Abstract
We apply the concept of formula treewidth and pathwidth to computation tree logic, linear temporal logic, and the full branching time logic. Several representations of formulas as graphlike structures are discussed, and corresponding notions of treewidth and pathwidth are introduced. As an application for such structures, we present a classification in terms of parametrised complexity of the satisfiability problem, where we make use of Courcelle’s famous theorem for recognition of certain classes of structures. Our classification shows a dichotomy between W[1]-hard and fixed-parameter tractable operator fragments almost independently of the chosen graph representation. The only fragments that are proven to be fixed-parameter tractable (FPT) are those that are restricted to the X operator. By investigating Boolean operator fragments in the sense of Post’s lattice, we achieve the same complexity as in the unrestricted case if the set of available Boolean functions can express the function “negation of the implication.” Conversely, we show containment in FPT for almost all other clones.
- E. Allen Emerson. 1990. Temporal and modal logic. In Handbook of Theoretical Computer Science (Vol. B), Jan van Leeuwen (Ed.). MIT Press, Cambridge, MA, 995--1072. Google ScholarDigital Library
- E. Allen Emerson and E. M. Clarke. 1981. Design and synthesis of synchronisation skeletons using branching time temporal logic. In Logic of Programs (Lecture Notes in Computer Science), Vol. 131. Springer Verlag, Berlin, 52--71. Google ScholarDigital Library
- E. Allen Emerson and J. Y. Halpern. 1985. Decision procedures and expressiveness in the temporal logic of branching time. J. Comput. System Sci. 30, 1 (1985), 1--24. Google ScholarDigital Library
- M. Bauland, E. Böhler, N. Creignou, S. Reith, H. Schnoor, and H. Vollmer. 2010. The complexity of problems for quantified constraints. Theor. Comput. Syst. 47 (2010), 454--490. Google ScholarDigital Library
- O. Beyersdorff, A. Meier, M. Mundhenk, T. Schneider, M. Thomas, and H. Vollmer. 2011. Model checking CTL is almost always inherently sequential. Logic. Methods Comput. Sci. 7, 2 (2011). Google ScholarCross Ref
- O. Beyersdorff, A. Meier, M. Thomas, and H. Vollmer. 2010. The complexity of reasoning for fragments of default logic. J. Logic Comput. (2010). Google ScholarDigital Library
- P. Blackburn, M. de Rijke, and Y. Venema. 2001. Modal Logic. Cambridge University Press, New York, NY. Google ScholarDigital Library
- E. Böhler, N. Creignou, M. Galota, S. Reith, H. Schnoor, and H. Vollmer. 2012. Complexity classifications for different equivalence and audit problems for Boolean circuits. Logic. Methods Comput. Sci. 8, 3:27 (2012), 1--25.Google Scholar
- C. Chekuri and A. Rajaraman. 1997. Conjunctive query containment revisited. In Database Theory—ICDT’97. Vol. 1186. Springer, Berlin, 56--70. Google ScholarDigital Library
- B. Courcelle and J. Engelfriet. 2012. Graph Structure and Monadic Second-order Logic, a Language Theoretic Approach. Cambridge University Press. Google ScholarDigital Library
- N. Creignou, A. Meier, H. Vollmer, and M. Thomas. 2012. The complexity of reasoning for fragments of autoepistemic logic. ACM Trans. Comput. Logic 13, 2 (April 2012), 1--22. Google ScholarDigital Library
- S. Demri and P. Schnoebelen. 2002. The complexity of propositional linear temporal logics in simple cases. Inform. Comput. 174, 1 (April 2002), 84--103. Google ScholarDigital Library
- R. G. Downey and M. R. Fellows. 1999. Parameterized Complexity. Springer-Verlag, Berlin. Google ScholarDigital Library
- M. Elberfeld, A. Jakoby, and T. Tantau. 2010. Logspace versions of the theorems of Bodlaender and Courcelle. In Proc. 51th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society. Google ScholarDigital Library
- M. J. Fischer and R. E. Ladner. 1979. Propositional dynamic logic of regular programs. J. Comput. Syst. Sci. 18, 2 (1979), 194--211. Google ScholarCross Ref
- J. Flum and M. Grohe. 2006. Parameterized Complexity Theory. Springer Verlag. Google ScholarDigital Library
- Haim Gaifman. 1982. On local and non-local properties. In Herbrand Symposium, Logic Colloquium’81. North-Holland, 105--135.Google ScholarCross Ref
