Efficient Interpolant Generation in Satisfiability Modulo Linear Integer Arithmetic

The problem of computing Craig interpolants in SAT and SMT has recently received a lot of interest, mainly for its applications in formal verification. Efficient algorithms for interpolant generation have been presented for some theories of interest — including that of equality and uninterpreted functions (EUF), linear arithmetic over the rationals (LA(Q)), and their combination— and they are successfully used within model checking tools. For the theory of linear arithmetic over the integers (LA(Z)), however, the problem of finding an interpolant is more challenging, and the task of developing efficient interpolant generators for the full theory LA(Z) is still the objective of ongoing research. In this article we try to close this gap. We build on previous work and present a novel interpolation algorithm for SMT(LA(Z)), which exploits the full power of current stateof-the-art SMT(LA(Z)) solvers. We demonstrate the potential of our approach with an extensive experimental evaluation of our implementation of the proposed algorithm in the MathSAT SMT solver.