Abstract. In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post’s lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P. 2000 ACM Subject Classification:

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Abstract—The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations—restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004). For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that in most applications, approaching CTL model checking by parallelism will not result in the desired speed up. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL +, and ECTL +. I.

The satisfiability problems for CTL and CTL ⋆ are known to be EXPTIME-complete, resp. 2EXPTIMEcomplete (Fischer and Ladner (1979), Vardi and Stockmeyer (1985)). For fragments that use less temporal or propositional operators, the complexity may decrease. This paper undertakes a systematic study of satisfiability for CTL- and CTL ⋆-formulae over restricted sets of propositional and temporal operators. We show that restricting the temporal operators yields satisfiability problems complete for 2EXPTIME, EXPTIME, PSPACE, and NP. Restricting the propositional operators either does not change the complexity (as determined by the temporal operators), or yields very low complexity like NC 1, TC 0, or NLOGTIME. Keywords: Temporal Logic, Satisfiability, Post’s Lattice.

Vol. 5 (1:1) 2009, pp. 1–1–21 www.lmcs-online.org

Linear Temporal Logic (LTL) was introduced by Pnueli in [Pnu77] as a formalism for reasoning about the properties and the behaviors of parallel programs and concurrent systems, and has widely been used for these purposes. Because of the need to perform reasoning tasks—such as deciding satisfiability, validity, or truth in a structure generated by binary