- G. Gottlob, R. Pichler, and F. Wei. 2010. Bounded treewidth as a key to tractability of knowledge representation and reasoning. Artif. Intell. 174, 1 (2010), 105--132. Google ScholarDigital Library
- J. Y. Halpern. 1995. The effect of bounding the number of primitive propositions and the depth of nesting on the complexity of modal logic. Artif. Intell. 75 (1995), 361--372. Google ScholarDigital Library
- E. Hemaspaandra, H. Schnorr, and I. Schnoor. 2010. Generalized modal satisfiability. J. Comput. Syst. Sci. 76 (2010), 561--578. Google ScholarDigital Library
- P. G. Kolaitis and M. Y. Vardi. 2000. Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci. 61, 2 (2000), 302--332. Google ScholarDigital Library
- S. Kripke. 1963. Semantical considerations on modal logic. In Acta Philosophica Fennica, Vol. 16. 84--94.Google Scholar
- H. Lewis. 1979. Satisfiability problems for propositional calculi. Math. Syst. Theor. 13 (1979), 45--53. Google ScholarCross Ref
- M. Lück. 2015. Quirky quantifiers: Optimal models and complexity of computation tree logic. CoRR abs/1510.08786 (2015).Google Scholar
- M. Lück, A. Meier, and I. Schindler. 2015. Parameterized complexity of CTL. In Language and Automata Theory and Applications. Lecture Notes in Computer Science, Vol. 8977. Springer, 549--560. Google ScholarCross Ref
- A. Meier, M. Mundhenk, T. Schneider, M. Thomas, V. Weber, and F. Weiss. 2010. The complexity of satisfiability for fragments of hybrid logic -- Part I. J. Appl. Logic 8, 4 (2010), 409--421. Google ScholarCross Ref
- A. Meier, M. Mundhenk, M. Thomas, and H. Vollmer. 2009. The complexity of satisfiability for fragments of CTL and CTL*. Int. J. Found. Comput. Sci. 20, 05 (2009), 901--918. Erratum See Meier et al. {2015}.Google ScholarCross Ref
- A. Meier, M. Mundhenk, M. Thomas, and H. Vollmer. 2015. Erratum: The complexity of satisfiability for fragments of CTL and CTL*. Int. J. Found. Comput. Sci. 26, 08 (2015), 1189--1190. Google ScholarCross Ref
- A. Meier, J. Schmidt, M. Thomas, and H. Vollmer. 2012. On the parameterized complexity of default logic and autoepistemic logic. In Proc. 6th International Conference on Language and Automata Theory and Applications (LATA)(LNCS), Vol. 7183. 389--400. Google ScholarDigital Library
- A. Meier and T. Schneider. 2013. Generalized satisfiability for the description logic ALC. Theor. Comput. Sci. 505, 0 (2013), 55--73. Theory and Applications of Models of Computation 2011. Google ScholarDigital Library
- N. Pippenger. 1997. Theories of Computability. Cambridge University Press. Google ScholarDigital Library
- A. Pnueli. 1977. The temporal logic of programs. In Proc. 18th Symposium on Foundations of Computer Science. IEEE Computer Society Press, 46--57. Google ScholarDigital Library
- E. Post. 1941. The two-valued iterative systems of mathematical logic. Ann. Math. Stud. 5 (1941), 1--122.Google Scholar
- M. Praveen. 2013. Does treewidth help in modal satisfiability? ACM Trans. Comput. Logic 14, 3 (2013), 18:1--18:32. Google ScholarDigital Library
- A. N. Prior. 1957. Time and Modality. Clarendon Press, Oxford.Google Scholar
- M. Samer and S. Szeider. 2006. A fixed-parameter algorithm for #SAT with parameter incidence treewidth. CoRR abs/cs/0610174 (2006).Google Scholar
- M. Samer and S. Szeider. 2010. Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76, 2 (2010), 103--114. Google ScholarDigital Library
Index Terms
- Parametrised Complexity of Satisfiability in Temporal Logic
Recommendations
The Complexity of Satisfiability for Fragments of CTL and CTL*;
The satisfiability problems for CTL and CTL^@? are known to be EXPTIME-complete, resp. 2EXPTIME-complete (Fischer and Ladner (1979), Vardi and Stockmeyer (1985)). For fragments that use less temporal or propositional operators, the complexity may ...
The tractability of model checking for LTL: The good, the bad, and the ugly fragments
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 ...
Lower Bounds for QBFs of Bounded Treewidth
LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceThe problem of deciding the validity (QSat) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful ...
Comments