  The problem of computing Craig Interpolants has recently received a lot of interest. In this paper, we address the problem of efficient generation of interpolants for some important fragments of first order logic, which are amenable for effective decision procedures, called Satisfiability Modulo Theory solvers.   We make the following contributions.   First, we provide interpolation procedures for several basic theories of interest: the theories of linear arithmetic over the rationals, difference logic over rationals and integers, and UTVPI over rationals and integers.   Second, we define a novel approach to interpolate combinations of theories, that applies to the Delayed Theory Combination approach.   Efficiency is ensured by the fact that the proposed interpolation algorithms extend state of the art algorithms for Satisfiability Modulo Theories. Our experimental evaluation shows that the MathSAT SMT solver can produce interpolants with minor overhead in search, and much more efficiently than other competitor solvers. 
Efficient Generation of Craig Interpolants in Satisfiability Modulo   Theories