Deciding Quantifier-Free Presburger Formulas using Finite Instantiation based on Parameterized Solution Bounds
CARNEGIE-MELLON UNIV PITTSBURGH PA SCHOOL OF COMPUTER SCIENCE
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Given a formula Phi in quantifier-free Presburger arithmetic, it is well known that, if there is a satisfying solution to Phi, there is one whose size, measured in bits, is polynomially bounded in the size of Phi. In this paper, we consider a special class of quantifier-free Presburger formulas in which most linear constraints are separation difference-bound constraints, and the non-separation constraints are sparse. This class has been observed to commonly occur in software verification problems. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of non-separation constraints, in addition to traditional measures of formula size. In particular, the number of bits needed per integer variable is linear in the number of non-separation 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. We present empirical evidence indicating that this method can greatly outperform other decision procedures.
- Operations Research
- Computer Programming and Software