research-article

The Complexity of Reasoning for Fragments of Autoepistemic Logic

Authors:

Nadia Creignou,

Heribert Vollmer,

Michael ThomasAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 13, Issue 2

Article No.: 17, Pages 1 - 22

https://doi.org/10.1145/2159531.2159539

Published: 01 April 2012 Publication History

Abstract

Autoepistemic logic extends propositional logic by the modal operator L. A formula φ that is preceded by an L is said to be “believed.” The logic was introduced by Moore in 1985 for modeling an ideally rational agent’s behavior and reasoning about his own beliefs. In this article we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of checking that a given set of formulae characterizes a stable expansion and that of counting the number of stable expansions of a given knowledge base. We improve the best known Δ₂^p-upper bound on the former problem to completeness for the second level of the Boolean hierarchy. To the best of our knowledge, this is the first paper analyzing counting problem for autoepistemic logic.

References

[1]

Bauland, M., Hemaspaandra, E., Schnoor, H., and Schnoor, I. 2006. Generalized modal satisfiability. In Proceedings of the 23nd Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 3884, Springer, 500--511.

Digital Library

[2]

Bauland, M., Mundhenk, M., Schneider, T., Schnoor, H., Schnoor, I., and Vollmer, H. 2009a. The tractability of model checking for LTL: The good, the bad, and the ugly fragments. Electr. Notes Theor. Comput. Sci. 231, 277--292.

Digital Library

[3]

Bauland, M., Schneider, T., Schnoor, H., Schnoor, I., and Vollmer, H. 2009b. The complexity of generalized satisfiability for linear temporal logic. Logic. Meth. Comput. Sci. 5, 1, 1--21.

[4]

Bauland, M., Böhler, E., Creignou, N., Reith, S., Schnoor, H., and Vollmer, H. 2010. The complexity of problems for quantified constraints. Theor. Comput. Syst. 47, 2, 454--490.

Digital Library

[5]

Beyersdorff, O., Meier, A., Thomas, M., and Vollmer, H. 2009a. The complexity of propositional implication. Inf. Process. Lett. 109, 18, 1071--1077.

Digital Library

[6]

Beyersdorff, O., Meier, A., Thomas, M., and Vollmer, H. 2009b. The complexity of reasoning for fragments of default logic. In Proceedings of the 12th International Conference on Theory and Applications of Satisfiability. Lecture Notes in Computer Science, vol. 5584, Springer, 51--64.

Digital Library

[7]

Böhler, E., Creignou, N., Reith, S., and Vollmer, H. 2003. Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory. SIGACT News 34, 4, 38--52.

Digital Library

[8]

Buntrock, G., Damm, C., Hertrampf, U., and Meinel, C. 1992. Structure and importance of logspace MOD-classes. Math. Syst. Theory 25, 223--237.

[9]

Creignou, N., Schmidt, J., and Thomas, M. 2010a. Complexity of propositional abduction for restricted sets of Boolean functions. In Proceedings of the 12th International Conference on Principles of Knowledge Representation and Reasoning. AAAI Press, 8--16.

[10]

Creignou, N., Schmidt, J., Thomas, M., and Woltran, S. 2010b. Sets of Boolean connectives that make argumentation easier. In Proceedings of the 12th European Conference on Logics in Artificial Intelligence. Lecture Notes in Computer Science, vol. 6341, Springer, 117--129.

Digital Library

[11]

Durand, A. and Hermann, M. 2008. On the counting complexity of propositional circumscription. Inf. Process. Lett. 106, 164--170.

Digital Library

[12]

Durand, A., Hermann, M., and Kolaitis, P. 2005. Subtractive reductions and complete problems for counting complexity classes. Theor. Comput. Sci. 340, 3, 496--513.

Digital Library

[13]

Gottlob, G. 1992. Complexity results for nonmonotonic logics. J. Log. Comput. 2, 3, 397--425.

[14]

Gottlob, G. 1995. Translating default logic into standard autoepistemic logic. J. ACM 42, 4, 711--740.

Digital Library

[15]

Hemaspaandra, L. and Vollmer, H. 1995. The Satanic notations: Counting classes beyond #P and other definitional adventures. ACM-SIGACT News 26, 1, 2--13.

