Interpolation-Based Semantic Gate Extraction and Its Applications to QBF Preprocessing

Abstract

We present a new semantic gate extraction technique for propositional formulas based on interpolation. While known gate detection methods are incomplete and rely on pattern matching or simple semantic conditions, this approach can detect any definition entailed by an input formula. As an application, we consider the problem of computing unique strategy functions from Quantified Boolean Formulas (QBFs) and Dependency Quantified Boolean Formulas (DQBFs). Experiments with a prototype implementation demonstrate that functions can be efficiently extracted from formulas in standard benchmark sets, and that many of these definitions remain undetected by syntactic gate detection. We turn this into a preprocessing technique by substituting unique strategy functions for input variables and test solver performance on the resulting instances. Compared to syntactic gate detection, we see a significant increase in the number of solved QBF instances, as well as a modest increase for DQBF instances.

Engineering an Efficient Boolean Functional Synthesis Engine

“…To grow Z further, we rely on the notion of definability, and iteratively identify the variables y i ∈ Y such that y i is definable in terms of rest of the variables such that its definition ψ i respects the dependency constraints imposed by the definitions of variables in Z. To extract the corresponding definitions, we rely on the Padoa's theorem (Lemma 1) to check whether y i is definable in terms of rest of the variables and then employ interpolation-based extraction of the corresponding definition [48].…”

“…We used Open-WBO [38] for unweighted MaxSAT queries, RC2 [26] for LexMaxSAT queries, and PicoSAT [10] to compute UnsatCore. Further, we used CryptoMiniSat [49] to find unates and a library based on UNIQUE [48] to extract unique Skolem functions. We used CMSGen [22] to sample the satisfying assignments of the specification.…”

“…These methods are fast but can only detect definitions from a pre-defined library of gates. By contrast, Manthan2 extracts the functions for uniquely defined variables using semantic gate extraction based on propositional interpolation [48]. This approach is computationally more expensive (each definability check requires a SAT call), but it is complete: whenever a variable y is defined in terms of a given set X of variables, the corresponding definition will be returned.…”

Certified DQBF Solving by Definition Extraction

“…Let ϕ and ψ be two formulae, ϕ entails ψ, denoted by ϕ ψ, if every assignment satisfying ϕ also satisfies ψ. A definition for a variable x by a set of variables X in a formula ϕ is a formula ψ with var (ψ) ⊆ X such that for every satisfying assignment σ of ϕ the equality σ(x) = ψ[σ] holds [40].…”

“…1 We believe these methods should be complemented with algorithms that directly reason at the level of Skolem functions [35]. A strong argument in favor of such an approach is the fact that DQBF instances often have a large fraction of unique Skolem functions that can be obtained by definition extraction, but the current solving paradigms have no direct way of exploiting this [40].…”