Context-Free Commutative Grammars with Integer Counters and Resets

TL;DR

This paper analyzes the computational complexity of reachability, coverability, and inclusion problems in extended context-free commutative grammars with integer counters and resets, revealing NP-completeness and high complexity bounds.

Contribution

It establishes the NP-completeness of reachability and coverability, and the coNEXP-completeness of inclusion, introducing a new a0_2^P-complete subset sum variant.

Findings

Reachability and coverability are NP-complete and inter-reducible.

Inclusion problem is coNEXP-complete, even with a single non-terminal.

The paper introduces a novel a0_2^P-complete subset sum variant.

Abstract

We study the computational complexity of reachability, coverability and inclusion for extensions of context-free commutative grammars with integer counters and reset operations on them. Those grammars can alternatively be viewed as an extension of communication-free Petri nets. Our main results are that reachability and coverability are inter-reducible and both NP-complete. In particular, this class of commutative grammars enjoys semi-linear reachability sets. We also show that the inclusion problem is, in general, coNEXP-complete and already $\Pi_2^\text{P}$-complete for grammars with only one non-terminal symbol. Showing the lower bound for the latter result requires us to develop a novel $\Pi_2^\text{P}$-complete variant of the classic subset sum problem.