Polyadic Dynamic Logics for HPSG Parsing

Abstract

Head-driven phrase structure grammar (HPSG) is one of the most prominent theories employed in deep parsing of natural language. Many linguistic theories are arguably best formalized in extensions of modal or dynamic logic (Keller, Feature logics, infinitary descriptions and grammar, 1993; Kracht, Linguistics Philos 18:401–458, 1995; Moss and Tiede, In: Blackburn, van Benthem, and Wolther (eds.) Handbook of modal logic, 2006), and HPSG seems to be no exception. Adequate extensions of dynamic logic have not been studied in detail, however; the most important aspect is the reference to sets of substructures. In this paper, an adequate extension is identified, and some important results are established: Satisfiability is highly undecidable, and model checking is shown to be in EXPTIME and PSPACE-hard. A fragment with polynomial time model checking procedures is identified; it is shown to cover considerable fragments of HPSG.

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References

  1. Benevides, M. (2003). Modal logics for finite graphs. In: R. de Queiroz (Ed.), Logic for concurrency and synchronisation (pp. 239–267). Norwell, Michigan: Kluwer.

    Google Scholar 

  2. Blackburn P. (1994) Structures, languages and translations: The structural approach to feature logic. In: Rupp C.J., Johnson R., Rosner M. (eds) Constraints, language and computation. Academic Press, London, pp 1–29

    Google Scholar 

  3. Blackburn P., Meyer-Viol W. (1994) Linguistics, logic and finite trees. Logic Journal of IGPL 2(1): 3–29

    Article  Google Scholar 

  4. Blackburn, P., Spaan, E. (1993). Decidability and undecidability in stand-alone feature logics. In Proceedings of the Sixth Conference of the European Chapter of the Association for Computational Linguistics (pp. 30–36). Utrecht, The Netherlands.

  5. Bredenkamp, A., Markantonatou, S., & Sadler, L. (1996). Lexical rules: What are they? In Proceedings of the 16th International Conference on Computational Linguistics (pp. 163–168). Copenhagen, Denmark.

  6. Carpenter B. (1991) The generative power of categorial grammars and head-driven phrase structure grammars with lexical rules. Computational Linguistics 17(3): 301–313

    Google Scholar 

  7. Chandra A., Kozen D., Stockmeyer L. (1981) Journal of the Association for Computing Machinery 28(1): 114–133

    Google Scholar 

  8. Danecki, S. (1984). Nondeterministic propositional dynamic logic with intersection is decidable. In A. Skowron (Ed.), Proceedings of the 5th Symposium on Computation Theory, Lecture Notes in Computer Science (Vol. 208, pp. 34–53). Zaborów, Poland: Springer.

  9. Fischer M.J., Ladner R.E. (1979) Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18(2): 194–211

    Article  Google Scholar 

  10. Franceschet M., de Rijke M. (2005) Model checking for hybrid logics. Journal of Applied Logic 4(3): 279–304

    Article  Google Scholar 

  11. Harel D. (1985) Recurring dominoes: Making the highly undecidable highly understandable. Annals of Discrete Mathematics 24: 51–72

    Google Scholar 

  12. Jaeger E., Francez N., Wintner S. (2005) Unification grammars and off-line parsability. Journal of Logic, Language and Information 14: 199–234

    Article  Google Scholar 

  13. Johnson M. (1988) Attribute-value logic and the theory of grammar. CSLI Publications, Stanford, California

    Google Scholar 

  14. Keller B. (1993) Feature logics, infinitary descriptions and grammar. CSLI Publications, Stanford, California

    Google Scholar 

  15. Kepser S. (2001) On the complexity of RSRL. Electronic Notes in Computer Science 53: 146–162

    Article  Google Scholar 

  16. Kepser, S., & Mönnich, U. (2003). Undecidability results for HPSG. In Algebraic methods in language processing 3 (pp. 1–12). Verona, Italy.

  17. King, P. (1989). A logical formalism for head-driven phrase structure grammar. Ph.D. thesis, University of Manchester. Doctoral thesis.

  18. King, P. (1999). Towards truth in head-driven phrase structure grammar. In V. Kordoni (Ed.), Tübingen studies in head-driven phrase structure grammar (Vol. 2, pp. 301–352). Tübingen, Germany: Arbeitspapiere des SFB 340, Bericht Nr. 132.

