Path Checking for MTL and TPTL over Data Words

TL;DR

This paper analyzes the computational complexity of path checking for MTL and TPTL over infinite data words, revealing P-completeness for MTL and PSPACE-completeness for TPTL, thus clarifying their verification difficulty.

Contribution

It provides the first precise complexity classifications for path checking of MTL and TPTL over infinite data words, depending on register variables and encoding.

Findings

MTL path checking is P-complete.

TPTL path checking is PSPACE-complete.

Results clarify model checking complexity for one-counter machine behaviors.

Abstract

Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are quantitative extensions of linear temporal logic, which are prominent and widely used in the verification of real-timed systems. It was recently shown that the path checking problem for MTL, when evaluated over finite timed words, is in the parallel complexity class NC. In this paper, we derive precise complexity results for the path-checking problem for MTL and TPTL when evaluated over infinite data words over the non-negative integers. Such words may be seen as the behaviours of one-counter machines. For this setting, we give a complete analysis of the complexity of the path-checking problem depending on the number of register variables and the encoding of constraint numbers (unary or binary). As the two main results, we prove that the path-checking problem for MTL is P-complete, whereas the path-checking problem for TPTL is PSPACE-complete. The results yield the precise complexity of model checking deterministic one-counter machines against formulae of MTL and TPTL.