High Performance Static Analysis for Industry

Formal Techniques for Safety-Critical Systems

“…This result generalizes from the unsorted to the order-sorted case, and from AC to ACCU axioms, a previous result by H. Comon-Lundh [29]. This is done in Theorem 8 below.…”

“…Generalizing a similar result in [29] for the unsorted and AC case, I show below that, when B Ω = ACCU -where ACCU stands for any combination of associativitycommutativity (AC), commutativity (C), and/or left-or right-identity (U ) axioms for some binary function symbols-satisfiability of QF formulas in T Ω/ACCU is decidable. But, generalizing again another result in [29], we can view such a satisfiability result as part of a broader one, namely, decidable satisfiability in T Σ,Π,Γ or, equivalently, in T Σ∪Π/ Γ when ((Σ, Π), Γ ) is an OS-compact equational OS-FO theory.…”

“…Note that this generalizes the notion of compact theory in [29] in four ways: (i) from unsorted to OS theories; (ii) by dealing with the phenomenon of possibly having some sorts finite and some infinite; (iii) by extending the notion from equational theories to OS-FO equational theories; and (iv) by including the case of computable finite initial models, because an OS-FO theory ((Σ, ∅), E) whose sorts are all finite and for which we can effectively compute representatives has decidable equality and finitary unification, and is OS-compact in a vacuous sort of way; e.g., the Boolean theory B of Example 1, and the theory N 2 in Example 5 are both OS-compact. I will illustrate with examples that extensions (i)-(iii) are needed in many useful applications.…”

“…The key theorem about OS-compact theories is again a generalization of a similar one in [29]. This corollary further "unpacks" how the satisfiability in T Ω,Δ,Γ0 of an Ω ∪Δdisjunction of disequalities D obtained in step (2) of the satisfiability decision procedure "unpacking" Corollary 1 can be checked in step (3) when ((Ω, Δ), Γ 0 ) is OS-compact, namely, we then replace D by the disjunction of all the representative ground instantiations D rep(β) of its finite sort variables, and then check whether at least one such D rep(β) is satisfiable by checking the B Ωconsistency of ( D rep(β))!…”

“…General conditions under which the initial algebra of constructors associated to an initial algebra T Σ/E B has decidable satisfiability and makes, in turn, satisfiability in T Σ/E B decidable are investigated. A key notion is that of an OS-compact theory, which generalizes in several ways that of a compact theory in [29]. In particular, it is shown that T Ω/B has decidable QF satisfiability for B any combination of associativity, commutativity and identity axioms, except associativity without commutativity; furthermore, various relevant examples of decidable initial algebras whose initial algebra of constructors are of the form T Ω/B are given.…”