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In this paper we consider two different views of the model checking problemsfor the Linear Temporal Logic (LTL). On the one hand, we consider the universalmodel checking problem for LTL, where one asks that for a given system and agiven formula all the runs of the system satisfy the formula. On the otherhand, the fair model checking problem for LTL asks that for a given system anda given formula almost all the runs of the system satisfy the formula. It wasshown that these two problems have the same theoretical complexity i.e.PSPACE-complete. The question arises whether one can find a fragment of LTL forwhich the complexity of these two problems differs. One such fragment wasidentified in a previous work, namely the Muller fragment. We address a similarcomparison for the prompt fragment of LTL (pLTL). pLTL extends LTL with anadditional operator, i.e. the prompt-eventually. This operator ensures theexistence of a bound such that liveness properties are satisfied within thisbound. We show that the corresponding Muller fragment for pLTL does not enjoythe same algorithmic properties with respect to the comparison considered. Wealso identify a new expressive fragment for which the fair model checking isfaster than the universal one.

Fairness and promptness in Muller formulas