Model Checking Vector Addition Systems with one zero-test

TL;DR

This paper introduces a novel algorithm for analyzing vector addition systems with one zero-test, enabling decision procedures for problems like place-boundedness and LTL model-checking by computing finite representations of their cover sets.

Contribution

It develops a new forward algorithm using refined and filtered covers to handle zero-tests, extending the analysis capabilities of vector addition systems.

Findings

Finite representation of the cover set is computable for systems with one zero-test.

The refined cover is a recursive set, despite undecidability of equality.

The filtered cover effectively supports decision procedures for key problems.

Abstract

We design a variation of the Karp-Miller algorithm to compute, in a forward manner, a finite representation of the cover (i.e., the downward closure of the reachability set) of a vector addition system with one zero-test. This algorithm yields decision procedures for several problems for these systems, open until now, such as place-boundedness or LTL model-checking. The proof techniques to handle the zero-test are based on two new notions of cover: the refined and the filtered cover. The refined cover is a hybrid between the reachability set and the classical cover. It inherits properties of the reachability set: equality of two refined covers is undecidable, even for usual Vector Addition Systems (with no zero-test), but the refined cover of a Vector Addition System is a recursive set. The second notion of cover, called the filtered cover, is the central tool of our algorithms. It inherits properties of the classical cover, and in particular, one can effectively compute a finite representation of this set, even for Vector Addition Systems with one zero-test.