Advertisement

Theory of Computing Systems

, Volume 50, Issue 3, pp 446–491 | Cite as

Trichotomies in the Complexity of Minimal Inference

  • Arnaud Durand
  • Miki Hermann
  • Gustav Nordh
Article

Abstract

We study the complexity of the propositional minimal inference problem. Although the complexity of this problem has been already extensively studied before because of its fundamental importance in nonmonotonic logics and commonsense reasoning, no complete classification of its complexity was found. We classify the complexity of four different and well-studied formalizations of the problem in the version with unbounded queries, proving that the complexity of the minimal inference problem for each of them has a trichotomy (between P, coNP-complete, and Π2P-complete). One of these results finally settles with a positive answer the trichotomy conjecture of Kirousis and Kolaitis (Theory Comput. Syst. 37(6):659–715, 2004). In the process we also strengthen and give a much simplified proof of the main result from Durand and Hermann (Proceedings 20th Symposium on Theoretical Aspects of Computer Science (STACS 2003), pp. 451–462, 2003).

Keywords

Boolean Function Minimal Model Conjunctive Normal Form Propositional Formula Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baker, K.A., Pixley, A.F.: Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems. Math. Z. 143(2), 165–174 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory. SIGACT News, Complexity Theory Column 42 34(4), 38–52 (2003) Google Scholar
  3. 3.
    Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with Boolean blocks, part II: Constraint satisfaction problems. SIGACT News, Complexity Theory Column 43 35(1), 22–35 (2004) Google Scholar
  4. 4.
    Böhler, E., Reith, S., Schnoor, H., Vollmer, H.: Bases for Boolean co-clones. Inf. Process. Lett. 96(2), 59–66 (2005) zbMATHCrossRefGoogle Scholar
  5. 5.
    Cadoli, M.: The complexity of model checking for circumscriptive formulae. Inf. Process. Lett. 44(3), 113–118 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cadoli, M., Lenzerini, M.: The complexity of propositional closed world reasoning and circumscription. J. Comput. Syst. Sci. 48(2), 255–310 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chen, Z.-Z., Toda, S.: The complexity of selecting maximal solutions. Inf. Comput. 119(2), 231–239 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001) zbMATHCrossRefGoogle Scholar
  9. 9.
    Dalmau, V.: Computational complexity of problems over generalized formulas. PhD thesis, Department de Llenguatges i Sistemes Informàtica, Universitat Politécnica de Catalunya, 2000 Google Scholar
  10. 10.
    de Kleer, J., Mackworth, A.K., Reiter, R.: Characterizing diagnoses and systems. Artif. Intell. 56(2–3), 197–222 (1992) zbMATHCrossRefGoogle Scholar
  11. 11.
    Durand, A., Hermann, M.: The inference problem for propositional circumscription of affine formulas is coNP-complete. In: Alt, H., Habib, M. (eds.) Proceedings 20th Symposium on Theoretical Aspects of Computer Science (STACS 2003). Lecture Notes in Computer Science, vol. 2607, pp. 451–462. Springer, Berlin (2003) Google Scholar
  12. 12.
    Durand, A., Hermann, M., Nordh, G.: Trichotomy in the complexity of minimal inference. In: Proceedings 24th Annual IEEE Symposium on Logic in Computer Science (LICS 2009), Los Angeles (CA, USA), pp. 387–396 (2009) Google Scholar
  13. 13.
    Eiter, T., Gottlob, G.: Propositional circumscription and extended closed-world reasoning are \(\mathrm{\Pi}_{2}^{\mathrm{p}}\)-complete. Theor. Comput. Sci. 114(2), 231–245 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Geiger, D.: Closed systems of functions and predicates. Pac. J. Math. 27(1), 95–100 (1968) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R.A., Bowen, K.A. (eds.) Proceeding 5th International Conference and Symposium on Logic Programming ICLP/SLP, Seatle (WA), pp. 1070–1080. MIT Press, Cambridge (1988) Google Scholar
  16. 16.
    Gelfond, M., Przymusinska, H., Przymusinski, T.C.: On the relationship between circumscription and negation as failure. Artif. Intell. 38(1), 75–94 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. Assoc. Comput. Mach. 44(4), 527–548 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Khachiyan, L., Boros, E., Elbassioni, K., Gurvich, V., Makino, K.: On the complexity of some enumeration problems for matroids. SIAM J. Discrete Math. 19(4), 966–984 (2005) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kirousis, L.M., Kolaitis, P.G.: On the complexity of model checking and inference in minimal models. In: Eiter, T., Faber, W., Truszczyński, M. (eds.) Proceedings 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNRM 2001), Vienna (Austria). Lecture Notes in Computer Science, vol. 2173, pp. 42–53. Springer, Berlin (2001) Google Scholar
