Higher Order Unification 30 Years Later

Extended Abstract
  • Gérard Huet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2410)

Abstract

The talk will present a survey of higher order unification, covering an outline of its historical development, a summary of its applications to three fields: automated theorem proving, and more generally engineering of proof assistants, programming environments and software engineering, and finally computational linguistics. It concludes by a presentation of open problems, and a few prospective remarks on promising future directions. This presentation assumes as background the survey by Gilles Dowek in the Handbook of automated theorem proving [28].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Amiot. The undecidability of the second order predicate unification problem. Archive for mathematical logic, 30:193–199, 1990.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dale Miller Amy Felty. Specifying theorem provers in a higher-order logic programming language. In Proceedings of the 9th Int. Conference on Automated Deduction (CADE-9), volume 310 of Lecture Notes in Computer Science, pages 61–80. Springer-Verlag, Argonne, Illinois, 1988.CrossRefGoogle Scholar
  3. 3.
    Penny Anderson. Represnting proof transformations for program optimization. In Proceedings of the 12th Int. Conference on Automated Deduction (CADE-12), volume 814 of Lecture Notes in Artificial Intelligence, pages 575–589, Nancy, France, 1994. Springer-Verlag.Google Scholar
  4. 4.
    P.B. Andrews. Resolution in type theory. The Journal of Symbolic Logic, 36(3):414–432, 1971.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    P.B. Andrews, M. Bishop, S. Issar, D. Nesmith, F. Pfenning, and H. Xi. Tps: a theorem proving system for classical type theory. J. of Automated Reasoning, 16:321–353, 1996.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. Avenhaus and C.A. Loría-Sáenz. Higher-order conditional rewriting and narrowing. In J.-P. Jouannaud, editor, International Conference on Constaints in Computational Logic, volume 845 of Lecture Notes in Computer Science, pages 269–284. Springer-Verlag, 1994.CrossRefGoogle Scholar
  7. 7.
    Yves Bekkers, Olivier Ridoux, and Lucien Ungaro. Dynamic memory management for sequential logic programming languages. In IWMM, pages 82–102, 1992.Google Scholar
  8. 8.
    C. Benzmüller. Extensional higher-order resolution. In Proceedings of the 15th Int. Conference on Automated Deduction (CADE-15), volume 1421 of Lecture Notes in Artificial Intelligence, pages 56–71. Springer-Verlag, Lindau, Germany, 1998.Google Scholar
  9. 9.
    C. Benzmüller. Equality and extensionality in automated higher-order theorem proving. Doctoral Dissertation, Universität des Saarlandes, Saarbrücken, 1999.Google Scholar
  10. 10.
    Frédéric Blanqui. Théorie des types et réécriture. Thèse de Doctorat, Université Paris-Sud, 2001.Google Scholar
  11. 11.
    P. Borovanský. Implementation of higher-order unification based on calculus of explicit substitution. In M. Bartošek, J. Staudek, and J. Wiedermann, editors, SOFSEM: Theory and Practice of Informatics, volume 1012 in Lecture Notes in Computer Science, pages 363–368. Springer-Verlag, 1995.Google Scholar
  12. 12.
    A. Boudet and E. Contejean. AC-unification of higher-order patterns. In G. Smolka, editor, Principles and Practice of Constraint Programming, volume 1330 of Lecture Notes in Computer Science, pages 267–281. Springer-Verlag, 1997.CrossRefGoogle Scholar
  13. 13.
    D. Briaud. Higher order unification as typed narrowing. Technical Report 96-R-112, Centre de recherche en informatique de Nancy, 1996.Google Scholar
  14. 14.
    Pascal Brisset and Olivier Ridoux. The architecture of an implementation of lambdaprolog: Prolog/mali. Technical Report RR-2392, INRIA, Oct. 1994.Google Scholar
  15. 15.
    M. Kohlhase, C. Gardent, and K. Konrad. Higher-order colored unification: A linguistic application. J. of Logic, Language and Information, 9:313–338, 2001.Google Scholar
  16. 16.
