Abstract
Reiter’s default logic formalizes nonmonotonic reasoning using default assumptions. The semantics of a given instance of default logic is based on a fixpoint equation defining an extension. Three different reasoning problems arise in the context of default logic, namely the existence of an extension, the presence of a given formula in an extension, and the occurrence of a formula in all extensions. Since the end of 1980s, several complexity results have been published concerning these default reasoning problems for different syntactic classes of formulas. We derive in this paper a complete classification of default logic reasoning problems by means of universal algebra tools using Post’s clone lattice. In particular we prove a trichotomy theorem for the existence of an extension, classifying this problem to be either polynomial, NP-complete, or Σ2P-complete, depending on the set of underlying Boolean connectives. We also prove similar trichotomy theorems for the two other algorithmic problems in connection with default logic reasoning.
Supported by ÉGIDE 05835SH, DAAD D/0205776 and DFG VO 630/5-1.
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References
Ben-Eliyahu-Zohary, R.: Yet some more complexity results for default logic. Artificial Intelligence 139(1), 1–20 (2002)
Böhler, E., et al.: Playing with Boolean blocks, parts I and II. SIGACT News, 34(4), 38–52 (2003) and 35(1), 22–35 (2004)
Böhler, E., et al.: Equivalence and isomorphism for Boolean constraint satisfaction. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 412–426. Springer, Heidelberg (2002)
Böhler, E., et al.: Bases for Boolean co-clones. Information Processing Letters 96(2), 59–66 (2005)
Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34(3), 720–742 (2005)
Cadoli, M., Eiter, T., Gottlob, G.: Default logic as a query language. IEEE Transactions on Knowledge and Data Engineering 9(3), 448–463 (1997)
Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. SIAM, Philadelphia (2001)
Creignou, N., Zanuttini, B.: A complete classification of the complexity of propositional abduction (Submitted October 2004)
Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman, New York (1979)
Geiger, D.: Closed systems of functions and predicates. Pacific Journal of Mathematics 27(1), 95–100 (1968)
Gottlob, G.: Complexity results for nonmonotonic logics. Journal of Logic and Computation 2(3), 397–425 (1992)
Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. Journal of the Association for Computing Machinery 44(4), 527–548 (1997)
Kautz, H.A., Selman, B.: Hard problems for simple default logics. Artificial Intelligence 49(1-3), 243–279 (1991)
Marek, W., Truszczyński, M.: Modal logic for default reasoning. Annals of Mathematics and Artificial Intelligence 1(1-4), 275–302 (1990)
Niemelä, I.: On the decidability and complexity of autoepistemic reasoning. Fundamenta Informaticae 17(1-2), 117–155 (1992)
Nordh, G.: A trichotomy in the complexity of propositional circumscription. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 257–269. Springer, Heidelberg (2005)
Nordh, G., Zanuttini, B.: Propositional abduction is almost always hard. In: Kaelbling, L.P., Saffiotti, A. (eds.) Proc. 19th IJCAI, Edinburgh, UK, pp. 534–539 (2005)
Pippenger, N.: Theories of Computability. Cambridge University Press, Cambridge (1997)
Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin (1979)
Post, E.L.: The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies 5, 1–122 (1941)
Reiter, R.: A logic for default reasoning. Artificial Intelligence 13(1-2), 81–132 (1980)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proc. 10th Symposium on Theory of Computing (STOC’78), San Diego, California, USA, pp. 216–226 (1978)
Stillman, J.: It’s not my default: the complexity of membership problems in restricted propositional default logics. In: Proc. 8th AAAI, Boston, MA, USA, pp. 571–578 (1990)
Stillman, J.: The complexity of propositional default logics. In: Proc. 10th AAAI, San Jose, California, USA, July 1992, pp. 794–799 (1992)
Wrathall, C.: Complete sets and the polynomial-time hierarchy. Theoretical Computer Science 3(1), 23–33 (1976)
Zhao, X., Ding, D.: Complexity results for 2CNF default theories. Fundamenta Informaticae 45(4), 393–404 (2001)
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Chapdelaine, P., Hermann, M., Schnoor, I. (2007). Complexity of Default Logic on Generalized Conjunctive Queries. In: Baral, C., Brewka, G., Schlipf, J. (eds) Logic Programming and Nonmonotonic Reasoning. LPNMR 2007. Lecture Notes in Computer Science(), vol 4483. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72200-7_7
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DOI: https://doi.org/10.1007/978-3-540-72200-7_7
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