Lectures

doc/07.tex

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+\section{Arrays and strings}
+
+An array can be represented as a pair: the length of the array and a mapping from indices to elements. If we denote
+$\mathscr E$ the set of elements then the set of all arrays $\mathscr A (\mathscr E)$ can be defined as follows:
+
+\[
+\mathscr A(\mathscr E) = \mathbb N \times (\mathbb N \to \mathscr E)
+\]
+
+For an array $(n, f)$ we assume $\dom{f}=[0\,..\,n-1]$. An element selection function:
+
+\[
+\begin{array}{c}
+  [\bullet] : \mathscr A (\mathscr E) \to \mathbb N \to \mathscr E\\[2mm]
+  (n, f) [i] = \left\{
+                  \begin{array}{rcl}
+                     f\;i &, & i < n\\
+                     \bot&,&\;\mbox{otherwise}
+                  \end{array}
+               \right.
+\end{array}
+\]
+
+\begin{comment}
+A set of (ASCII) characters:
+
+\[
+\mathscr C = [0\,..\,255]
+\]
+
+A string is an array of characters:
+
+\[
+\mathscr S = \mathscr A (\mathscr C)
+\]
+
+Adding strings and arrays to the language changes the set of values the programs operate on:
+\end{comment}
+
+We represent arrays by \emph{references}. Thus, we introduce a (linearly) ordered set of locations
+
+\[
+\mathscr L = \{l_0, l_1, \dots\}
+\]
+
+Now, the set of all values the programs operate on can be described as follows:
+
+\[
+    \mathscr V = \mathbb Z \uplus \mathscr L 
+\]
+
+Here, every value is either an integer, or a reference (some location). The disjoint union ``$\uplus$'' makes it possible to
+unambiguously discriminate between the shapes of each value. To access arrays, we introduce an abstraction of memory:
+
+
+\[
+    \mathscr M = \mathscr L \to \mathscr A\,(\mathscr V)
+\]
+
+We now add two more components to the states: a memory function $\mu$ and the first free memory location $l_m$. 
+
+%Enriching the value set extends the definitions of state (which now has to map variable names to values) and
+%stack (similarly), but does not affect the definition of input/output streams, read and write primitives, etc.
+
+\subsection{Adding arrays/strings on expression level}
+
+On expression level, abstractly/concretely:
+
+\[
+\begin{array}{rclll}
+\mathscr E += &      &\mathscr E \mathtt{[} \mathscr E \mathtt{]} & (\llang{$a$ [$e$]})               & \mbox{taking an element}\\
+              & \mid & \mathtt{[} \mathscr E^* \mathtt{]}         & (\llang{[$e_1$, $e_2$,.., $e_k$]}) & \mbox{creating an array}\\
+              & \mid & \mathscr E \mathtt{.length}                & ($e$\mathtt{.length})             & \mbox{taking the length}
+\end{array}
+\]
+
+In addition, in a concrete syntax we supply two special forms for strings: \llang{'$x$'} as a denotation for integer code of ASCII symbol
+$x$, and \llang{"..."}~--- a string constant.
+
+The semantics of enriched expressions is modified as follows. First, we add two additional premises to the rule for binary operators:
+
+\setsubarrow{_{\mathscr E}}
+\[\trule{\begin{array}{c}
+            \withenv{\Phi}{\trans{c}{A}{c^\prime=\inbr{\_,\, \_,\, \_,\, a}}}\\
+            \withenv{\Phi}{\trans{c^\prime}{B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, b}}}\\
+            a\in\mathbb Z,\,b\in\mathbb Z
+          \end{array}
+        }
+        {\withenv{\Phi}{\trans{c}{A\otimes B}{\inbr{\sigma^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, a\oplus b}}}}
+        \ruleno{Binop}
+\]
+       
+These two premises ensure that both operand expressions are evaluated into integer values. Second, we have to add the rules for new
+kinds of expressions (see Figure~\ref{array_expressions}).