    A. Church. A formulation of the simple theory of types. The Journal of Symbolic Logic, 5(1):56–68, 1940.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    H. Comon and Y. Jurski. Higher-order matching and tree automata. In M. Nielsen and W. Thomas, editors, Conference on Computer Science Logic, volume 1414 of Lecture Notes in Computer Science, pages 157–176. Springer-Verlag, 1997.CrossRefGoogle Scholar
  18. 18.
    D.J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114:273–298, 1993.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    D.J. Dougherty and P. Johann. A combinatory logic approach to higher-order E-unification. In D. Kapur, editor, Conference on Automated Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 79–93. Springer-Verlag, 1992.Google Scholar
  20. 20.
    G. Dowek. Démonstration automatique dans le calcul des constructions. Thèse de Doctorat, Université de Paris VII, 1991.Google Scholar
  21. 21.
    G. Dowek. L’indécidabilité du filtrage du troisième ordre dans les calculs avec types dépendants ou constructeurs de types (the undecidability of third order pattern matching in calculi with dependent types or type constructors). Comptes Rendus à l’Académie des Sciences, I, 312(12):951–956, 1991. Erratum, ibid. I, 318, 1994, p. 873.MATHMathSciNetGoogle Scholar
  22. 22.
    G. Dowek. A second-order pattern matching algorithm in the cube of typed λ-calculi. In A. Tarlecki, editor, Mathematical Foundation of Computer Science, volume 520 of Lecture notes in computer science, pages 151–160. Springer-Verlag, 1991.Google Scholar
  23. 23.
    G. Dowek. A complete proof synthesis method for the cube of type systems. Journal of Logic and Computation, 3(3):287–315, 1993.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    G. Dowek. The undecidability of pattern matching in calculi where primitive recursive functions are representable. Theoretical Computer Science, 107:349–356, 1993.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    G. Dowek. The undecidability of typability in the lambda-pi-calculus. In M. Bezem and J.F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 in Lecture Notes in Computer Science, pages 139–145. Springer-Verlag, 1993.CrossRefGoogle Scholar
  26. 26.
    G. Dowek. A unification algorithm for second-order linear terms. Manuscript, 1993.Google Scholar
  27. 27.
    G. Dowek. Third order matching is decidable. Annals of Pure and Applied Logic, 69:135–155, 1994.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    G. Dowek. Higher-Order Unification and Matching, chapter 16, pages 1009–1062. Elsevier, 2001.Google Scholar
  29. 29.
    G. Dowek, T. Hardin, and C. Kirchner. Higher-order unification via explicit substitutions. Information and Computation, 157:183–235, 2000.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    G. Dowek, T. Hardin, C. Kirchner, and F. Pfenning. Unification via explicit substitutions: the case of higher-order patterns. In M. Maher, editor, Joint International Conference and Symposium on Logic Programming, pages 259–273. The MIT Press, 1996.Google Scholar
  31. 31.
    C.M. Elliott. Higher-order unification with dependent function types. In N. Dershowitz, editor, Internatinal Conference on Rewriting Techniques and Applications, volume 355 of Lecture Notes in Computer Science, pages 121–136. Springer-Verlag, 1989.Google Scholar
  32. 32.
    C.M. Elliott. Extensions and applications of higher-order unification. PhD thesis, Carnegie Mellon University, 1990.Google Scholar
  33. 33.
    W.M. Farmer. A unification algorithm for second-order monadic terms. Annals of Pure and applied Logic, 39:131–174, 1988.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    W.M. Farmer. Simple second order languages for which unification is undecidable. Theoretical Computer Science, 87:25–41, 1991.MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Amy Felty. Implementing tactics and tacticals in a higher-order logic programming language. Journal of Automated Reasoning, 11:43–81, 1993.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    C. Gardent. Deaccenting and higher-order unification. J. of Logic, Language and Information, 9:313–338, 2000.MATHCrossRefGoogle Scholar
  37. 37.
    C. Gardent and M. Kohlhase. Focus and higher-order unification. In 16th International Conference on Computational Linguistics, Copenhagen, 1996.Google Scholar
  38. 38.
    C. Gardent and M. Kohlhase. Higher-order coloured unification and natural language semantics. In ACL, editor, 34th Annual Meeting of the Association for Computational Linguistics, Santa Cruz, 1996.Google Scholar
  39. 39.