  20. 20.
    Kirousis, L.M., Kolaitis, P.G.: The complexity of minimal satisfiability problems. Inf. Comput. 187(1), 20–39 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kirousis, L.M., Kolaitis, P.G.: A dichotomy in the complexity of propositional circumscription. Theory Comput. Syst. 37(6), 695–715 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    McCarthy, J.: Circumscription—a form of non-monotonic reasoning. Artif. Intell. 13(1–2), 27–39 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    McCarthy, J.: Applications of circumscription to formalizing common-sense knowledge. Artif. Intell. 28(1), 89–116 (1986) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nordh, G.: A trichotomy in the complexity of propositional circumscription. In: Baader, F., Voronkov, A. (eds.) Proceedings 11th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2004), Montevideo (Uruguay). Lecture Notes in Computer Science, vol. 3452, pp. 257–269. Springer, Berlin (2005) Google Scholar
  25. 25.
    Nordh, G., Jonsson, P.: An algebraic approach to the complexity of propositional circumscription. In: Proceedings 19th IEEE Symposium on Logic in Computer Science (LICS 2004), Turku (Finland), pp. 367–376 (2004) CrossRefGoogle Scholar
  26. 26.
    Nordh, G., Zanuttini, B.: Frozen Boolean partial co-clones. In: Proceedings 39th International Symposium on Multiple-Valued Logic (ISMVL 2009), Naha (Okinawa, Japan), pp. 120–125. IEEE Computer Society, Los Alamitos (2009) Google Scholar
  27. 27.
    International Workshop on Mathematics of Constraint Satisfaction. Open problems list. http://www.cs.rhul.ac.uk/home/green/mathscsp/slides/problems/OpenProblemsList.pdf, March 2006
  28. 28.
    Papadimitriou, C.H.: On selecting a satisfying truth assignment (extended abstract). In: Proceedings 32nd IEEE Symposium on Foundations of Computer Science (FOCS’91), San Juan (Puerto Rico), pp. 163–169. IEEE Computer Society, Los Alamitos (1991) Google Scholar
  29. 29.
    Pippenger, N.: Theories of Computability. Cambridge University Press, Cambridge (1997) zbMATHGoogle Scholar
  30. 30.
    Pöschel, R.: Galois connection for operations and relations. In: Denecke, K., Erné, M., Wismath, S.L. (eds.) Galois Connections and Applications, pp. 231–258. Kluwer Academic, Norwell (2004) Google Scholar
  31. 31.
    Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin (1979) Google Scholar
  32. 32.
    Post, E.L.: The two-valued iterative systems of mathematical logic. Ann. Math. Stud. 5, 1–122 (1941) MathSciNetGoogle Scholar
  33. 33.
    Robertson, N., Seymour, P.D.: Graph minors XIII: the disjoint paths problem. J. Comb. Theory, Ser. B 63(1), 65–110 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Romov, B.A.: The algebras of partial functions and their invariants. Cybern. Syst. Anal. 17(2), 157–167 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings 10th Symposium on Theory of Computing (STOC’78), San Diego (CA, USA), pp. 216–226 (1978) CrossRefGoogle Scholar
  36. 36.
    Schnoor, H., Schnoor, I.: Partial polymorphisms and constraint satisfaction problems. In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.) Complexity of Constraints. Lecture Notes in Computer Science, vol. 5250, pp. 229–254. Springer, Berlin (2008) CrossRefGoogle Scholar
  37. 37.
    Thomas, M.: The complexity of circumscriptive inference in Post’s lattice. In: Erdem, E., Lin, F., Schaub, T. (eds.) Proceedings 10th International Conference om Logic Programming and Nonmonotonic Reasoning (LPNMR 2009), Potsdam (Germany). Lecture Notes in Computer Science, vol. 5753, pp. 290–302. Springer, Berlin (2009) Google Scholar
  38. 38.
    Thomas, M., Vollmer, H.: Complexity of non-monotonic logics. Bull. Eur. Assoc. Theor. Comput. Sci. 102, 53–82 (2010) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Equipe de Logique MathématiqueInstitut de Mathématiques de Jussieu (CNRS UMR 7586)Paris cedex 05France
  2. 2.LIX (CNRS UMR 7161)École PolytechniquePalaiseau cedexFrance
  3. 3.Department of Computer and Information SciencesLinköpings UniversitetLinköpingSweden

Personalised recommendations