a, \sigma) -> \bot}{\text{if }\mathcal{A}|[a|] \sigma = \bot}
\]
\[
  \inference{}{(\<while>\ b\ \<do>\ S, \sigma) -> \bot}{\text{if } \mathcal{B}|[b|] \sigma = \bot}
\]
\[
  \inference{}{(\<if>\ b\ \<then>\ S_1\ \<else>\ S_2, \sigma) -> \bot}{\text{if } \mathcal{B}|[b|] \sigma = \bot}
\]

To ensure that the error can make it through the end, we also add the error propagation rule:

\[
  \inference{}{(S, \bot) -> \bot}
\]

\section{Type system}

We suppose now that every variable in the program is declared with a type satisfying the following syntax:
\[
  t\quad::=\quad\text{int }|\ \{f_1 : t_1;\ \dots\ ; f_n : t_n\}
\]

We note $\Gamma \vdash S$ the type judgment for our language, defined by the following typing rules:

\[
  \inference{}{\Gamma \vdash n : int}
\]
\[
  \inference{(x : T) \in \Gamma}{\Gamma \vdash x : T}
\]
\[
  \inference{\Gamma \vdash a_1 : int & \Gamma \vdash a_2 : int}{\Gamma \vdash a_1\ o\ a_2 : int}
\]
\[
  \inference{\Gamma \vdash e_1 : T_1 & \dots & \Gamma \vdash e_n : T_n}{\Gamma \vdash \{f_1 = e_1 ; \dots ; f_n = e_n\} : \{f_1 : T_1 ; \dots f_n : T_n\}}
\]
\[
  \inference{\Gamma \vdash \{f : T\}}{\Gamma \vdash e.f : T}
\]
\[
  \inference{\Gamma \vdash e_2 : T & \Gamma \vdash e_1 : \{f_1 : T_1 ; \dots ; f_n : T_n\}}{\Gamma \vdash e_1[f \mapsto e_2] : \{f : T ; f_1 : T_1 ; \dots ; f_n : T_n\}}
\]

% TODO : règle sub-typing
% TODO : booléens… (autre photo)

\section{Static analysis}

$RF : Lab -> Var -> \mathcal{P}(Field)$

% TODO : définir init

\[
  \inference{RF(l) \subseteq RF (init(S)) & RF \vdash (S, l) & RF(l) \subseteq RF(l')}{RF \vdash (\<while>\ [b]^l\ \<do>\ S, l')}
\]
\[
  \inference{RF(l) \subseteq RF(init(S_1)) & RF \vdash (S_1, l') \\ RF(l) \subseteq RF(init(S_2)) & RF \vdash (S_2, l')}{RF \vdash (\<if>\ [b]^l\ \<then>\ S_1\ \<else>\ S_2, l')}
\]
\[
  \inference{RF(l) \subseteq RF(l')}{RF \vdash ([skip]^l, l')}
\]
\[
  \inference{RF \vdash (S_1, init(S_2)) & RF \vdash (S_2, l')}{RF \vdash (S_1 ; S_2, l')}
\]
\[
  \inference{RF(l)[x \mapsto \{f_1, \dots, f_n\}] \subseteq RF(l')}{RF \vdash ([x := \{f_1 = e_1 ; \dots ; f_n = e_n\}]^l, l')}
\]
\[
  % TODO : il faudrait évaluer statiquement e_1 pour être moins pessimiste
  \inference{RF(l)[x \mapsto \{f\}] \subseteq RF(l')}{RF \vdash ([x := e_1[f \mapsto e_2]]^l, l')}
\]
% autres affectations
\[
  \inference{RF(l)[x \mapsto \emptyset] \subseteq RF(l')}{RF \vdash ([x := n]^l, l')}
\]
\[
  \inference{RF(l)[x \mapsto RF(l)(y)] \subseteq RF(l')}{RF \vdash ([x := y]^l, l')}
\]
\[
  \inference{RF(l)[x \mapsto \emptyset] \subseteq RF(l')}{RF \vdash ([x := e_1\ o\ e_2]^l, l')}
\]
\[
  \inference{RF(l)[x \mapsto \emptyset] \subseteq RF(l')}{RF \vdash ([x := e.f]^l, l')}
\]

\end{document}