    W.D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225–230, 1981.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    J. Goubault. Higher-order rigid E-unification. In F. Pfenning, editor, 5th International Conference on Logic Programming and Automated Reasoning, volume 822 in Lecture Notes in Artificial Intelligence, pages 129–143. Springer-Verlag, 1994.Google Scholar
  41. 41.
    W.E. Gould. A matching procedure for ω-order logic. Scientific report 4, AFCRL 66-781 (contract AF 19 (628)-3250) AD 646 560.Google Scholar
  42. 42.
    J.R. Guard. Automated logic for semi-automated mathematics. Scientific report 1, AFCRL 64-411 (contract AF 19 (628)-3250) AS 602 710, 1964.Google Scholar
  43. 43.
    M. Hagiya. Programming by example and proving by example using higher-order unification. In M.E. Stickel, editor, Conference on Automated Deduction, volume 449 in Lecture Notes in Computer Science, pages 588–602. Springer-Verlag, 1990.Google Scholar
  44. 44.
    M. Hagiya. Higher-order unification as a theorem proving procedure. In K. Furukawa, editor, International Conference on Logic Programming, pages 270–284. MIT Press, 1991.Google Scholar
  45. 45.
    John Hannan and Dale Miller. Uses of higher-order unification for implementing programs transformers. In R.A. Kowalski an K.A. Bowen, editor, International Conference and Symposium on Logic Programming, pages 942–959, 1988.Google Scholar
  46. 46.
    G. Huet. The undecidability of unification in third order logic. Information and Control, 22:257–267, 1973.CrossRefMathSciNetMATHGoogle Scholar
  47. 47.
    G. Huet. Résolution d’équations dans les langages d’ordre 1,2,..., ω. Thèse d’État, Université de Paris VII, 1976.Google Scholar
  48. 48.
    G. Huet and B. Lang. Proving and applying program transformations expressed with second order patterns. Acta Informatica, 11:31–55, 1978.MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Gérard Huet. Constrained resolution: A complete method for higher order logic. PhD thesis, Case Western University, Aug. 1972.Google Scholar
  50. 50.
    D.C. Jensen and T. Pietrzykowski. Mechanizing ω-order type theory through unification. Theoretical Computer Science, 3:123–171, 1976.CrossRefMathSciNetGoogle Scholar
  51. 51.
    Frank Pfenning, Joëlle Despeyroux, and Carsten Schürmann. Primitive recursion for higher-order syntax. In R. Hindley, editor, Proceedings of the 3rd Int. Conference on Typed Lambda Calculus and Applications (TLCA’ 97), volume 1210 of Lecture Notes in Computer Science, pages 147–163, Nancy, France, 1997. Springer-Verlag.Google Scholar
  52. 52.
    Patricia Johann and Michael Kohlhase. Unification in an extensional lambda calculus with ordered function sorts and constant overloading. In Alan Bundy, editor, Conference on Automated Deduction, volume 814 in Lecture Notes in Artificial Intelligence, pages 620–634. Springer-Verlag, 1994.Google Scholar
  53. 53.
    C. Kirchner and Ch. Ringeissen. Higher-order equational unification via explicit substitutions. In M. Hanus, J. Heering, and K. Meinke, editors, Algebraic and Logic Programming, International Joint Conference ALP’97-HOA’97, volume 1298 of Lecture Notes in Computer Science, pages 61–75. Springer-Verlag, 1997.CrossRefGoogle Scholar
  54. 54.
    K. Kwon and G. Nadathur. An instruction set for higher-order hereditary harrop formulas (extended abstract). In Proceedings of the Workshop on the lambda Prolog Programming Language, UPenn Technical Report MS-CIS-92-86, 1992.Google Scholar
  55. 55.
    Peter Lee, Frank Pfenning, John Reynolds, Gene Rollins, and Dana Scott. Research on semantically based program-design environments: The ergo project in 1988. Technical Report CMU-CS-88-118, Computer Science Department, Carnegie Mellon University, 1988.Google Scholar
  56. 56.
    J. Levy. Linear second order unification. In H. Ganzinger, editor, Rewriting Techniques and Applications, volume 1103 of Lecture Notes in Computer Science, pages 332–346. Springer-Verlag, 1996.Google Scholar
  57. 57.
