Program equivalence by circular reasoning

Abstract

We propose a logic and a deductive system for stating and automatically proving the equivalence of programs in deterministic languages having a rewriting-based operational semantics. The deductive system is circular in nature and is proved sound and weakly complete; together, these results say that, when it terminates, our system correctly solves the program-equivalence problem as we state it. We show that our approach is suitable for proving the equivalence of both terminating and non-terminating programs, and also the equivalence of both concrete and symbolic programs. The latter are programs in which some statements or expressions are symbolic variables. By proving the equivalence between symbolic programs, one proves in one shot the equivalence of (possibly, infinitely) many concrete programs obtained by replacing the variables by concrete statements or expressions. We also report on a prototype implementation of the proposed deductive system in the K Framework.Key-words: Program Equivalence, Circular Reasoning, K framework. * University of Iasi, Romania † Inria Lille Nord Europe Equivalence de Programmes par Raisonnement Circulaire Résumé :Nous proposons une logique et un système déductif pour exprimer et prouver automatiquement l'équivalence de programmes dans des langages déterministes munis de sé-mantiques opérationnelles définies par réécriture. Le système déductif proposé est de nature circulaire; nous démontrons qu'il est correct et faiblement complet. Ces deux résultats signifient que notre système résout correctement le problème d'équivalence de programmes tels que nous l'avons posé. Nous montrons que ce système fonctionne autant pour des programmes qui terminent que pour des programmes qui ne terminent pas. Les programmes dits symboliques, dans lesquels certaines expressions ou instructions restent non-interprétés, peuvent également être traités par notre approche. La démonstration d'une équivalence entre deux programmes symboliques revient à démontrer l'équivalence entre une infinité potentielle de programmes concrets, qui sont des instances des programmes symboliques obtenues en remplaçant les variables symboliques par des instructions ou des expressions concrètes. Enfin, nous décrivons une implé-mentation prototype de notre système déductif dans la K framework.

Executing and verifying higher-order functional-imperative programs in Maude

“…By contrast, in a deep embedding programs have to stay within the bounds of the guest language's syntax and semantics. The same dichotomy (shallow vs. deep embeddings) distinguishes the present paper with our previous work on language-parametric symbolic execution, program verification and program equivalence [11,12,13,14,15].…”

A language-independent proof system for full program equivalence

“…They use operational semantics (of a specific language they designed, called LPL) and proof systems, and formally prove their proof system's soundness. Symbolic programs are considered in [25] but for a different notion of program equivalence. In [14] a classification of equivalence relations used in program-equivalence research is given, one of which is full equivalence.…”

“…Finally, our own related work [25] gives a proof system for another equivalence relation parameterized by an observation relation and uses other technical mechanisms. The equivalence relation is based on bisimulation, with each process having some part of its state observable.…”

“…The equivalence relation is based on bisimulation, with each process having some part of its state observable. Therefore, the equivalence relation in [25] is better suited for interactive processes. Also, the proof system presented in [25] is based on derivatives and is more operational than the proof system here.…”

Reverse engineering language product lines from existing DSL variants

“…Whereas comparison of signatures can be performed by syntactic comparison of the signature elements -i.e., checking if the names, return types, visibility rules-, comparison of bodies can be arbitrary difficult. If we try to compare the behavior of the domain-specific actions, then we will have to address the semantic equivalence problem, which is known to be undecidable [32]. To address this issue, we conceive bodies comparison in terms of its abstract syntax tree as proposed by Biegel et al [33].…”