Advertisement

Journal of Logic, Language and Information

, Volume 12, Issue 1, pp 13–45 | Cite as

A Two-Variable Fragment of English

  • Ian Pratt-Hartmann
Article

Abstract

Controlled languages are regimented fragments of natural languagedesigned to make the processing of natural language more efficient andreliable. This paper defines a controlled language, E2V, whose principalgrammatical resources include determiners, relative clauses, reflexivesand pronouns. We provide a formal syntax and semantics for E2V, in whichanaphoric ambiguities are resolved in a linguistically natural way. Weshow that the expressive power of E2V is equal to that of thetwo-variable fragment of first-order logic. It follows that the problemof determining the satisfiability of a set of E2V sentences is NEXPTIMEcomplete. We show that E2V can be extended in various ways withoutcompromising these complexity results; however, relaxing our policy onanaphora resolution renders the satisfiability problem for E2Vundecidable. Finally, we argue that our results have a bearing on thebroader philosophical issue of the relationship between natural andformal languages.

controlled languages logic natural language specification two-variable fragment 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andréka, H., van Benthem, J., and Németi, I., 1998, “Modal languages and bounded fragments of predicate logic,” Journal of Philosophical Logic 27, 217–274.Google Scholar
  2. Börger, E., Grädel, E., and Gurevich, Y., 1997, The Classical Decision Problem, Perspectives in Mathematical Logic, Berlin: Springer-Verlag.Google Scholar
  3. Cowper, E.A., 1992, A Concise Introduction to Syntactic Theory, Chicago, IL: University of Chicago Press.Google Scholar
  4. de Nivelle, H., 2000, “An overview of resolution decision procedures,” pp. 115–130 in Formalizing the Dynamics of Information, M. Faller, S. Kaufmann, and M. Pauly, eds., Stanford, CA: CSLI Publications.Google Scholar
  5. de Nivelle, H. and Pratt-Hartmann, I., 2001, “A resolution-based decision procedure for the twovariable fragment with equality,” pp. 211–225 in Automated Reasoning, T.N.R. Goré, A. Leitsch, eds., Lecture Notes in Computer Science, Vol. 2083, Berlin: Springer-Verlag.Google Scholar
  6. Englebretsen, G.: 1981, Three Logicians, Assen: Van Gorcum.Google Scholar
  7. Fantechi, A., Gnesi, S., Ristori, G., Carenini, M., Vanocchi, M., and Moreschini, P., 1994, “Assisting requirement formalization by means of natural language translation,” Formal Methods in System Design 4, 243–263.Google Scholar
  8. Fitch, F.B., 1973, “Natural deduction rules for English,” Philosophical Studies 24, 89–104.Google Scholar
  9. Fuchs, N., Schwertl, U., and Schwitter, R., 1999a, “Attempto controlled English - Not just another logic specification language,” pp. 1–20 in Logic-Based Program Synthesis and Transformation, P. Flener, ed., Lecture Notes in Computer Science, Vol. 1559, Berlin: Springer-Verlag.Google Scholar
  10. Fuchs, N.E., Schwertel, U., and Torge, S., 1999b, “Controlled natural language can replace first-order logic,” pp. 295–298 in 14th IEEE International Conference on Automated Software Engineering, New York: IEEE Computer Society Press.Google Scholar
  11. Grädel, E., 1999, “On the restraining power of guards,” Journal of Symbolic Logic 64, 1719–1742.Google Scholar
  12. Grädel, E. and Otto, M., 1999, “On logics with two variables,” Theoretical Computer Science 224, 73–113.Google Scholar
  13. Hintikka, J., 1974, “Quantifiers vs quantification theory,” Inquiry 5, 153–77.Google Scholar
  14. Holt, A., 1999, “Formal verification with natural language specifications: Guidelines, experiments and lessons so far,” South African Computer Journal 24, 253–257.Google Scholar
  15. Holt, A. and Klein, E., 1999, “A semantically-derived subset of English for hardware verification,” pp. 451–456 in Proceedings of the 37th Annual Meeting of the Association for Computational Linguistics, New Brunswick, NJ: Association for Computational Linguistics.Google Scholar
  16. Humberstone, I.L., 1983, “Inaccessible worlds,” Notre Dame Journal of Formal Logic 24, 346–352.Google Scholar
  17. Humberstone, I.L., 1987, “The modal logic of all and only,” Notre Dame Journal of Formal Logic 28, 177–188.Google Scholar
  18. Kamp, H. and Reyle, U., 1993, From Discourse to Logic: Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory, Studies in Linguistics and Philosophy, Vol. 42, Dordrecht: Kluwer Academic Publishers.Google Scholar
  19. Lutz, C. and Sattler, U., 2001, “The complexity of reasoning with Boolean modal logics,” in Advances in Modal Logics, Vol. 3, F.Wolter, H. Wansing, M. de Rijke, and M. Zakharyaschev, eds., Stanford, CA: CSLI Publications, forthcoming.Google Scholar
  20. Macias, B. and Pulman, S., 1995, “A method for controlling the production of specifications in natural language,” The Computer Journal 38, 310–318.Google Scholar
  21. McAllester, D.A. and Givan, R., 1992, “Natural language syntax and first-order inference,” Artificial Intelligence 56, 1–20.Google Scholar
  22. Mortimer, M., 1975, “On languages with two variables,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21, 135–140.Google Scholar
  23. Muskens, R., 1996, “Combining Montague semantics and discourse representation,” Linguistics and Philosophy 19, 143–186.Google Scholar
  24. Nelken, R. and Francez, N., 1996, “Translating natural language system specifications into temporal logic via DRT,” Technical Report LCL-96-––2, Laboratory for Computational Linguistics, Department of Computer Science, Technion, Israel Institute of Technology.Google Scholar
  25. Pacholski, L., Szwast, W., and Tendera, L., 1997, “Complexity of two-variable logic with counting,” pp. 318–327 in 12th IEEE Symposium on Logic in Computer Science, New York: IEEE Computer Society Press.Google Scholar
  26. Pratt-Hartmann, I.: 2000, “On the semantic complexity of some fragments of English,” Technical Report UMCS-00-––5-–1, Department of Computer Science, University of Manchester.Google Scholar
  27. Purdy, W.C., 1991, “A logic for natural language,” Notre Dame Journal of Formal Logic 32, 409–425.Google Scholar
  28. Sommers, F., 1982, The Logic of Natural Language, Oxford: Clarendon Press.Google Scholar
  29. Suppes, P., 1979, “Logical inference in English: A preliminary analysis,” Studia Logica 38, 375–391.Google Scholar
  30. Vadera, S. and Meziane, F., 1994, “From English to formal specifications,” The Computer Journal 37, 753–763.Google Scholar
  31. Walker, A., McCord, M., Sowa, J.F., and Wilson, W.G., 1987, Knowledge Systems and Prolog: A Logical Approach to Expert Systems and Natural Language Processing, Reading, MA: Addison Wesley.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Ian Pratt-Hartmann
    • 1
  1. 1.Department of Computer ScienceUniversity of ManchesterManchesterU.K.

Personalised recommendations