# Accession Number:

## ADA461078

# Title:

## Deciding Quantifier-Free Presburger Formulas using Finite Instantiation based on Parameterized Solution Bounds

# Descriptive Note:

## Technical rept.

# Corporate Author:

## CARNEGIE-MELLON UNIV PITTSBURGH PA SCHOOL OF COMPUTER SCIENCE

# Personal Author(s):

# Report Date:

## 2003-12-01

# Pagination or Media Count:

## 20.0

# Abstract:

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.

# Descriptors:

# Subject Categories:

- Operations Research
- Computer Programming and Software