Continuous One-Counter Automata

TL;DR

This paper investigates the reachability problem in continuous one-counter automata, providing complexity results for various configurations, including polynomial-time decidability and placement within the NC2 class.

Contribution

It introduces the model of continuous one-counter automata and establishes complexity bounds for the reachability problem under different conditions.

Findings

Reachability with global bounds is in NC2.

General reachability is decidable in polynomial time.

Parametric updates and tests are decidable in the polynomial hierarchy.

Abstract

We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper and lower bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: (1) We prove that the reachability problem for COCA with global upper and lower bound tests is in NC2; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is decidable in the polynomial hierarchy for COCA with parametric counter updates and bound tests.