Proof Tree Preserving Interpolation

Abstract

Abstract. Craig interpolation in SMT is difficult because, e. g., theory combination and integer cuts introduce mixed literals, i. e., literals containing local symbols from both input formulae. In this paper, we present a scheme to compute Craig interpolants in the presence of mixed literals. Contrary to existing approaches, this scheme neither limits the inferences done by the SMT solver, nor does it transform the proof tree before extracting interpolants. Our scheme works for the combination of uninterpreted functions and linear arithmetic but is extendable to other theories. The scheme is implemented in the interpolating SMT solver SMTInterpol.

A Survey of Satisfiability Modulo Theory

“…13 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 A SMT-solver, in contrast, produces a monolithic proof of unsatisfiability of (23): it mixes variables from different x i , that is, in program analysis, from different times of the execution of the program. It is however possible to obtain instead a sequence I i satisfying (24) by post-processing that proof [12,13,49].…”

“…It is easy to see that if one can find interpolants for arbitrary conjunctions A, B such that A ⇒ ¬B, one can find them between arbitrary quantifier-free formulas, by putting them into DNF. Because such a procedure would be needlessly costly due to disjunctive normal forms, the usual approach is to post-process a DPLL(T) proof that A( x, y)∧B( y, z) is unsatisfiable [12,13]. First, interpolants are derived for all theory lemmas: each lemma expresses that a conjunction of atoms from the original formula is unsatisfiable, these atoms can thus be divided into a conjunction α of atoms from A and a conjunction β of atoms from B, and an interpolant I is derived for α ⇒ ¬β.…”

“…A SMT-solver, in contrast, produces a monolithic proof of unsatisfiability of ( 23): it mixes variables from different x i , that is, in program analysis, from different times of the execution of the program. It is however possible to obtain instead a sequence I i satisfying (24) by post-processing that proof [12,13,49].…”

“…This poses a problem for interpolation: if interpolating for A(x, y) ∧ B(y, z) over linear real arithmetic, we can rely on all atomic propositions being linear inequalities either over x, y or y, z, but here, we have new atomic propositions that can involve both x, z. Special theory-dependent methods are needed to get rid of these new propositions when processing the DPLL(T) proof into an interpolant [12,13].…”

Analysis of Petri Nets and Transition Systems

“…http://ojalgo.org/). Later, SMTInterpol [11] was used. N 3 is created via ojAlgo; in other releases, a different solution of lts can and will be obtained.…”