Digital Library

[16]

Hermann, M. and Pichler, R. 2007. Counting complexity of propositional abduction. In Proceedings of the 20th International Joint Conference on Artificial Intelligence. 417--422.

Digital Library

[17]

Hertrampf, U., Reith, S., and Vollmer, H. 2000. A note on closure properties of logspace MOD-classes. Inf. Process. Lett. 75, 3, 91--93.

Digital Library

[18]

Jakl, M., Pichler, R., Rümmele, S., and Woltran, S. 2008. Fast counting with bounded treewidth. In Proceedings of the 15th International Conference on Logic Programming, Artificial Intelligence and Reasoning. Lecture Notes in Computer Science, vol. 5330, Springer, 436--450.

Digital Library

[19]

Janhunen, T. 1999. On the intertranslatability of non-monotonic logics. Ann. Math. Artif. Intell. 27, 1--4, 79--128.

Digital Library

[20]

Lewis, H. 1979. Satisfiability problems for propositional calculi. Math. Syst. Theory 13, 45--53.

[21]

Lifschitz, V. 1985. Computing circumscription. In Proceedings of the 9th International Joint Conference on Artificial Intelligence. Morgan Kaufman, 121--127.

Digital Library

[22]

Lifschitz, V. and Schwarz, G. 1993. Extended logic programs as autoepistemic theories. In Proceedings of the 2nd International Conference on Logic Programming and Nonmonotonic Reasoning. 101--114.

Digital Library

[23]

Marek, V. W. and Truszczynski, M. 1991. Autoepistemic logic. J. ACM 38, 3, 588--619.

Digital Library

[24]

McCarthy, J. 1980. Circumscription--A form of non-monotonic reasoning. Artif. Intell. 13, 27--39.

Digital Library

[25]

McDermott, D. and Doyle, J. 1980. Non-monotonic logic I. Artif. Intell. 13, 41--72.

Digital Library

[26]

Meier, A., Mundhenk, M., Thomas, M., and Vollmer, H. 2008. The complexity of satisfiability for fragments of CTL and CTL*. Electr. Notes Theor. Comput. Sci. 223, 201--213.

Digital Library

[27]

Moore, R. C. 1985. Semantical considerations on modal logic. Artif. Intell. 25, 75--94.

Digital Library

[28]

Niemelä, I. 1991. Towards automatic autoepistemic reasoning. In Proceedings of the 2nd European Conference on Logics in Artificial Intelligence. Lecture Notes in Computer Science, vol. 478, 428--443.

Digital Library

[29]

Niemelä, I. 1993. Autoepistemic logic as a unified basis for nonmonotonic reasoning. Ph.D. thesis, Digital Systems Laboratory, Helsinki University of Technology, Espoo, Finland.

[30]

Papadimitriou, C. 1994. Computational Complexity. Addison-Wesley.

[31]

Papadimitriou, C. and Wolfe, D. 1988. The complexity of facets resolved. J. Comput. Syst. Sci. 37, 1, 2--13.

Digital Library

[32]

Pippenger, N. 1997. Theories of Computability. Cambridge University Press.

Digital Library

[33]

Post, E. 1941. The two-valued iterative systems of mathematical logic. Ann. Math. Stud. 5, 1--122.

[34]

Reiter, R. 1980. A logic for default reasoning. Artif. Intell. 13, 81--132.

Digital Library

[35]

Reith, S. 2003. On the complexity of some equivalence problems for propositional calculi. In Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 2747, Springer, 632--641.

[36]

Schnoor, H. 2010. The complexity of model checking for Boolean formulas. Int. J. Found. Comput. Sci. 21, 3, 289--309.

[37]

Thomas, M. 2009. The complexity of circumscriptive inference in Post’s lattice. In Proceedings of the 10th International Conference on Logic Programming and Nonmonotonic Reasoning. Lecture Notes in Computer Science, vol. 5753, Springer, 290--302.

Digital Library

[38]

Thomas, M. 2010. On the complexity of fragments of nonmonotonic logics. Ph.D. thesis, Leibniz Universität Hannover.

