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- \documentclass[11pt]{article}
- \usepackage{fullpage}
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- \usepackage{microtype}
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- \usepackage[T1]{fontenc}
- \usepackage{lmodern}
- \usepackage{csquotes}
- %% 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{Small-step Semantic}
- \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, o \in \{+, -, \times\} \\
- \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:
- \[
- \inference{}{(S, \bot) -> \bot}
- \]
- Finally, note that according to the exercise statement, we suppose that all variables are already defined in $\sigma$,
- so that no error can occur because of an undefined variable.
- \section{Type system}
- We suppose now that every variable in the program is declared with a type satisfying the following syntax:
- \[
- t\quad::=\quad\text{int }|\ \{f_1 : t_1;\ \dots\ ; f_n : t_n\}
- \]
- \subsection{Definition}
- We note $\Gamma \vdash S$ the type judgment for our language, defined by the following typing rules:
- \[
- \inference{}{\Gamma \vdash n : int}
- \]
- \[
- \inference{(x : T) \in \Gamma}{\Gamma \vdash x : T}
- \]
- \[
- \inference{\Gamma \vdash a_1 : int & \Gamma \vdash a_2 : int}{\Gamma \vdash a_1\ o\ a_2 : int}{o \in \{+, -, \times\}}
- \]
- Records
- \[
- \inference{\Gamma \vdash e_1 : T_1 & \dots & \Gamma \vdash e_n : T_n}{\Gamma \vdash \{f_1 = e_1 ; \dots ; f_n = e_n\} : \{f_1 : T_1 ; \dots; f_n : T_n\}}
- \]
- Field assignment
- \[
- \inference[append ]{\Gamma \vdash e_2 : T & \Gamma \vdash e_1 : \{f_1 : T_1; \dots; f_n : T_n\}}{\Gamma \vdash e_1[f \mapsto e_2] : \{f : T; f_1 : T_1; \dots; f_n : T_n\}}
- \]
- \[
- \inference[override ]{\Gamma \vdash e_2 : T & \Gamma \vdash e_1 : \{f_1 : T_1; \dots; f_i : T_i; \dots; f_n : T_n\}}{\Gamma \vdash e_1[f_i \mapsto e_2] : \{f_1 : T_1; \dots; f_i : T; \dots; f_n : T_n\}}
- \]
- Field lookup
- \[
- \inference{\Gamma \vdash e : \{f : T\}}{\Gamma \vdash e.f : T}
- \]
- As is, the last rule can seems too restrictive.
- Indeed, an expression $e$ does not have to match \emph{exactly} the type $\{f : T\}$ to provide the field $f$.
- It can contains some other fields.
- We want the type $\{f : T\}$ to express \enquote{any record with a field $f$ of type $T$}.
- To do this, we introduce the following sub-typing relation:
- \[
- \inference{}{\{f_1 : T_1; \dots ; f_n : T_n; g_1 : S_1; \dots ; g_k : S_k\} <: \{f_1 : T_1; \dots ; f_n : T_n\}}
- \]
- % FIXME on avait noté la relation <: dans l'autre sens
- Note that in order to assure the reflexivity of our relation,
- a type $S$ is a sub-type of $T$ if all the fields of $T$ are included in $S$.
- We also introduce a new subsumption typing rule:
- \[
- \inference{\Gamma \vdash e : S & S <: T}{\Gamma \vdash e : T}
- \]
- Assignment
- \[
- \inference{\Gamma \vdash x : T & \Gamma \vdash a : T}{\Gamma \vdash x := a}
- \]
- Sequence
- \[
- \inference{\Gamma \vdash S_1 & \Gamma \vdash S_2}{\Gamma \vdash (S_1 ; S_2)}
- \]
- While loop
- \[
- \inference{\Gamma \vdash b & \Gamma \vdash S}{\Gamma \vdash \<while>\ b\ \<do>\ S}
- \]
- Condition
- \[
- \inference{\Gamma \vdash b & \Gamma \vdash S_1 & \Gamma \vdash S_2}{\Gamma \vdash \<if>\ b\ \<then>\ S_1\ \<else>\ S_2}
- \]
- Booleans
- \[
- \inference{}{\Gamma \vdash \<true>}
- \]
- \[
- \inference{}{\Gamma \vdash \<false>}
- \]
- \[
- \inference{\Gamma \vdash b}{\Gamma \vdash \<not>\ b}
- \]
- \[
- \inference{\Gamma \vdash b_1 & \Gamma \vdash b_2}{\Gamma \vdash b_1\ o\ b_2}{o \in \{\<and>, \<or>\}}
- \]
- \[
- \inference{\Gamma \vdash a_1 : int & \Gamma \vdash a_2 : int}{\Gamma \vdash a_1 \le a_2}
- \]
- \[
- \inference{\Gamma \vdash a_1 : T & \Gamma \vdash a_2 : T}{\Gamma \vdash a_1 = a_2}
- \]
- Note: for the equality test, the rule works for \<int> as well as for records.
