Abstract
By a fragment of a natural language we mean a subset of thatlanguage equipped with semantics which translate its sentences intosome formal system such as first-order logic. The familiar conceptsof satisfiability and entailment can be defined for anysuch fragment in a natural way. The question therefore arises, for anygiven fragment of a natural language, as to the computational complexityof determining satisfiability and entailment within that fragment. Wepresent a series of fragments of English for which the satisfiabilityproblem is polynomial, NP-complete, EXPTIME-complete,NEXPTIME-complete and undecidable. Thus, this paper represents a casestudy in how to approach the problem of determining the logicalcomplexity of various natural language constructions. In addition, wedraw some general conclusions about the relationship between naturallanguage and formal logic.
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References
Andréka, H., van Benthem, J., and Németi, I., 1998, “Modal languages and bounded fragments of predicate logic,” Journal of Philosophical Logic 27, 217-274.
Börger, E., Grädel, E., and Gurevich, Y., 1997, The Classical Decision Problem, Perspectives in Mathematical Logic, Berlin: Springer-Verlag.
Cowper, E.A., 1992, A Concise Introduction to Syntactic Theory, Chicago, IL: University of Chicago Press.
Englebretsen, G., 1981, Three Logicians, Assen: Van Gorcum.
Fitch, F.B., 1973, “Natural deduction rules for English,” Philosophical Studies 24, 89-104.
Grädel, E., 1999, “On the restraining power of guards,” Journal of Symbolic Logic 64, 1719-1742.
Grädel, E. and Otto, M., 1999, “On logics with two variables,” Theoretical Computer Science 224, 73-113.
Harel, D., Kozen, D., and Tiuryn, J., 2000, Dynamic Logic, Cambridge, MA.: MIT Press.
Heim, I. and Kratzer, A., 1998, Semantics in Generative Grammar, Oxford: Blackwell.
Hintikka, J., 1974, “Quantifiers vs quantification theory,” Inquiry 5, 153-77.
McAllester, D.A. and Givan, R., 1992, “Natural language syntax and first-order inference,” Artificial Intelligence 56, 1-20.
Mortimer, M., 1975, “On languages with two variables,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21, 135-140.
Pratt-Hartmann, I., 2000, “On the semantic complexity of some fragments of English,” Technical Report UMCS-00-5-1, University of Manchester Department of Computer Science, Manchester.
Pratt-Hartmann, I., 2003, “A two-variable fragment of English,” Journal of Logic, Language and Information 12, 13-45.
Purdy, W.C., 1991, “A logic for natural language,” Notre Dame Journal of Formal Logic 32, 409-425.
Sommers, F., 1982, The Logic of Natural Language, Oxford: Clarendon Press.
Suppes, P., 1979, “Logical inference in English: A preliminary analysis,” Studia Logica 38, 375-391.
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Pratt-Hartmann, I. Fragments of Language. Journal of Logic, Language and Information 13, 207–223 (2004). https://doi.org/10.1023/B:JLLI.0000024735.97006.5a
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DOI: https://doi.org/10.1023/B:JLLI.0000024735.97006.5a