Model Checking Probabilistic Timed Automata with One or Two Clocks

TL;DR

This paper analyzes the computational complexity of model-checking probabilistic timed automata with one or two clocks, providing complexity classifications for various subclasses and logic fragments.

Contribution

It establishes the complexity bounds for PCTL model-checking problems on one- and two-clock probabilistic timed automata, including subclasses with restricted timing and probability constraints.

Findings

PCTL model-checking is PTIME-complete for one-clock automata.

Model-checking is EXPTIME-complete for automata with two clocks.

Restricted subclasses with certain timing and probability constraints are PTIME-complete.

Abstract

Probabilistic timed automata are an extension of timed automata with discrete probability distributions. We consider model-checking algorithms for the subclasses of probabilistic timed automata which have one or two clocks. Firstly, we show that PCTL probabilistic model-checking problems (such as determining whether a set of target states can be reached with probability at least 0.99 regardless of how nondeterminism is resolved) are PTIME-complete for one-clock probabilistic timed automata, and are EXPTIME-complete for probabilistic timed automata with two clocks. Secondly, we show that, for one-clock probabilistic timed automata, the model-checking problem for the probabilistic timed temporal logic PCTL is EXPTIME-complete. However, the model-checking problem for the subclass of PCTL which does not permit both punctual timing bounds, which require the occurrence of an event at an exact time point, and comparisons with probability bounds other than 0 or 1, is PTIME-complete for one-clock probabilistic timed automata.