Craig interpolants are widely used in program verification as a means of abstraction. In this paper, we (i) introduce Partial Variable Assignment Interpolants (PVAIs) as a generalization of Craig interpolants. A variable assignment focuses computed interpolants by restricting the set of clauses taken into account during interpolation. PVAIs can be for example employed in the context of DAG interpolation, in order to prevent unwanted out-of-scope variables to appear in interpolants. Furthermore, we (ii) present a way to compute PVAIs for propositional logic based on an extension of the Labeled Interpolation Systems, and (iii) analyze the strength of computed interpolants and prove the conditions under which they have the path interpolation property.

Model checking based on Craig's interpolants ultimately relies on efficient engines, such as SMT-Solvers, to log proofs of unsatisfiability and to derive the desired interpolant by means of a set of algorithms known in literature. These algorithms, however, are designed for proofs that do not contain mixed predicates. In this paper we present a technique for transforming the propositional proof produced by an SMT-Solver in such a way that mixed predicates are eliminated. We show a number of cases in which mixed predicates arise as a consequence of state-of-the-art solving procedures (e.g. lemma on demand, theory combination, etc.). In such cases our technique can be applied to allow the reuse of known interpolation algorithms. We demonstrate with a set of experiments that our approach is viable.