Abstract
Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value Structures unify amounts to testing for modal satisfiability. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted.
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References
Balcázar, J., Díaz, J. and Gabarró, J., 1988,Structural Complexity I, EATCS Monographs on Theoretical Computer Science 11, Berlin: Springer Verlag.
Ben-Ari, M., Halpern, J. and Pnueli, A., 1982, “Deterministic propositional dynamic logic: finite models, complexity and completeness”,Journal of Computer and System Sciences 25, 402–417.
Berger, R., 1966, “The undecidability of the dominoe problem”,Mem. Amer. Math. Soc. 66, 1–72.
Blackburn, P., 1989, “Nominal tense logic”, to appear inNotre Dame Journal of Formal Logic.
Blackburn, P., 1991, “Modal logic and attribute value structures”, to appear inDiamonds and Defaults, M. de Rijke, ed., Studies in Logic, Language and Information, Dordrecht: Kluwer.
Brink, C. and Schmidt, R., 1992, “Subsumption computed algebraically”,Computers Math. Applic. 23, 329–342.
Bull, R., 1970, “An approach to tense logic”,Theoria 12, 171–182.
Carpenter, B., 1992,The Logic of Typed Feature Structures, Cambridge: Cambridge University Press.
Evans, R., 1987, “Towards a formal specification for defaults in GPSG”, inCategories, Polymorphism and Unification, E. Klein and J. van Benthem, eds., Edinburgh: Centre for Cognitive Science and Amsterdam: ITLI.
Evans, R. and Gazdar, G., 1989, “The semantics of DATR”,AISB-89, Proceedings of the Seventh Conference of the Society for the Study of Artificial Intelligence and the Simulation of Behaviour, A. Cohn, ed., London: Pitman.
Fischer, M. and Ladner, R., 1979, “Propositional dynamic logic of regular programs”,Journal of Computer and System Sciences 18, 194–211.
Gargov, G., Passy, S. and Tinchev, T., 1986, “Modal environment for boolean speculations (preliminary report)”, inin Mathematical Logic and its Applications. Proceedings of the 1986 Gödel Summer School and Conference Bulgaria, D. Skordev, ed., New York: Plenum Press.
Gargov, G. and Passy, S., 1988, “Determinism and looping in combinatory PDL”,Theoretical Computer Science 61, pages 259–277.
Gargov, G. and Goranko, V., 1989, “Modal logic with names”, to appear inJournal of Philosophical Logic.
Gazdar, G., Klein, E., Pullum, G. and Sag, I., 1985,Generalized Phrase Structure Grammar, Oxford: Basil Blackwell.
Gazdar, G., Pullum G., Carpenter, R., Klein, E., Hukari, T. and Levine, R., 1988, “Category structures”,Computational Linguistics 14, pages 1–19.
Goldblatt, R., 1989, “Varieties of complex algebras”,Annals of Pure and Applied Logic 44, 173–242.
Goranko, V. and Passy, S., 1992, “Using the universal modality: gains and questions”,Journal of Logic and Computation 2, 5–30.
Halpern, J. and Moses, M., 1985, “A guide to the modal logics of knowledge and belief”, pp. 480–490 inProceedings 9th International Joint Conference on Artificial Intelligence.
Harel, D., 1983, “Recurring dominoes: making the highly undecidable highly understandable”, pp. 177–194 inProc. of the Conference on Foundations of Computing Theory, Springer Lecture Notes in Computer Science158.
Harel, D., 1984, “Dynamic logic”, inHandbook of Philosophical Logic 2, D. Gabbay and F. Guenthner, eds., Dordrecht: Reidel.
Harel, D., 1986, “Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness”,Journal of the ACM 33(1), 224–248.
van der Hoek, W. and de Rijke, M., 1991, “Generalized quantifiers and modal logic”, to appear inJournal of Logic, Language and Information.
Hughes, G. and Cresswell, M., 1984,A Companion to Modal Logic, London: Methuen & Co. Ltd.
