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sav08:relational_semantics_of_procedures [2009/05/26 17:45] vkuncak |
sav08:relational_semantics_of_procedures [2009/05/26 23:01] vkuncak |
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How can we define semantics for program with recursive procedures? | How can we define semantics for program with recursive procedures? | ||
+ | |||
+ | We consider first the case of procedures without parameters; adding parameters is straightforward. | ||
===== One Recursive Procedure ===== | ===== One Recursive Procedure ===== | ||
Line 60: | Line 62: | ||
In fact, $F^n(\emptyset)$ represents the result of inlining the recursive procedure $n$ times. Therefore, if $s_0$ is the initial state such | In fact, $F^n(\emptyset)$ represents the result of inlining the recursive procedure $n$ times. Therefore, if $s_0$ is the initial state such | ||
that the computation in the procedure terminates within $n$ steps, then $(s_0,s) \in F^n(\emptyset)$ iff $(s_0,s) \in r_{move}$. | that the computation in the procedure terminates within $n$ steps, then $(s_0,s) \in F^n(\emptyset)$ iff $(s_0,s) \in r_{move}$. | ||
+ | |||
+ | In this example, we have | ||
+ | \[ | ||
+ | r_{move} = \{((x,y),(x',y').\ (x > 0 \land x'=0 \land y'=x+y) \lor (x \le 0 \land x'=x \land y'=y) \} | ||
+ | \] | ||
===== Multiple Mutually Recursive Procedures ===== | ===== Multiple Mutually Recursive Procedures ===== | ||
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\] | \] | ||
- | The results extend to any number of call sites and any number of mutually recursive procedures, we just consider a function $G : {\cal R}^d \to {\cal R}^d$ where $d$ is the number of procedures; this mapping describes how, given one approximation of procedure semantics, compute a better approximation that is correct for longer executions. We have operations $\sqsubseteq$ and $\sqcup$ analogous to $\subseteq$ and $\cup$ (indeed, the domain ${\cal R}^d$, that is $(2^{S\times S})^d$, is isomorphic to $2^{S\times S \times \{1,2,\ldots,d\}}$). | + | We define lattice structure on ${\cal R}^2$ by |
+ | \[ | ||
+ | (r_1,r_2) \sqsubseteq (r'_1,r'_2) \ \mbox{ iff } \ (r_1 \subseteq r'_1) \land (r_2 \subseteq r'_2) | ||
+ | \] | ||
+ | \[ | ||
+ | (r_1,r_2) \sqcup (r'_1,r'_2) = (r_1 \cup r'_1, r_2 \cup r'_2) | ||
+ | \] | ||
+ | Note that: | ||
+ | \[ | ||
+ | G(\emptyset,\emptyset) = ([\![assume(x==0)]\!] \circ [\![wasEven=true]\!], [\![assume(x==0)]\!] \circ [\![wasEven=false]\!]) | ||
+ | \] | ||
+ | \[ | ||
+ | G(G(\emptyset,\emptyset)) = | ||
+ | \begin{array}[t]{@{}l@{}} | ||
+ | \bigg( ([\![assume(x==0)]\!] \circ [\![wasEven=true]\!]) \cup | ||
+ | ([\![assume(x!=0)]\!] \circ [\![x=x-1]\!] \circ G(\emptyset,\emptyset).\_2)\ , \\ | ||
+ | ([\![assume(x==0)]\!] \circ [\![wasEven=false]\!]) \cup | ||
+ | ([\![assume(x!=0)]\!] \circ [\![x=x-1]\!] \circ G(\emptyset,\emptyset).\_1) \bigg) | ||
+ | \end{array} | ||
+ | \] | ||
+ | where $p.\_1$ and $p.\_2$ denote first, respectively, second, element of the pair $p$. | ||
- | The meaning of all procedures is therefore the tuple $p_* \in {\cal R}^n$ given by $\bigsqcup\limits_{n \ge 0} G^n(\emptyset,\ldots,\emptyset)$. | + | The results extend to any number of call sites and any number of mutually recursive procedures, we just consider a function $G : {\cal R}^d \to {\cal R}^d$ where $d$ is the number of procedures; this mapping describes how, given one approximation of procedure semantics, compute a better approximation that is correct for longer executions. |
+ | |||
+ | We define $\sqsubseteq$ and $\sqcup$ analogous to $\subseteq$ and $\cup$ (indeed, the domain ${\cal R}^d$, that is $(2^{S\times S})^d$, is isomorphic to $2^{S\times S \times \{1,2,\ldots,d\}}$). | ||
+ | |||
+ | The meaning of all procedures is then the tuple $p_* \in {\cal R}^n$ given by $\bigsqcup\limits_{n \ge 0} G^n(\emptyset,\ldots,\emptyset)$. | ||
+ | |||
+ | === Example === | ||
+ | |||
+ | In the example above, we can prove that | ||
+ | \[ | ||
+ | r_{even} = \{((x,wasEven),(x',wasEven').\ x \ge 0 \land x' = 0 \land | ||
+ | (wasEven' \leftrightarrow (x \mod 2 = 0)) | ||
+ | \] | ||
+ | \[ | ||
+ | r_{odd} = \{((x,wasEven),(x',wasEven').\ x \ge 0 \land x' = 0 \land | ||
+ | (wasEven' \leftrightarrow (x \mod 2 \ne 0)) | ||
+ | \] | ||
- | **Remark:** $p_*$ is the least fixpoint of $G$, so by [[sav08:Tarski's Fixpoint Theorem]], it is also the least $p$ such that $G(p) \sqsubseteq p$. Thus, $G(p) \sqsubseteq p$ implies $p_* \sqsubseteq p$. | + | === Remark === |
+ | $p_*$ is the least fixpoint of $G$, so by [[sav08:Tarski's Fixpoint Theorem]], it is also the least $p$ such that $G(p) \sqsubseteq p$. Thus, $G(p) \sqsubseteq p$ implies $p_* \sqsubseteq p$. | ||