Monodic Fragments of Probabilistic First-Order Logic

  • Jean Christoph Jung
  • Carsten Lutz
  • Sergey Goncharov
  • Lutz Schröder
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8573)


By classical results of Abadi and Halpern, validity for probabilistic first-order logic of type 2 (ProbFO) is \(\Pi^2_1\)-complete and thus not recursively enumerable, and even small fragments of ProbFO are undecidable. In temporal first-order logic, which has similar computational properties, these problems have been addressed by imposing monodicity, that is, by allowing temporal operators to be applied only to formulas with at most one free variable. In this paper, we identify a monodic fragment of ProbFO and show that it enjoys favorable computational properties. Specifically, the valid sentences of monodic ProbFO are recursively enumerable and a slight variation of Halpern’s axiom system for type-2 ProbFO on bounded domains is sound and complete for monodic ProbFO. Moreover, decidability can be obtained by restricting the FO part of monodic ProbFO to any decidable FO fragment. In some cases, which notably include the guarded fragment, our general constructions result in tight complexity bounds.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abadi, M., Halpern, J.: Decidability and expressiveness for first-order logics of probability. Inf. Comput. 112, 1–36 (1994)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Andréka, H., van Benthem, J., Németi, I.: Back and forth between modal logic and classical logic. Logic J. IGPL 3, 685–720 (1995)CrossRefMATHGoogle Scholar
  3. 3.
    Artin, E., Schreier, O.: Algebraische Konstruktion reeller Körper. Abh. Math. Sem. Univ. Hamburg 5, 85–99 (1927)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bacchus, F.: Representing and reasoning with probabilistic knowledge - a logical approach to probabilities. MIT Press (1990)Google Scholar
  5. 5.
    Bacchus, F., Grove, A., Koller, D., Halpern, J.: From statistics to beliefs. In: Artificial Intelligence, AAAI 1992, pp. 602–608. AAAI Press/The MIT Press (1992)Google Scholar
  6. 6.
    Bárány, V., ten Cate, B., Segoufin, L.: Guarded negation. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 356–367. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Enderton, H.B.: A mathematical introduction to logic. Academic Press (1972)Google Scholar
  8. 8.
    Gabbay, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional modal logics: theory and applications. Studies in Logic, vol. 148. Elsevier (2003)Google Scholar
  9. 9.
    Getoor, L., Taskar, B.: Introduction to Statistical Relational Learning. MIT Press (2007)Google Scholar
  10. 10.
    Grädel, E., Kolaitis, P., Vardi, M.: On the decision problem for two-variable first-order logic. Bull. Symb. Log. 3, 53–69 (1997)CrossRefMATHGoogle Scholar
  11. 11.
    Gutiérrez-Basulto, V., Jung, J., Lutz, C., Schröder, L.: A closer look at the probabilistic description logic prob-\(\mathcal{EL}\). In: Artificial Intelligence, AAAI 2011. AAAI Press (2011)Google Scholar
  12. 12.
    Halpern, J.: An analysis of first-order logics of probability. Artif. Intell. 46, 311–350 (1990)CrossRefMATHGoogle Scholar
  13. 13.
    Harel, D.: Recurring dominoes: making the highly undecidable highly understandable. Ann. Discrete Math. 24, 51–72 (1985)MATHMathSciNetGoogle Scholar
  14. 14.
    Hodkinson, I.: Monodic packed fragment with equality is decidable. Stud. Log. 72, 185–197 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Hodkinson, I.: Complexity of monodic guarded fragments over linear and real time. Ann. Pure Appl. Logic 138, 94–125 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hodkinson, I., Kontchakov, R., Kurucz, A., Wolter, F., Zakharyaschev, M.: On the computational complexity of decidable fragments of first-order linear temporal logics. In: Proc. TIME-ICTL 2003, pp. 91–98. IEEE Computer Society (2003)Google Scholar
  17. 17.
    Hodkinson, I., Wolter, F., Zakharyaschev, M.: Decidable fragment of first-order temporal logics. Ann. Pure Appl. Logic 106, 85–134 (2000)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Hodkinson, I., Wolter, F., Zakharyaschev, M.: Monodic fragments of first-order temporal logics: 2000-2001 A.D. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 1–23. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Koller, D., Halpern, J.: Irrelevance and conditioning in first-order probabilistic logic. In: Proc. AAAI/IAAI 1096, pp. 569–576. AAAI Press / The MIT Press (1996)Google Scholar
  20. 20.
    Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. In: Principles of Knowledge Representation and Reasoning, KR 2010. AAAI Press (2010)Google Scholar
  21. 21.
    Richardson, M., Domingos, P.: Markov logic networks. Machine Learning 62, 107–136 (2006)CrossRefGoogle Scholar
  22. 22.
    Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 334–344. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Wolter, F., Zakharyaschev, M.: Axiomatizing the monodic fragment of first-order temporal logic. Ann. Pure Appl. Logic 118, 133–145 (2002)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jean Christoph Jung
    • 1
  • Carsten Lutz
    • 1
  • Sergey Goncharov
    • 2
  • Lutz Schröder
    • 2
  1. 1.Universität BremenGermany
  2. 2.Friedrich-Alexander-Universität Erlangen-NürnbergGermany

Personalised recommendations