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Parametrised Complexity of Satisfiability in Temporal Logic

Parametrised Complexity of Satisfiability in Temporal Logic Parametrised Complexity of Satisfiability in Temporal Logic ¨ ¨ MARTIN LUCK, ARNE MEIER, and IRENA SCHINDLER, Leibniz Universitat Hannover 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. CCS Concepts: · Theory of computation Complexity theory and logic; Complexity classes; Additional Key Words and Phrases: Parametrised complexity, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Parametrised Complexity of Satisfiability in Temporal Logic

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References (38)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2017 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/3001835
Publisher site
See Article on Publisher Site

Abstract

Parametrised Complexity of Satisfiability in Temporal Logic ¨ ¨ MARTIN LUCK, ARNE MEIER, and IRENA SCHINDLER, Leibniz Universitat Hannover 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. CCS Concepts: · Theory of computation Complexity theory and logic; Complexity classes; Additional Key Words and Phrases: Parametrised complexity,

Journal

ACM Transactions on Computational Logic (TOCL)Association for Computing Machinery

Published: Jan 20, 2017

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