On Boundedness Problems for Pushdown Vector Addition Systems

TL;DR

This paper investigates the boundedness of reachability sets in pushdown vector addition systems, providing decidability results for one-dimensional cases using derivation tree analysis.

Contribution

It introduces a decidability result for counter boundedness in one-dimensional pushdown vector addition systems, employing a novel small witness property.

Findings

Decidability of counter boundedness in exponential time for 1D systems

Development of a small witness property for derivation trees

Analysis of reachability set boundedness in pushdown vector addition systems

Abstract

We study pushdown vector addition systems, which are synchronized products of pushdown automata with vector addition systems. The question of the boundedness of the reachability set for this model can be refined into two decision problems that ask if infinitely many counter values or stack configurations are reachable, respectively. Counter boundedness seems to be the more intricate problem. We show decidability in exponential time for one-dimensional systems. The proof is via a small witness property derived from an analysis of derivation trees of grammar-controlled vector addition systems.