+
+\begin{figure}
+  \[\trule{\begin{array}{c}
+             \withenv{\Phi}{\trans{c}{E}{c^\prime=\inbr{\_, \_, \_, l}}}\\
+             \withenv{\Phi}{\trans{c^\prime}{J}{\inbr{s^{\prime\prime}=\inbr{\_,\, \_,\, \_,\, \mu,\,\_},\, i^{\prime\prime},\, o^{\prime\prime},\, j}}}\\
+             l\in \mathscr L,\,(n,\,f)=\mu\,\;l\,j\in\mathbb N,\,j<n
+           \end{array}}
+          {\withenv{\Phi}{\trans{c}{E\,\mathtt{[}J\mathtt{]}}{\inbr{s^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, f\,j}}}}
+          \ruleno{ArrayElement}
+  \]
+  \[\trule{\withenv{\Phi}{\trans{c_j}{e_j}{c_{j+1}=\inbr{\inbr{\sigma_g^{j+1},\,S^{j+1},\,\sigma_l^{j+1},\,\mu^{j+1},\,l_{m_{j+1}}}, i_{j+1}, o_{j+1}, a_j}}},\,j\in [0..k]}
+          {\withenv{\Phi}{\trans{c_0}{\mathtt{[}e_0, e_1,...,e_k\mathtt{]}}{\inbr{\inbr{\sigma_g^{j+1},\,S^{j+1},\,\sigma_l^{j+1},\,\mu^{j+1}[l_{m_{k+1}}\gets (k,\,\lambda i\,.\,a_i)],\, l_{m_{k+1}+1}}, i_{k+1}, o_{k+1}, l_{m_{k+1}}}}}}
+          \ruleno{ArrayConstructor}
+  \]
+  \[\trule{\begin{array}{c}
+              \withenv{\Phi}{\trans{c}{E}{\inbr{s^{\prime\prime}=\inbr{\_,\,\_,\,\_,\,\mu,\,\_}, i^{\prime\prime}, o^{\prime\prime}, l}}}\\
+              l\in\mathscr L,\,(n,\,f)=\mu\;l
+            \end{array}
+          }
+          {\withenv{\Phi}{\trans{c}{E\mathtt{.length}}{\inbr{s^{\prime\prime}, i^{\prime\prime}, o^{\prime\prime}, n}}}}
+          \ruleno{ArrayLength}
+  \]
+  \caption{Big-step Operational Semantics for Array Expressions}
+  \label{array_expressions}
+\end{figure}
+
+\subsection{Adding arrays/strings on statement level}
+
+On statement level, we add the single construct:
+
+\[
+\mathscr S += \mathscr E \mathtt{[} \mathscr E \mathtt{]} \llang{:=} \mathscr E
+\]
+
+This construct is interpreted as an assignment to an element of an array. The semantics of this construct is described by the following rule:
+
+\[
+\trule{\setsubarrow{_{\mathscr E}}
+       \begin{array}{c}
+         \withenv{\Phi}{\trans{c}{E}{c^\prime=\inbr{\_,\_,\_, (n,\,f)}}}\\
+         \withenv{\Phi}{\trans{c^\prime}{J}{c^{\prime\prime}=\inbr{\_,\_,\_,j}}}\\
+         \withenv{\Phi}{\trans{c^{\prime\prime}}{F}{c^{\prime\prime\prime}=\inbr{s^{\prime\prime},i^{\prime\prime},o^{\prime\prime},y}}}\\
+         j\in\mathbb N,\,j<n\\
+         \setsubarrow{}\withenv{\llang{skip},\,\Phi}{\trans{\inbr{\mathtt{update}(s^{\prime\prime}, E, (n,\,f [j\gets y])),i^{\prime\prime},o^{\prime\prime},\hbox{---}}}{K}{\widetilde{c}}}
+       \end{array}
+      }
+      {\setsubarrow{}\withenv{K,\,\Phi}{\trans{c}{E\mathtt{[}J\mathtt{]}\llang{:=}F}{\widetilde{c}}}}
+      \ruleno{ArrayAssign}
+\]