Tractability of model-checking for LTL: the good, the bad, and the ugly fragments

by    M. Bauland, M. Mundhenk, Th. Schneider, H. Schnoor, I. Schnoor, H. Vollmer

Preprint series: 07-04, Reports on Computer Science

03B44 Temporal logic

Abstract: In a seminal paper from 1985, Sistla and Clarke showed
that the model-checking problem 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 systematically studies the model-checking problem for LTL formulae over restricted sets
of propositional and temporal operators. For almost all combinations of temporal and propositional
operators, we determine whether the model-checking problem is tractable (in polynomial time) or intractable (NP-hard).
We then focus on the tractable cases, showing that they all
are NL-complete or even logspace solvable.
This leads to a surprising gap in complexity between tractable and intractable cases.
It is worth noting that our analysis covers an infinite set of problems, since
there are infinitely many sets of propositional operators.

Keywords: temporal logic, LTL, fragments, complexity of model-checking

Upload: 2007-11-05

Update: 2007 -11 -06

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