Kaplan, K. and Bresnan, J., 1982, “Lexical functional grammar: a formal system for grammatical representation”, pp. 173–281 inThe Mental Representation of Grammatical Relations, Joan Bresnan, ed., Cambridge: MIT Press.
Karttunen, L., 1984, “Features and values”, pp. 28–33 inProceedings of the 10th International Conference on Computational Linguistics and the 22nd Annual Meeting of the Association for Computational Linguistics, Stanford, California.
Kasper, R. and Rounds, W., 1986, “A logical semantics for feature structures”, pp. 257–266 inProceedings of the 24th Annual Meeting of the Association for Computational Linguistics, Columbia University, New York.
Kasper, R. and Rounds, W., 1990, “The logic of unification in grammar”,Linguistics and Philosophy 13, 33–58.
Keller, B., 1991,Feature Logics, Infinitary Descriptions and the Logical Treatment of Grammar, PhD thesis, School of Cognitive and Computing Sciences, University of Sussex, United Kingdom.
Kracht, M., 1989, “On the logic of category definition”,Computational Linguistics 15, 111–113.
Ladner, R., 1977, “The computational complexity of provability in systems of modal logic”,SIAM Journal on Computing 6(3), 467–480.
Moss, L., 1991, “Completeness theorems for logics of feature structures”, Indiana University Logic Group Preprint No. IULG-91-2. To appear inProceedings of the MSRI Workshop on Logic From Computer Science, Y. Moschovakis, ed., Berlin: Springer Verlag.
Passy, S. and Tinchev, T., 1985, “PDL with data constants”,Information Processing Letters 20, 35–41.
Passy, S. and Tinchev, T., 1991, “An essay in combinatory dynamic logic”,Information and Computation 93, 263–332.
Pereira, F and Shieber, S., 1984, “The semantics of grammar formalisms seen as computer languages”, pp. 123–129 inProceedings of the 10th International Conference on Computational Linguistics and the 22nd Annual Meeting of the Association for Computational Linguistics, Stanford, California.
Pollard, C. and Sag, I.,Information-Based Syntax and Semantics: Volume 1 — Fundamentals, CSLI Lecture Notes13, Stanford.
Pratt, V., 1979, “Models of program logics”, pp. 115–122 inProceedings of the 20th IEEE Symposium on Foundations of Computer Science.
Prior, A., 1967,Past, Present and Future, Oxford: Oxford University Press.
Reape, M., 1991,An Introduction to the Semantics of Unification-Based Grammar Formalisms, DYANA deliverable R3.2.A, Edinburgh: Centre for Cognitive Science.
Robinson, R., 1971, “Undecidability and nonperiodicity for tilings of the plane”,Inventiones Math. 12, 177–209.
Roorda, D., 1991, “Dyadic modalities and Lambek calculus”, to appear inDiamonds and Defaults, M. de Rijke, ed., Studies in Logic, Language and Information, Dordrecht: Kluwer.
Rounds, W. and Kasper, R., 1986, “A complete logical calculus for record structures representing linguistic information”, inProceedings of the 15th Annual Symposium on Logic in Computer Science, Cambridge MA.
Schild, K., 1990, “A correspondence theory for terminological logics: preliminary report”, inProceedings of the 12th International Joint Conference on Artificial Intelligence, Sydney, Australia.
Shieber, S., 1986,An Introduction to Unification-Based Approaches to Grammar, CSLI Lecture Notes4, Stanford.
Sistla, A. and Clarke E, 1985, “The complexity of propositional linear temporal logics”,J. Assoc. Comput. Mach. 32, 733–749.
Smolka, G., 1989, “Feature constraint logics for unification grammars”, IWBS Report 93, Stuttgart: IBM Deutschland.
van Benthem, J., 1984, “Correspondence theory”, inHandbook of Philosophical Logic 2, D. Gabbay and F. Guenthner, eds., Dordrecht: Reidel.
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Blackburn, P., Spaan, E. A modal perspective on the computational complexity of attribute value grammar. J Logic Lang Inf 2, 129–169 (1993). https://doi.org/10.1007/BF01050635
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DOI: https://doi.org/10.1007/BF01050635