Abstract
We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching. We obtain this way already known NP-hardness bounds for the order two and three and a new result that the fourth order matching is NEXPTIME-hard
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Henk Barendregt, Lambda Calculi with Types, Handbook of Logic in Comput. Sci., vol. 2, S. Abramsky, D.M. Gabbay, T.S.E. Maibaum, eds., Clarendon Press, Oxford, 1992, 118–310.
Lewis D. Baxter, The Complexity of Unification, Ph.D. Thesis, University of Waterloo, 1976.
Paul Bernays, Moses Schönfinkel, Zum Entscheidungsproblem der mathematischen Logik, Math. Annalen, 99 (1928) 342–372.
Egon Börger, Erich Grädel, Yuri Gurevich, The Classical Decision Problem, Springer-Verlag, 1997.
Hubert Comon, Yan Jurski, Higher order pattern matching and tree automata, Proc. 11th Int’l Workshop on Comput. Sci. Logic, CSL’97, Aarhus, Denmark, August 23-29, 1997, M. Nielsen, W. Thomas, eds., LNCS 1414, Springer-Verlag, 1998.
Gilles Dowek, Third order matching is decidable, Proc. 7th IEEE Symp. Logic in Comput. Sci., LICS’92, Santa Cruz, California, June 22-25, 1992, 2–10, also in Annals of Pure and Applied Logic, 69 (1994), 135-155.
Gilles Dowek, The undecidability of pattern matching in calculi where primitive recursive functionals are representable, Theoret. Comput. Sci., 107(1993) 349–56.
Cynthia Dwork, Paris C. Kanellakis, John C. Mitchell, On the sequential nature of unification, J. Logic Programming, 1(1) (1984) 35–50.
Warren D. Goldfarb, Note on the undecidability of the second order unification problem, Theoret. Comput. Sci. 13(1981) 225–230.
Gerard P. Huet, A unification algorithm for typed λ-calculus, Theoret. Comput. Sci., 1 (1) (1975) 27–57.
Gerard P. Huet, Résolution d’équations dans des langages d’ordre 1, 2,..., w. Thèse de Doctorat d’État, University of Paris, 1976.
Gerard P. Huet, Bernard Lang, Proving and applying program transformations expressed with second order patterns, Acta Informatica, 11 (1978) 31–55.
Ralph Loader, The undecidability of λ-definability, Church Memorial Volume, A. Anderson, M. Zeleny eds., Kluwer Acad. Press, to appear.
Harry G. Mairson, A simple proof of a theorem of Statman, Theoret. Comput. Sci., 103 (1992) 387–394.
Dale Miller, A logic programming language with lambda-abstraction, function variables, and simple unification, J. Logic and Comput., 1(4) (1991) 497–536.
Vincent Padovani, Filtrage d’ordre supérieur, PhD Thesis, Université Paris VII, 1996.
Christos H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994.
John Alan Robinson, A machine-oriented logic based on the resolution principle, J. ACM, 12(1) (1965) 23–41.
Aleksy Schubert, Linear interpolation for the higher order matching problem, Technical Report of the Institute of Informatics, Warsaw University, TR 96-16 (237), also in Proc. 7th Int’l Joint Conf. Theory and Practice of Software Development, TAPSOFT’97, Lille, France, April 14-18, 1997, M. Bidoit, M. Dauchet, eds., LNCS 1214, Springer-Verlag, 1997.
Richard Statman, Intuitionistic propositional logic is polynomial-space complete, Theoret. Comput. Sci., 9 (1979) 67–72.
Richard Statman, The typed λ-calculus is not elementary recursive, Theoret. Comput. Sci., 9 (1979) 73–81.
Richard Statman, Completeness, invariance and λ-definability, J. Symbolic Logic, 47(1) (1982) 17–26.
Richard Statman, On the existence of closed terms in the typed λ-calculus II: transformations of unification problems, Theoret. Comput. Sci., 15 (1981) 329–338.
Sergei Vorobyov, The “Hardest” Natural Decidable Theory, Proc. 12th Annual IEEE Symp. Logic in Comput. Sci., LICS’97, Warsaw, Poland, June 29-July 2, 1997, 294–305.
David A. Wolfram, The decidability of higher-order matching, Proc. 6th Int’lWorkshop on Unification, Schloß Dagstuhl, Germany, July 29-31, 1992.
David A. Wolfram, The Clausal Theory of Types, Cambridge Tracts in Theor. Comput. Sci. vol. 36, Cambridge University Press, 1993.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wierzbicki, T. (1999). Complexity of the Higher Order Matching. In: Automated Deduction — CADE-16. CADE 1999. Lecture Notes in Computer Science(), vol 1632. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48660-7_6
Download citation
DOI: https://doi.org/10.1007/3-540-48660-7_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66222-8
Online ISBN: 978-3-540-48660-2
eBook Packages: Springer Book Archive