Proving Theorems by Program Transformation

Abstract

In this paper we present an overview of the unfold/fold proof method, a method for proving theorems about programs, based on program transformation. As a metalanguage for specifying programs and program properties we adopt constraint logic programming (CLP), and we present a set of transformation rules (including the familiar unfolding and folding rules) which preserve the semantics of CLP programs. Then, we show how program transformation strategies can be used, similarly to theorem proving tactics, for guiding the application of the transformation rules and inferring the properties to be proved. We work out three examples: (i) the proof of predicate equivalences, applied to the verification of equality between CCS processes, (ii) the proof of first order formulas via an extension of the quantifier elimination method, and (iii) the proof of temporal properties of infinite state concurrent systems, by using a transformation strategy that performs program specialization. IntroductionProgram transformation is a methodology that allows the programmer to separate the correctness concern and the efficiency concern when developing programs [4]. An initial, maybe inefficient, program whose correctness with respect to a given specification can easily be proved, is transformed, possibly in several steps, into an efficient program by applying correctness preserving transformations.Although its main objective is the improvement of efficiency, it has long been recognized that program transformation can also be used as a methodology for proving program properties and, more in general, for proving theorems. Indeed, in the case of functional or logic programming, programs can be regarded as theories consisting of sets of equations and logical implications, respectively, which are associated with models defined by a suitable program semantics (either least or greatest models). Thus, transforming programs can be regarded as an activity by which one deduces consequences of theories, that is, theorems which hold in the models defined by the given semantics. In this setting, the elementary transformation steps, often called transformation rules, can be regarded as inference rules, and composite transformations, often called transformation strategies, can be regarded as theorem proving tactics.The view of program transformation as a theorem proving activity was first suggested in the seminal paper by Burstall and Darlington [4], where some equivalences between functions defined by recursive equations are proved by applying unfolding and folding transformations. Given a function definition f (x) = D[x], the unfolding rule consists in replacing a function call f (t) occurring in the right hand side of a program equation by the expression D[t]. The folding rule is the inverse of the unfolding rule, and consists in replacing an occurrence of the expression D[t] by the function call f (t). In order to prove the equivalence of two functions, say f and g, Burstall and Darlington proposed a method, which we will call the unfold/fold proof me...

Analysis and Transformation of Constrained Horn Clauses for Program Verification

“…As already mentioned in Section 4, constrained Horn clauses have recently been applied for modelling programs written in many different programming languages. Besides programs, CHCs have also been used for encoding more abstract computational models of various kinds, including Petri nets (Fribourg andOlsén 1997, Leuschel andLehmann 2000), timed automata (Jaffar et al 2004), linear hybrid automata (Banda and Gallagher 2009), concurrent systems Podelski 1999, Fioravanti et al 2001b;2013a), parameterized systems (Roychoudhury et al 2000), process algebras (Fioravanti et al 2013b), and business processes (De .…”

Analysis and Transformation of Constrained Horn Clauses for Program Verification

“…As already mentioned in Section 4, constrained Horn clauses have recently been applied for modelling programs written in many different programming languages. Besides programs, CHCs have also been used for encoding more abstract computational models of various kinds, including Petri nets (Fribourg andOlsén 1997, Leuschel andLehmann 2000), timed automata (Jaffar et al 2004), linear hybrid automata (Banda and Gallagher 2009), concurrent systems (Delzanno and Podelski 1999, Fioravanti et al 2001b, 2013a, parameterized systems (Roychoudhury et al 2000), process algebras (Fioravanti et al 2013b) and business processes .…”