    Jordi Levy and Margus Veanes. On unification problems in restricted second-order languages. In Annual Conf. of the European Ass. of Computer Science Logic (CSL98), Brno, Czech Republic, 1998.Google Scholar
  58. 58.
    Jordi Levy and Mateu Villaret. Linear second-order unification and context unification with tree-regular constraints. In Proceedings of the 11th Int. Conf. on Rewriting Techniques and Applications (RTA 2000), volume 1833 of Lecture Notes in Computer Science, pages 156–171, Norwich, UK, 2000. Springer-Verlag.CrossRefGoogle Scholar
  59. 59.
    R. Loader. The undecidability of λ-definability. To appear in Church memorial volume, 1994.Google Scholar
  60. 60.
    R. Loader. Higher order β matching is undecidable. Private communication, 2001.Google Scholar
  61. 61.
    C.L. Lucchesi. The undecidability of the unification problem for third order languages. Technical Report CSRR 2060, Department of applied analysis and computer science, University of Waterloo, 1972.Google Scholar
  62. 62.
    Stuart M. Shieber Mary Dalrymple and Fernando C.N. Pereira. Ellipsis and higher-order unification. Linguistic and Philosophy, 14:399–452, 1991.CrossRefGoogle Scholar
  63. 63.
    R. Mayr and T. Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192:3–29, 1998.MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Raymond McDowell and Dale Miller. Reasoning with higher-order abstract syntax in a logical framework. ACM Transactions on Computational Logic (TOCL), 3:80–136, 2002.CrossRefMathSciNetGoogle Scholar
  65. 65.
    Dale Miller. Abstractions in logic programming. In P. Odifreddi, editor, Logic and computer science, pages 329–359. Academic Press, 1991.Google Scholar
  66. 66.
    Dale Miller. Unification under a mixed prefix. Journal of Symbolic Computation, 14:321–358, 1992.MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Dale Miller and Gopalan Nadathur. A logic programming approach to manipulating formulas and programs. In IEEE Symposium on Logic Programming, pages 247–255, San Francisco, California, 1986.Google Scholar
  68. 68.
    Dale Miller and Gopalan Nadathur. Some uses of higher-order logic in computational linguistics. In 24th Annual Meeting of the Association for Computational Linguistics, pages 247–255, 1986.Google Scholar
  69. 69.
    O. Müller and F. Weber. Theory and practice of minimal modular higher-order E-unification. In A. Bundy, editor, Conference on Automated Deduction, volume 814 in Lecture Notes in Artificial Intelligence, pages 650–664. Springer-Verlag, 1994.Google Scholar
  70. 70.
    Gopalan Nadathur. An explicit substitution notation in a λprolog implementation. In First International Workshop on Explicit Substitutions, 1998.Google Scholar
  71. 71.
    Gopalan Nadathur. A Higher-Order Logic as the Basis for Logic Programming. PhD thesis, University of Pennsylvania, Dec. 1986.Google Scholar
  72. 72.
    Gopalan Nadathur and Dale Miller. Higher-order horn clauses. J. of the Association for Computing Machinery, 37:777–814, 1990.MATHMathSciNetGoogle Scholar
  73. 73.
    Gopalan Nadathur and Dale Miller. Higher-order logic programming. In D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Handbook of logic in artificial intelligence and logic programming, volume 5, pages 499–590. Clarendon Press, 1998.Google Scholar
  74. 74.
    Gopalan Nadathur and D.J. Mitchell. System description: A compiler and abstract machine based implementation of λprolog. In Conference on Automated Deduction, 1999.Google Scholar
  75. 75.
    T. Nipkow. Higher-order critical pairs. In Logic in Computer Science, pages 342–349, 1991.Google Scholar
  76. 76.
    T. Nipkow. Higher-order rewrite systems. In Proceedings of the 6th Int. Conf. on Rewriting Techniques and Applications (RTA-95), volume 914 of Lecture Notes in Computer Science, pages 256–256, Keiserlautern, Germany, 1995. Springer-Verlag.Google Scholar
  77. 77.
    T. Nipkow and Ch. Prehofer. Higher-order rewriting and equational reasoning. In W. Bibel and P.H. Schmitt, editors, Automated Deduction-A Basis for Applications, volume 1, pages 399–430. Kluwer, 1998.Google Scholar
  78. 78.
