Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds

TL;DR

This paper introduces a parameterized solution bound for a special class of quantifier-free Presburger formulas, enabling more efficient decision procedures by bounding solution sizes based on formula sparseness and non-difference constraints.

Contribution

The paper derives a new, parameterized solution bound for sparse, difference-based Presburger formulas, improving decision procedure efficiency in software verification contexts.

Findings

Solution size bound is linear in non-difference constraints

Decision procedure outperforms existing methods empirically

Bound depends on sparseness and non-zero coefficients

Abstract

Given a formula in quantifier-free Presburger arithmetic, if it has a satisfying solution, there is one whose size, measured in bits, is polynomially bounded in the size of the formula. In this paper, we consider a special class of quantifier-free Presburger formulas in which most linear constraints are difference (separation) constraints, and the non-difference constraints are sparse. This class has been observed to commonly occur in software verification. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of non-difference constraints, in addition to traditional measures of formula size. In particular, we show that the number of bits needed per integer variable is linear in the number of non-difference constraints and logarithmic in the number and size of non-zero coefficients in them, but is otherwise independent of the total number of linear constraints in the formula. The derived bound can be used in a decision procedure based on instantiating integer variables over a finite domain and translating the input quantifier-free Presburger formula to an equi-satisfiable Boolean formula, which is then checked using a Boolean satisfiability solver. In addition to our main theoretical result, we discuss several optimizations for deriving tighter bounds in practice. Empirical evidence indicates that our decision procedure can greatly outperform other decision procedures.