Automatic Constrained Rewriting Induction towards Verifying Procedural Programs

Rewriting modulo SMT and open system analysis

“…Again, all this is understandable given their focus on inductive theorem proving of universal formulas. One similarity between the work in [41] and our work is that, to handle input-output in an imperative language, they allow, as we do, extra variables in the righthand sides of rewrite rules. In general, while approaches such as in [5,12,[27][28][29][38][39][40] address symbolic reasoning for equational theorem proving purposes, or apply these techniques to imperative program analysis and verification, even allowing sometimes extra variables in the right-hand sides of equations, e.g., [41,63,64], theses approaches are quite different from ours because of their predominant focus on equational reasoning for proving, often inductively, universal formulas, and/or on applications to, typically sequential, programming languages.…”

“…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. 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.…”

“…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.…”

“…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.…”

Generalized Rewrite Theories and Coherence Completion

“…Symbolic methods are used to reason about concurrent systems specified by rewrite theories in many ways, including: (i) cryptographic protocol verification, e.g., [10], (ii) logical LTL model checking, e.g., [2], (iii) rewriting modulo SMT and related approaches, e.g., [22,1], (iv) inductive theorem proving and program verification, e.g., [12,16], and (v) reachability logic theorem proving, e.g., [25,17,24]. One key issue is that the rewrite theories used in several of these approaches go beyond the standard notion of rewrite theory in, say [3], and also beyond the executability requirements in, say, [8].…”