Skolem Functions for Factored Formulas

TL;DR

This paper introduces a new CEGAR-based algorithm for generating Skolem functions from factored propositional formulas, improving scalability and performance over existing methods by exploiting the formula's structure.

Contribution

The paper presents a novel CEGAR-style algorithm that leverages the factored form of formulas for efficient Skolem function generation, avoiding the need for QBF satisfiability proofs.

Findings

Generates smaller Skolem functions.

Outperforms state-of-the-art approaches on large benchmarks.

Exploits factored structure for scalability.

Abstract

Given a propositional formula F(x,y), a Skolem function for x is a function \Psi(y), such that substituting \Psi(y) for x in F gives a formula semantically equivalent to \exists F. Automatically generating Skolem functions is of significant interest in several applications including certified QBF solving, finding strategies of players in games, synthesising circuits and bit-vector programs from specifications, disjunctive decomposition of sequential circuits etc. In many such applications, F is given as a conjunction of factors, each of which depends on a small subset of variables. Existing algorithms for Skolem function generation ignore any such factored form and treat F as a monolithic function. This presents scalability hurdles in medium to large problem instances. In this paper, we argue that exploiting the factored form of F can give significant performance improvements in practice when computing Skolem functions. We present a new CEGAR style algorithm for generating Skolem functions from factored propositional formulas. In contrast to earlier work, our algorithm neither requires a proof of QBF satisfiability nor uses composition of monolithic conjunctions of factors. We show experimentally that our algorithm generates smaller Skolem functions and outperforms state-of-the-art approaches on several large benchmarks.