- % Domains
- \[
- n \in Num
- \]
- \[
- x \in Var
- \]
- \[
- e,e_1,e_2,a,a_1,a_2 \in AExpr
- \]
- \[
- b,b_1,b_2 \in BExpr
- \]
- \[
- f,f_1,\dots,f_n,f_i,g_1,\dots,g_k, \in Field % Field n'est pas défini…
- \]
- \[
- S,S_1,S_2 \in Statement
- \]
- \subsection{Correctness}
- \[\forall \Gamma P,\quad \Gamma \vdash P \; => \;\forall \sigma,\; (P, \sigma) \not\reduce \bot\]
- \begin{proof}
- By induction on the path of $\reduce$, and by induction on the structure/type of the statement at each step of the path.
- \end{proof}
- \section{Static analysis}
- We propose a constraint based static analysis to determine at each instruction which field is defined in each variables.
- Our static analysis is incarnated by the function $RF$ (\enquote{Reachable Fileds}).
- \[RF : Lab -> Var -> \mathcal{P}(Field)\]
- where $Lab$ is the set of labels.
- One unique label is assigned to each instruction.
- For the sake of simplicity, we note $[I]^l$ the instruction of label $l$.
- We also define an $init$ function returning the first label of a statement where:
- \begin{align*}
- &init([\<skip>]^l) = l\\
- &init([x := a]^l) = l\\
- &init(S_1;S_2) = init(S_1)\\
- &init(\<while>\ [b]^l\ \<do>\ S) = l \\
- &init(\<if>\ [b]^l\ \<then>\ S_1\ \<else>\ S_2) = l
- \end{align*}
- \subsection{Definition}
- \[
- \inference{RF(l) \subseteq RF (init(S)) & RF \vdash (S, l) & RF(l) \subseteq RF(l')}{RF \vdash (\<while>\ [b]^l\ \<do>\ S, l')}
- \]
- \[
- \inference{RF(l) \subseteq RF(init(S_1)) & RF \vdash (S_1, l') \\ RF(l) \subseteq RF(init(S_2)) & RF \vdash (S_2, l')}{RF \vdash (\<if>\ [b]^l\ \<then>\ S_1\ \<else>\ S_2, l')}
- \]
- \[
- \inference{RF(l) \subseteq RF(l')}{RF \vdash ([skip]^l, l')}
- \]
- \[
- \inference{RF \vdash (S_1, init(S_2)) & RF \vdash (S_2, l')}{RF \vdash (S_1 ; S_2, l')}
- \]
- \[
- \inference{RF(l)[x \mapsto \{f_1, \dots, f_n\}] \subseteq RF(l')}{RF \vdash ([x := \{f_1 = e_1 ; \dots ; f_n = e_n\}]^l, l')}
- \]
- \[
- % TODO : il faudrait évaluer statiquement e_1 pour être moins pessimiste
- \inference{RF(l)[x \mapsto \{f\}] \subseteq RF(l')}{RF \vdash ([x := e_1[f \mapsto e_2]]^l, l')}
- \]
- % autres affectations
- \[
- \inference{RF(l)[x \mapsto \emptyset] \subseteq RF(l')}{RF \vdash ([x := n]^l, l')}
- \]
- \[
- \inference{RF(l)[x \mapsto RF(l)(y)] \subseteq RF(l')}{RF \vdash ([x := y]^l, l')}
- \]
- \[
- \inference{RF(l)[x \mapsto \emptyset] \subseteq RF(l')}{RF \vdash ([x := e_1\ o\ e_2]^l, l')}
- \]
- \[
- \inference{RF(l)[x \mapsto \emptyset] \subseteq RF(l')}{RF \vdash ([x := e.f]^l, l')}
- \]
- \subsection{Example}
- Example.
- \subsection{Termination}
- In each premise of each rule, the size of the statement is \emph{strictly} decreasing,
- assuring the termination of the analysis.
- \subsection{Safety}
- This is a pessimistic static analysis.
- \end{document}
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