A New Numerical Abstract Domain Based on Difference-Bound Matrices

Abstract

This paper presents a new numerical abstract domain for static analysis by abstract interpretation. This domain allows us to represent invariants of the form (x-y<=c) and (+/-x<=c), where x and y are variables values and c is an integer or real constant. Abstract elements are represented by Difference-Bound Matrices, widely used by model-checkers, but we had to design new operators to meet the needs of abstract interpretation. The result is a complete lattice of infinite height featuring widening, narrowing and common transfer functions. We focus on giving an efficient O(n2) representation and graph-based O(n3) algorithms - where n is the number of variables|and claim that this domain always performs more precisely than the well-known interval domain. To illustrate the precision/cost tradeoff of this domain, we have implemented simple abstract interpreters for toy imperative and parallel languages which allowed us to prove some non-trivial algorithms correct.