A New Roadmap for Linking Theories of Programming

Abstract

Formal methods advocate the crucial role played by the algebraic approach in specification and implementation of programs. Traditionally, a top-down approach (with denotational model as its origin) links the algebra of programs with the denotational representation by establishment of the soundness and completeness of the algebra against the given model, while a bottom-up approach (a journey started from operational model) introduces a variety of bisimulations to establish the equivalence relation among programs, and then presents a set of algebraic laws in support of program analysis and verification. This paper proposes a new roadmap for linking theories of programming. Our approach takes an algebra of programs as its foundation, and generates both denotational and operational representations from the algebraic refinement relation.

Appendix

Lemma 3.1

If $(s,\,G(Q))\rightarrow ^{*}(t,\,\epsilon )$, then either $(s,\,G(\bot ))\uparrow$ or $(s,\,G(\bot ))\rightarrow ^{*}(t,\,\epsilon )$.

Proof:

Induction on the structure of G.

Base case: $G(Q)=Q$. The conclusion follows from From Rule (6)

$$\begin{aligned} (s,\,\bot )\rightarrow (s,\,\bot ) \end{aligned}$$

in Definition 3.2.

Inductive step:

• (1) $G(Q)=G_1(Q)\sqcap G_2(Q)$.

  

• (2) $G(Q)=G_1(Q)\lhd b\rhd G_2(X)$

  

• (3) $G(Q)=G_1(Q);G_2(Q)$

  

• (4) $G(Q)=\mu X\bullet P(Q,\,X)$

  

Lemma 3.2

• (1) $(s,\,F(P))\uparrow \,\Rightarrow \,(s,\,\mathcal{F}(\bot ))\uparrow$

• (2) $(s,\,F(\mu X\bullet P(X))\uparrow \,\Rightarrow \, (s,\,F(P(\mu X\bullet P(X))))\uparrow$

Proof

(1). Based on induction on the structure of F.

Base case: $F(X)=X$. The conclusion follows from the rule (6).

Inductive Step:

Proof of (2): Similar to (1).