Playing in the grey area of proofs

Abstract

Interpolation is an important technique in verification and static analysis of programs. In particular, interpolants extracted from proofs of various properties are used in invariant generation and bounded model checking. A number of recent papers studies interpolation in various theories and also extraction of smaller interpolants from proofs. In particular, there are several algorithms for extracting of interpolants from so-called local proofs. The main contribution of this paper is a technique of minimising interpolants based on transformations of what we call the "grey area" of local proofs. Another contribution is a technique of transforming, under certain common conditions, arbitrary proofs into local ones.Unlike many other interpolation techniques, our technique is very general and applies to arbitrary theories. Our approach is implemented in the theorem prover Vampire and evaluated on a large number of benchmarks coming from first-order theorem proving and bounded model checking using logic with equality, uninterpreted functions and linear integer arithmetic. Our experiments demonstrate the power of the new techniques: for example, it is not unusual that our proof transformation gives more than a tenfold reduction in the size of interpolants.

Complete instantiation-based interpolation

“…Recent works have explored techniques to influence the quality of computed interpolants, e.g., by reducing the size of unsatisfiability proofs from which interpolants are generated [27], restricting the language in which interpolants can be expressed [1,30], or by controlling the interpolant strength [16]. Our technique of guiding the proof search of the interpolation procedure through user-defined abstraction functions is orthogonal to these approaches.…”

Beautiful Interpolants

“…For example, [14,32,28] focused on the process of extracting interpolants of varying strengths from refutation proofs. Hoder et al [19] proposed an algorithm that produces syntactically smaller interpolants by applying transformations to refutation proofs. Jhala and McMillan [20] described a modified theory solver that yields interpolants in a bounded language.…”

A First Class Boolean Sort in First-Order Theorem Proving and TPTP

“…These properties usually are expressed in combined theories of various data structures, such as integers and arrays, and hence require reasoning with both theories and quantifiers. Recent approaches in interpolation and loop invariant generation [14,12,10] present initial results of using first-order theorem provers for generating quantified program properties. First-order theorem provers can also be used to generate program properties with quantifier alternations [12]; such properties could not be generated fully automatically by any previously known method.…”