Problems concerning fairness and temporal logic for conflict-free petri nets

Abstract

In this paper, we examine the complexity of the fair nontermination problem for conflict-free Petri nets under several definitions of fairness. For each definition of fairness, we are able to show the problem to be complete for either NP, PTIME, or NLOGSPACE. We then address the question of whether these results extend to the more general model checking problem with respect to the temporal logic for Petri nets introduced by Suzuki. Since many of the model checking problems concerning finite state systems can be reduced to a version of the fair nontermination problem, it would seem plausible that the model checking problem for conflict-free Petri nets would be decidable. However, it turns out that unless the logic is severely restricted, model checking is undecidable for conflict-free Petri nets. In particular, the problem is undecidable even when formulas are of the form Gƒ (“invariantly ƒ”) where ƒ contains no temporal logic operators. On the other hand, we show that model checking for conflict-free Petri nets is NP-complete for L̃ ( F, X)—the logic restricted to the operators F (eventually), X (next time), ∧, and ∨, with negations allowed only on the predicates.