    Tobias Nipkow and Zhenyu Qian. Modular higher-order e-unification. In Proceedings of the 4th Int. Conf. on Rewriting Techniques and Applications (RTA), volume 488 of Lecture Notes in Computer Science, pages 200–214. Springer-Verlag, 1991.Google Scholar
  79. 79.
    Tobias Nipkow and Zhenyu Qian. Reduction and unification in lambda calculi with a general notion of subtype. Journal of Automated Reasoning, 12:389–406, 1994.MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    V. Padovani. Fourth-order matching is decidable. Manuscript, 1994.Google Scholar
  81. 81.
    V. Padovani. Decidability of all minimal models. In S. Berardi and M. Coppo, editors, Types for Proof and Programs 1995, volume 1158 in Lecture Notes in Computer Science, pages 201–215. Springer-Verlag, 1996.Google Scholar
  82. 82.
    V. Padovani. Filtrage d’ordre supérieur. Thèse de Doctorat, Université de Paris VII, 1996.Google Scholar
  83. 83.
    Larry C. Paulson. Isabelle: The next 700 theorem provers. In P. Odifreddi, editor, Logic and computer science, pages 361–385. Academic Press, 1991.Google Scholar
  84. 84.
    F. Pfenning. Partial polymorphic type inference and higher-order unification. In Conference on Lisp and Functional Programming, pages 153–163, 1988.Google Scholar
  85. 85.
    F. Pfenning. Logic programming in the LF logical framework. In G. Huet and G. Plotkin, editors, Logical frameworks, pages 149–181. Cambridge University Press, 1991.Google Scholar
  86. 86.
    F. Pfenning. Unification and anti-unification in the calculus of constructions. In Logic in Computer Science, pages 74–85, 1991.Google Scholar
  87. 87.
    F. Pfenning and I. Cervesato. Linear higher-order pre-unification. In Logic in Computer Science, 1997.Google Scholar
  88. 88.
    Frank Pfenning and Carsten Schürmann. System description: Twelf — A metalogical framework for deductive systems. In Proceedings of the 16th Int. Conference on Automated Deduction (CADE-16), volume 1632 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1999.Google Scholar
  89. 89.
    T. Pietrzykowski. A complete mecanization of second-order type theory. J. of the Association for Computing Machinery, 20:333–364, 1973.MATHMathSciNetGoogle Scholar
  90. 90.
    G. Plotkin. Building-in equational theories. Machine Intelligence, 7:73–90, 1972.MathSciNetMATHGoogle Scholar
  91. 91.
    C. Prehofer. Decidable higher-order unification problems. In A. Bundy, editor, Conference on Automated Deduction, volume 814 of Lecture Notes in Artificial Intelligence, pages 635–649. Springer-Verlag, 1994.Google Scholar
  92. 92.
    C. Prehofer. Higher-order narrowing. In Logic in Computer Science (LICS’94, pages 507–516, 1994.Google Scholar
  93. 93.
    C. Prehofer. Solving higher-order equations: From logic to programming. Doctoral thesis, Technische Universität München. Technical Report TUM-19508, Institut für Informatik, TUM, München, 1995.Google Scholar
  94. 94.
    C. Prehofer. Solving Higher-Order Equations: From Logic to Programming. Progress in theoretical coputer science. Birkhäuser, 1998.Google Scholar
  95. 95.
    S. Pulman. Higher-order unification and the interpretation of focus. Linguistics and Philosophy, 20:73–115, 1997.CrossRefGoogle Scholar
  96. 96.
    D. Pym. Proof, search, and computation in general logic. Doctoral thesis, University of Edinburgh, 1990.Google Scholar
  97. 97.
    Z. Qian. Higher-order equational logic programming. In Principle of Programming Languages, pages 254–267, 1994.Google Scholar
  98. 98.
    Z. Qian and K. Wang. Higher-order equational E-unification for arbitrary theories. In K. Apt, editor, Joint International Conference and Symposium on Logic Programming, 1992.Google Scholar
  99. 99.
