Abstract
The paper describes an implementation of instantiation-based Craig interpolation for quantified formulae. The implementation is based on the CSIsat interpolating solver. The tool supports interpolation for formulae with linear real arithmetic, uninterpreted functions and quantifiers. The paper suggests usage of an external decision procedure (namely CVC3) for quantified formula instantiation. It describes how the CSIsat and CVC3 tools were modified in order to support quantified formulae interpolation. The paper also contains results for benchmarking the modified CSIsat tool on a set of tests obtained by randomly splitting tasks from the SMTLIB library as well as on a small collection of specific dedicated interpolation tasks generated by encoding several manually specified parameterized error trace patterns with quantified logical formulae. The approach to interpolation considered in the paper is based on the recently proposed extension of the McMillan’s algorithm for resolution-based interpolant generation that was suggested earlier for implementation in the SMTInterpol interpolating solver by its developers. The extended algorithm additionally requires a set of quantified subformula instances sufficient for unsatisfiability proof of the initial formula and produces possibly quantified interpolants. A proper implementation of the algorithm could potentially be used in predicate abstraction-based verification tools for obtaining abstraction predicates from counterexamples by Craig interpolation. Though the evaluation presented in the paper showed that the considered implementation turned out to be too inefficient for this purpose due to significant repetitive proof overhead, which arose from combining a more efficient and advanced solver with a significantly less efficient one (in CVC3+CSIsat combination CSIsat is much less efficient than CVC3).
Highlights
Among several currently predominant model checking techniques, predicate abstraction is one of the most widespread and successful approaches
The first one was obtained from the SMT-LIB[3] benchmark set by dividing the unsatisfiable formulae from the AUFLIA (AUFLIA stands for Arrays, Uninterpreted Functions and Linear Integer Arithmetic) and AUFLIRA (AUFLIRA stands for Arrays, Uninterpreted Functions, and Linear Integer and Real Arithmetic) logics randomly into two sub-formulae at the top-level conjunctions
In this paper we proposed a relatively easy approach to implement an instantiation-based interpolating decision procedure for quantified formulae based on an existing interpolating solver and a modern SMT solver with quantifier support
Summary
Among several currently predominant model checking techniques, predicate abstraction is one of the most widespread and successful approaches. As soon as the counterexample is spurious, its weakest precondition and a path formula are unsatisfiable, if build precisely enough This fact is used by the tools to derive new predicates and refine current abstraction. The getting of quantified interpolants for such formulae is acceptable This way we see that SLAM2 with logical memory model for pointers needs only a good SMT solver with the support of quantifiers and unsatisfiable core extraction. Long as the extended McMillan algorithm’s implementation in case of a priori given necessary quantifier instantiations is reasonably easy, this implementation might be as well used for preliminary benchmarking the logical memory model efficiency in static software verifiers using interpolation for abstraction refinement This way we decided to implement the extended McMillan’s algorithm based on some existing interpolating prover and an SMT solver with quantifier support. We decided to estimate the efficiency of the new tool on specially generated benchmarks simulating the interpolation tasks a real model checker could give to our tool
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