An Improved Tight Closure Algorithm for Integer Octagonal Constraints

TL;DR

This paper introduces a new efficient O(n^3) algorithm for computing the tight closure of integer octagonal constraints, enhancing the tools for constraint solving in software and hardware verification.

Contribution

It presents a fully justified, improved algorithm for tight closure computation in UTVPI integer constraints, with polynomial complexity.

Findings

Algorithm operates in O(n^3) time

Ensures complete tight closure for integer octagonal constraints

Enhances constraint solving efficiency in verification tasks

Abstract

Integer octagonal constraints (a.k.a. ``Unit Two Variables Per Inequality'' or ``UTVPI integer constraints'') constitute an interesting class of constraints for the representation and solution of integer problems in the fields of constraint programming and formal analysis and verification of software and hardware systems, since they couple algorithms having polynomial complexity with a relatively good expressive power. The main algorithms required for the manipulation of such constraints are the satisfiability check and the computation of the inferential closure of a set of constraints. The latter is called `tight' closure to mark the difference with the (incomplete) closure algorithm that does not exploit the integrality of the variables. In this paper we present and fully justify an O(n^3) algorithm to compute the tight closure of a set of UTVPI integer constraints.