    Z. Qian and K. Wang. Modular AC unification of higher-order patterns. In J.-P. Jouannaud, editor, International Conference on Constaints in Computational Logic, volume 845 of Lecture Notes in Computer Science, pages 105–120. Springer-Verlag, 1994.CrossRefGoogle Scholar
  100. 100.
    J.A. Robinson. New directions in mechanical theorem proving. In A.J.H. Morrell, editor, International Federation for Information Processing Congress, 1968, pages 63–67. North Holland, 1969.Google Scholar
  101. 101.
    J.A. Robinson. A note on mechanizing higher order logic. Machine Intelligence, 5:123–133, 1970.Google Scholar
  102. 102.
    H. Saïdi. Résolution d’équations dans le système T de Gödel. Mémoire de DEA, Université de Paris VII, 1994.Google Scholar
  103. 103.
    M. Schmidt-Schauß. Unification of stratified second-order terms. Technical Report 12, J.W.Goethe-Universität, Frankfurt, 1994.Google Scholar
  104. 104.
    M. Schmidt-Schauß. Decidability of bounded second order unification. Technical Report 11, J.W.Goethe-Universität, Frankfurt, 1999.Google Scholar
  105. 105.
    M. Schmidt-Schauß and K.U. Schulz. Solvability of context equations with two context variables is decidable. In H. Ganzinger, editor, Conference on Automated Deduction, volume 1632 in Lecture Notes in Artificial Intelligence, pages 67–81, 1999.Google Scholar
  106. 106.
    A. Schubert. Linear interpolation for the higher order matching problem. In M. Bidoit and M. Dauchet, editors, Theory and Practice of Software Development, volume 1214 of Lecture Notes in Computer science, pages 441–452. Springer-Verlag, 1997.CrossRefGoogle Scholar
  107. 107.
    A. Schubert. Second-order unification and type inference for Church-style polymorphism. In Principle of Programming Languages, pages 279–288, 1998.Google Scholar
  108. 108.
    W. Snyder. Higher-order E-unification. In M. E. Stickel, editor, Conference on Automated Deduction, volume 449 of Lecture Notes in Artificial Intelligence, pages 573–587. Springer-Verlag, 1990.Google Scholar
  109. 109.
    W. Snyder and J. Gallier. Higher order unification revisited: Complete sets of tranformations. Journal of Symbolic Computation, 8(1 & 2):101–140, 1989. Special issue on unification. Part two.MathSciNetMATHCrossRefGoogle Scholar
  110. 110.
    W. Snyder and J. Gallier. Higher-order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8:101–140, 1989.MathSciNetCrossRefMATHGoogle Scholar
  111. 111.
    J. Springintveld. Algorithms for type theory. Doctoral thesis, Utrecht University, 1995.Google Scholar
  112. 112.
    J. Springintveld. Third-order matching in presence of type constructors. In M. Dezani-Ciancagliani and G. Plotkin, editors, Typed Lambda Calculi and Applications, volume 902 of Lecture Notes in Computer Science, pages 428–442. Springer-Verlag, 1995.CrossRefGoogle Scholar
  113. 113.
    J. Springintveld. Third-order matching in the polymorphic lambda calculus. In G. Dowek, J. Heering, K. Meinke, and B. Mßoller, editors, Higher-order Algebra, Logic and Term Rewriting, volume 1074 of Lecture Notes in Computer Science, pages 221–237. Springer-Verlag, 1995.Google Scholar
  114. 114.
    Fernando C.N. Pereira Stuart, M. Shieber, and Mary Dalrymple. Interactions of scope and ellipsis. Linguistics and Philosophy, 19:527–552, 1996.CrossRefGoogle Scholar
  115. 115.
    D.A. Wolfram. The clausal theory of types. Doctoral thesis, University of Cambridge, 1989.Google Scholar
  116. 116.
    M. Zaionc. The regular expression description of unifier set in the typed λ-calculus. Fundementa Informaticae, X:309–322, 1987.MathSciNetGoogle Scholar
  117. 117.
    M. Zaionc. Mechanical procedure for proof construction via closed terms in typed λ-calculus. Journal of Automated Reasoning, 4:173–190, 1988.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Gérard Huet
    • 1
  1. 1.INRIA-RocquencourtLe Chesnay Cedex

Personalised recommendations