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

Program Verification of Numerical Computation

These notes outline a formal method for program verification of numerical computation. It forms the basis of the software package VPC in its initial phase of development. Much of the style of presentation is in the form of notes that outline the definitions and rules upon which VPC is based. The initial motivation of this project was to address some practical issues of computation, especially of numerically intensive programs that are commonplace in computer models. The project evolved into a wider area for program construction as proofs leading to a model of inference in a more general sense. Some basic results of machine arithmetic are derived as a demonstration of VPC

The two-variable fragment with counting and equivalence

We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a sin-gle distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NEXPTIME-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable. Copyright line will be provided by the publisher

North-Holland SMALL SOLUTIONS OF LINEAR DIOPHANTINE EQUATIONS

Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi < ~ X for n- m variables and xi ~< (n- m + 1)Y for m variables. This improves previous results of the authors and others. 1