Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds

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.Comment: 26 page

The First-Order Theory of Sets with Cardinality Constraints is Decidable

We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is undecidable. Our language allows relating the cardinalities of sets to the values of integer variables, and can distinguish finite and infinite sets. We use quantifier elimination to show the decidability and obtain an elementary upper bound on the complexity. Precise program analyses can use our decidability result to verify representation invariants of data structures that use an integer field to represent the number of stored elements.Comment: 18 page

Quantifier-Free Boolean Algebra with Presburger Arithmetic is NP-Complete

Boolean Algebra with Presburger Arithmetic (BAPA) combines1) Boolean algebras of sets of uninterpreted elements (BA)and 2) Presburger arithmetic operations (PA). BAPA canexpress the relationship between integer variables andcardinalities of unbounded finite sets and can be used toexpress verification conditions in verification of datastructure consistency properties.In this report I consider the Quantifier-Free fragment ofBoolean Algebra with Presburger Arithmetic (QFBAPA).Previous algorithms for QFBAPA had non-deterministicexponential time complexity. In this report I show thatQFBAPA is in NP, and is therefore NP-complete. My resultyields an algorithm for checking satisfiability of QFBAPAformulas by converting them to polynomially sized formulasof quantifier-free Presburger arithmetic. I expect thisalgorithm to substantially extend the range of QFBAPAproblems whose satisfiability can be checked in practice

On decidability within the arithmetic of addition and divisibility

Abstract. The arithmetic of natural numbers with addition and divisibility has been shown undecidable as a consequence of the fact that multiplication of natural numbers can be interpreted into this theory, as shown by J. Robinson [Rob49]. The most important decidable subsets of the arithmetic of addition and divisibility are the arithmetic of addition, proved by M. Presburger (1) | in the paper, we show the existence of a quantifier elimination procedure which always leads to formulas of Presburger arithmetic. We generalize the L | is sketched in the end of the paper