Obtaining Finite Local Theory Axiomatizations via Saturation

Abstract

In this paper we present a method for obtaining local sets of clauses from possibly non-local ones. For this, we follow the work of Basin and Ganzinger and use saturation under a version of ordered resolution. In order to address the fact that saturation can generate infinite sets of clauses, we use constrained clauses and show that a link can be established between saturation and locality also for constrained clauses: This often allows us to give a finite representation of possibly infinite saturated sets of clauses

On Invariant Synthesis for Parametric Systems

“…Ideas similar to those used in the melting calculus [HW09] (used e.g. in [HS13,HS14]) could be used to obtain…”

“…Even if we cannot guarantee that assumption (A5) holds, it could theoretically be possible to identify situations in which we can transform candidate invariants which do not define local extensions into equivalent formulae which define local extensions -e.g. using the results in [HS13]. If all candidate invariants I generated in Algorithm 2 are ground, assumption (A5) is not needed.…”

Deciding Local Theory Extensions via E-matching

“…The second characterization of local theory extensions is proof-theoretic and states that a set of axioms is a local theory extension if it is saturated under (ordered) resolution [4]. This characterization can be used to automatically compute local theory extensions from non-local ones [20]. Note that the locality property depends both on the base theory as well as the specific axiomatization of the theory extension.…”

“…Local theory extensions are a class of such extensions that can be decided using finite quantifier instantiation of the extension axioms. This class is attractive because it is characterized by proof and model-theoretic properties that abstract from the intricacies of specific quantifier instantiation techniques [15,20,36]. Also, many well-known theories that are important in verification but not commonly supported by SMT solvers are in fact local theory extensions, even if they have not been presented as such in the literature.…”

Symbol Elimination for Parametric Second-Order Entailment Problems (with Applications to Problems in Wireless Network Theory)

“…The main challenge when using saturation approaches for symbol elimination is the fact that the saturated sets might be infinite. Sometimes finite representations of possibly infinite sets of clauses exist: for this, Horbach and Weidenbach introduced a melting calculus [27], later used in [25,26] and [16]. Similar aspects were explored in the study of acceleration for program verification modulo Presburger arithmetic by Boigelot, Finkel and Leroux [14,17], in relationship with array systems by [4], or in the study of constrained Horn clauses (cf.…”