[39]

Valiant, L. 1979. The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 3, 411--421.

Cited By

Fichte JMeier ASchindler I(2024)Strong Backdoors for Default LogicACM Transactions on Computational Logic10.1145/365502425:3(1-24)Online publication date: 30-Mar-2024
https://dl.acm.org/doi/10.1145/3655024
Mahmood YMeier ASchmidt J(2020)Parameterized complexity of abduction in Schaefer’s frameworkJournal of Logic and Computation10.1093/logcom/exaa079Online publication date: 29-Dec-2020
https://doi.org/10.1093/logcom/exaa079
Mahmood YMeier ASchmidt J(2020)Parameterised Complexity of Abduction in Schaefer’s FrameworkLogical Foundations of Computer Science10.1007/978-3-030-36755-8_13(195-213)Online publication date: 4-Jan-2020
https://dl.acm.org/doi/10.1007/978-3-030-36755-8_13
Show More Cited By

Index Terms

The Complexity of Reasoning for Fragments of Autoepistemic Logic
1. Computing methodologies
  1. Artificial intelligence
    1. Knowledge representation and reasoning
      1. Nonmonotonic, default reasoning and belief revision
2. Theory of computation
  1. Logic

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cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 13, Issue 2

April 2012

267 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2159531

Issue’s Table of Contents

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 April 2012

Accepted: 01 April 2011

Revised: 01 December 2010

Received: 01 June 2010

Published in TOCL Volume 13, Issue 2

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Cited By

Fichte JMeier ASchindler I(2024)Strong Backdoors for Default LogicACM Transactions on Computational Logic10.1145/365502425:3(1-24)Online publication date: 30-Mar-2024
https://dl.acm.org/doi/10.1145/3655024
Mahmood YMeier ASchmidt J(2020)Parameterized complexity of abduction in Schaefer’s frameworkJournal of Logic and Computation10.1093/logcom/exaa079Online publication date: 29-Dec-2020
https://doi.org/10.1093/logcom/exaa079
Mahmood YMeier ASchmidt J(2020)Parameterised Complexity of Abduction in Schaefer’s FrameworkLogical Foundations of Computer Science10.1007/978-3-030-36755-8_13(195-213)Online publication date: 4-Jan-2020
https://dl.acm.org/doi/10.1007/978-3-030-36755-8_13
Lück MMeier ASchindler I(2017)Parametrised Complexity of Satisfiability in Temporal LogicACM Transactions on Computational Logic10.1145/300183518:1(1-32)Online publication date: 20-Jan-2017
https://dl.acm.org/doi/10.1145/3001835
Fichte JMeier ASchindler I(2016)Strong Backdoors for Default LogicTheory and Applications of Satisfiability Testing – SAT 201610.1007/978-3-319-40970-2_4(45-59)Online publication date: 11-Jun-2016
https://doi.org/10.1007/978-3-319-40970-2_4
Creignou NVollmer H(2015)Parameterized Complexity of Weighted Satisfiability ProblemsFundamenta Informaticae10.5555/2734202.2734203136:4(297-316)Online publication date: 1-Oct-2015
https://dl.acm.org/doi/10.5555/2734202.2734203
Beyersdorff O(2013)The complexity of theorem proving in autoepistemic logicProceedings of the 16th international conference on Theory and Applications of Satisfiability Testing10.1007/978-3-642-39071-5_27(365-376)Online publication date: 8-Jul-2013
https://dl.acm.org/doi/10.1007/978-3-642-39071-5_27
Creignou NVollmer H(2012)Parameterized complexity of weighted satisfiability problemsProceedings of the 15th international conference on Theory and Applications of Satisfiability Testing10.1007/978-3-642-31612-8_26(341-354)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1007/978-3-642-31612-8_26
Meier ASchmidt JThomas MVollmer H(2012)On the parameterized complexity of default logic and autoepistemic logicProceedings of the 6th international conference on Language and Automata Theory and Applications10.1007/978-3-642-28332-1_33(389-400)Online publication date: 5-Mar-2012
https://dl.acm.org/doi/10.1007/978-3-642-28332-1_33

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