Path Checking for MTL and TPTL over Data Words

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 formulas of MTL and TPTL.

An efficient algorithm for monitoring practical TPTL specifications

“…A similar complexity result (PSPACEhard) for Raskin's TPTL semantics is obtained for integer time variables in [11]. It is mentioned in [11] that in order to obtain a polynomial time algorithm for TPTL monitoring (path checking), we need to fix the number of time variables. In other words, if the number of time variables is bounded then the finite trace monitoring will be polynomial to the size of the TPTL formula.…”

“…Since TPTL subsumes MTL, it is expected that the monitoring problem of TPTL is computationally more complex [11]. It has been proven that monitoring of a finite trace with respect to Alur's TPTL specification is PSPACE-hard [18].…”

“…In other words, if the number of time variables is bounded then the finite trace monitoring will be polynomial to the size of the TPTL formula. However, in [11], the authors did not provide any applicable algorithm for TPTL monitoring and they focused only on the complexity class.…”

A survey of challenges for runtime verification from advanced application domains (beyond software)

“…Solutions to these research efforts exist (see for instance [66,85,106,149,152,220,221]). We refer to [164] for a recent survey on this topic.…”

“…Impact of utility on monitoring complexity. The existing work on the complexity of monitoring [85,152,220] (called path checking in this context) only considers the problem of providing a single Boolean verdict in an offline manner. Tight complexity bounds for the online monitoring problem or other variants of the problem with different output utility (e.g., a verdict stream) have not yet been established.…”