  19. King P., Simov K., Aldag B. (1999) The complexity of modellability in finite and computable signatures of a constraint logic for head-driven phrase structure grammar. Journal of Logic, Language and Information 8(1): 83–110

    Article  Google Scholar 

  20. Kracht M. (1995) Is there a genuine modal perspective on feature structures? Linguistics & Philosophy 18: 401–458

    Article  Google Scholar 

  21. Lange M. (2006) Model checking propositional dynamic logic with all extras. Journal of Applied Logic 4: 39–49

    Article  Google Scholar 

  22. Lange M., Lutz C. (2005) 2-ExpTime lower bounds for propositional dynamic logics with intersection. Journal of Symbolic Logic 70(4): 1072–1086

    Article  Google Scholar 

  23. Moens, M., Calder, J., Klein, E., Reape, M., & Zeevat, H. (1989). Expressing generalizations in unification-based grammar formalisms. In Proceedings of the 4th Conference of the European Chapter of the Association for Computational Linguistics. Manchester, England.

  24. Moss L., Tiede H. (2006) Applications of modal logic in linguistics. In: Blackburn P., van Benthem J., Wolther F. (eds) Handbook of modal logic. Elsevier, Amsterdam, The Netherlands, pp 1031–1072

    Google Scholar 

  25. Pollard C. (1999) Strong generative capacity in HPSG. In: Webelhuth G., Koenig J.-P., Kathol A. (eds) Lexical and constructional aspects of linguistic explanation. CSLI Publications, Stanford, California, pp 281–297

    Google Scholar 

  26. Pollard C., Sag I. (1987) Information-based syntax and semantics. The University of Chicago Press, Chicago, Illinois

    Google Scholar 

  27. Pollard C., Sag I. (1994) Head-driven phrase structure grammar. The University of Chicago Press, Chicago, Illinois

    Google Scholar 

  28. Reape, M. (1992). A formal theory of word order: A case study of West Germanic. Ph.D. thesis, University of Edinburgh, Edinburgh, Scotland.

  29. Reape M. (1994) A feature value logic with intensionality, nonwellfoundedness and functional and relational dependencies. In: Rupp C.J., Rosner M., Johnson R. (eds) Constraints, language and computation. Academic Press, San Francisco, California, pp 77–110

    Google Scholar 

  30. Richter, F. (2004). A mathematical formalism for linguistic theories with an application in head-driven phrase structure grammar. Ph.D. thesis, Universität Tübingen, Tübingen, Germany.

  31. Rounds B. (1994) Feature logics. In: van Benthem J., ter Meulen A. (eds) Handbook of logic and language. Elsevier, Amsterdam, The Netherlands, pp 1–61

    Google Scholar 

  32. Sag I., Fodor J. (1994) Extraction without traces. In: Aranovich R., Byrne W., Preuss S., Senturia M. (eds) Proceedings of the 13th West Coast Conference on Formal Linguistics. CSLI Publications, Stanford, California, pp 365–384

    Google Scholar 

  33. Shieber S. (1992) Constraint-based grammar formalisms. MIT Press, Cambridge, Massachusetts

    Google Scholar 

  34. Søgaard, A. (2006). Logical investigations of certain feature-based theories of natural language. In Proceedings of The Joint Human Language Technology Conference and the North American Chapter of the Association for Computational Linguistics 2006, Companion Volume (pp. 239–242). New York, NY.

  35. Trautwein, M. (1995). Computational pitfalls in tractable grammar formalisms. Ph.D. thesis, University of Amsterdam, Amsterdam, The Netherlands. ILLC Dissertation Series DS-1995-15.

  36. Warshall S. (1962) A theorem on Boolean matrices. Journal of the Association for Computing Machinery 9(1): 11–12

    Google Scholar 

  37. Wedekind J. (1997) Approaches to unification in grammar. In: Blackburn P., de Rijke M. (eds) Specifying syntactic structure. CSLI Publications, Stanford, California, pp 245–280

    Google Scholar 

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Correspondence to Anders Søgaard.

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Søgaard, A., Lange, M. Polyadic Dynamic Logics for HPSG Parsing. J of Log Lang and Inf 18, 159–198 (2009). https://doi.org/10.1007/s10849-008-9061-x

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Keywords

  • Dynamic logic
  • HPSG
  • Model checking
  • Parsing