Complexity of the Higher Order Matching

  • ToMasz Wierzbicki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1632)


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


Normal Form Free Variable Propositional Variable Boolean Formula Propositional Formula 
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  1. 1.
    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.Google Scholar
  2. 2.
    Lewis D. Baxter, The Complexity of Unification, Ph.D. Thesis, University of Waterloo, 1976.Google Scholar
  3. 3.
    Paul Bernays, Moses Schönfinkel, Zum Entscheidungsproblem der mathematischen Logik, Math. Annalen, 99 (1928) 342–372.CrossRefzbMATHGoogle Scholar
  4. 4.
    Egon Börger, Erich Grädel, Yuri Gurevich, The Classical Decision Problem, Springer-Verlag, 1997.Google Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    Gilles Dowek, The undecidability of pattern matching in calculi where primitive recursive functionals are representable, Theoret. Comput. Sci., 107(1993) 349–56.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cynthia Dwork, Paris C. Kanellakis, John C. Mitchell, On the sequential nature of unification, J. Logic Programming, 1(1) (1984) 35–50.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Warren D. Goldfarb, Note on the undecidability of the second order unification problem, Theoret. Comput. Sci. 13(1981) 225–230.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gerard P. Huet, A unification algorithm for typed λ-calculus, Theoret. Comput. Sci., 1 (1) (1975) 27–57.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gerard P. Huet, Résolution d’équations dans des langages d’ordre 1, 2,..., w. Thèse de Doctorat d’État, University of Paris, 1976.Google Scholar
  12. 12.
    Gerard P. Huet, Bernard Lang, Proving and applying program transformations expressed with second order patterns, Acta Informatica, 11 (1978) 31–55.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ralph Loader, The undecidability of λ-definability, Church Memorial Volume, A. Anderson, M. Zeleny eds., Kluwer Acad. Press, to appear.Google Scholar
  14. 14.
    Harry G. Mairson, A simple proof of a theorem of Statman, Theoret. Comput. Sci., 103 (1992) 387–394.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dale Miller, A logic programming language with lambda-abstraction, function variables, and simple unification, J. Logic and Comput., 1(4) (1991) 497–536.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Vincent Padovani, Filtrage d’ordre supérieur, PhD Thesis, Université Paris VII, 1996.Google Scholar
  17. 17.
    Christos H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994.Google Scholar
  18. 18.
    John Alan Robinson, A machine-oriented logic based on the resolution principle, J. ACM, 12(1) (1965) 23–41.zbMATHCrossRefGoogle Scholar
  19. 19.
    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.Google Scholar
  20. 20.
    Richard Statman, Intuitionistic propositional logic is polynomial-space complete, Theoret. Comput. Sci., 9 (1979) 67–72.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Richard Statman, The typed λ-calculus is not elementary recursive, Theoret. Comput. Sci., 9 (1979) 73–81.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Richard Statman, Completeness, invariance and λ-definability, J. Symbolic Logic, 47(1) (1982) 17–26.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    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.Google Scholar
  25. 25.
    David A. Wolfram, The decidability of higher-order matching, Proc. 6th Int’lWorkshop on Unification, Schloß Dagstuhl, Germany, July 29-31, 1992.Google Scholar
  26. 26.
    David A. Wolfram, The Clausal Theory of Types, Cambridge Tracts in Theor. Comput. Sci. vol. 36, Cambridge University Press, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • ToMasz Wierzbicki
    • 1
  1. 1.University of WroclawGermany

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