Rewriting Induction + Linear Arithmetic = Decision Procedure

Abstract

Abstract. This paper presents new results on the decidability of inductive validity of conjectures. For this, a class of term rewrite systems (TRSs) with built-in linear integer arithmetic is introduced and it is shown how these TRSs can be used in the context of inductive theorem proving. The proof method developed for this couples (implicit) inductive reasoning with a decision procedure for the theory of linear integer arithmetic with (free) constructors. The effectiveness of the new decidability results on a large class of conjectures is demonstrated by an evaluation of the prototype implementation Sail2.

Rewriting modulo SMT and open system analysis

“…More recently, C. Kop and N. Nishida [40] have proposed a way to unify the ideas regarding equational rewriting with logical constraints and have proposed in [41] an inductive method of proving properties of programs in an imperative language by their notion of symbolic rewriting modulo decidable constraints. The main difference with our approach is that, as in [29], their notion of symbolic rewriting is universal, and therefore completely different from our existential notion in Definition 7; furthermore, in [41] termination of the rewrite theory is required for inductive reasoning, whereas no termination is required at all in our setting. Again, all this is understandable given their focus on inductive theorem proving of universal formulas.…”

“…In particular, they extended the dependency pair framework to handle termination of equational specifications with semantic data structures and evaluation strategies in the Maude functional sublanguage. The same authors used the idea of combining rewriting induction and linear arithmetic over constrained terms [29]. Their aim is to obtain equational decision procedures that can handle semantic data types represented by the constrained built-ins.…”

“…The main difference between their work and rewriting modulo SMT presented in this paper is that the notion of symbolic rewriting modulo decidable constraints is completely different. According to Definition 14 in [29], a symbolic rewrite step u; φ R v; φ with a rule l → r if ϕ in their sense requires a matching substitution θ such that φ ⇒ (ϕθ) is T E 0 -valid. This is a universal notion of symbolic rewriting with constraints completely different from our existential notion in Definition 7, which is based instead on constrain satisfiability.…”

“…This is a universal notion of symbolic rewriting with constraints completely different from our existential notion in Definition 7, which is based instead on constrain satisfiability. This difference is understandable by observing that the goal in [29] is to prove universal formulas about equational specifications by inductive theorem proving, whereas our goal is very different, namely, to prove existential reachability formulas about a concurrent system specified by a rewrite theory. More recently, C. Kop and N. Nishida [40] have proposed a way to unify the ideas regarding equational rewriting with logical constraints and have proposed in [41] an inductive method of proving properties of programs in an imperative language by their notion of symbolic rewriting modulo decidable constraints.…”

“…Rewriting modulo SMT techniques can then be applied to increase the power of rewrite-based equational reasoning for (Σ, E) such as, for instance, inductive theorem proving [29,39,40], termination checking [28,61], and procedural verification [41]. However, the full power of rewriting modulo SMT, including its model checking capabilities, can be better exploited when applied to concurrent open systems.…”