un peu de texte. Definition plus précise de Value.

dm.tex

@@ -38,42 +38,54 @@
 \begin{document}
 
 \author{Timoth\'ee Haudebourg \and Thibaut Marty}
-\title{PAS}
+\title{Homework PAS/SDL}
+\date{November 18, 2016}
 
 \maketitle
 
 \section{Exercise 1}
 
-\subsection{Denotational Semantic for expressions}
+\subsection{Domains}
 
-% $State = Var -> ((Field \rightharpoonup Value) \cup Value)$
+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 &= int \cup (Field \rightharpoonup Value) \\
-  State &= Var -> Value
+	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|] \sigma \\
+  \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} |[ \{f_1 = e_1 ; \dots ; f_n = e_n\} |] \sigma f &=
-    \begin{cases}
-      \mathcal{A}|[ e_1 |] \sigma \text{ if } f = f_1 \\
-      \hspace{1cm} \vdots \\
-      \mathcal{A}|[ e_n |] \sigma \text{ if } f = f_n
-    \end{cases} \\
   \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 records, we also note $\mathcal{A}|[e|] \sigma = \bot$.
+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 note $\mathcal{B}$ the denotational semantic of booleans expressions.
+We define $->$ the structural operational semantics for commands as:
+
+\[ -> : (Statement \times State) \times ((Statement \times State) \cup State) \]
 
+\subsubsection{Regular execution}
 For all $\sigma \neq \bot$:
 \[
   \inference{}{(skip, \sigma) -> \sigma}
@@ -100,17 +112,36 @@ For all $\sigma \neq \bot$:
   \inference{}{(\<if>\ b\ \<then>\ S_1\ \<else>\ S_2, \sigma) -> (S_2, \sigma)}{\text{if } \mathcal{B}|[b|] \sigma = \text{ff}}
 \]
 
-Error cases: % TODO
+\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.
 
 \[
-  \inference{}{(S, \bot) -> \bot}
+  \inference{}{(x := a, \sigma) -> \bot}{\text{if }\mathcal{A}|[a|] \sigma = \bot}
 \]
 \[
-  \inference{}{(x := a, \sigma) -> \bot}{\text{if }\mathcal{A}|[a|] \sigma = \bot}
+  \inference{}{(\<while>\ b\ \<do>\ S, \sigma) -> \bot}{\text{if } \mathcal{B}|[b|] \sigma = \bot}
+\]
+\[
+  \inference{}{(\<if>\ b\ \<then>\ S_1\ \<else>\ S_2, \sigma) -> \bot}{\text{if } \mathcal{B}|[b|] \sigma = \bot}
+\]
+
+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:
+
 \[
   \inference{}{\Gamma \vdash n : int}
 \]