Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms

  • D. E. Rydeheard
  • J. G. Stell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 283)


We provide a framework for equational deduction based on category theory. Firstly, drawing upon categorical logic, we show how the compositional structure of equational deduction is captured by a 2-category. Using this formulation, algorithms for solving equations are derived from general constructions in category theory. The basic unification algorithm arises from constructions of colimits. We also consider solving equations in the presence of term rewriting systems and the combination of unification algorithms.


Category Theory Equational Theory Deductive System Unification Algorithm Automate Deduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Burstall R.M. (1980) Electronic Category Theory. Proc. Ninth Annual Symposium on the Mathematical Foundations of Computer Science. Rydzyua, Poland.Google Scholar
  2. Burstall R.M. and Goguen J.A. (1981) An Informal Introduction to Specifications using Clear. In ‘The correctness problem in computer science', Eds. Boyer and Moore. Academic Press, London.Google Scholar
  3. Burstall R.M. and Landin P.J. (1969) Programs and Their Proofs: An Algebraic Approach. Machine Intelligence 4. Edinburgh Univ. Press. pp 17–44.Google Scholar
  4. Colmerauer A. et al. (1973) Etude et realisation d'un système PROLOG. Convention de Research IRIA-Sesori No. 77030.Google Scholar
  5. Eriksson L.H. (1984) Synthesis of a Unification Algorithm in a Logic Programming Calculus. Journal of Logic Programming, 1,1.Google Scholar
  6. Fay M. (1979) First-Order Unification in an Equational Theory. 4th Workshop on Automated Deduction, Texas.Google Scholar
  7. Goldblatt R. (1979) Topoi: The Categorial Analysis of Logic. Studies in Logic and the Foundations of Mathematics, Vol. 98, North-Holland.Google Scholar
  8. Gordon M.J.C., Milner R. and Wadsworth C.P. (1979) Edinburgh LCF. LNCS Lecture Notes in Computer Science, Springer-Verlag. 78.Google Scholar
  9. Harper R., Honsell F. and Plotkin G. (1987) A Framework for Defining Logics. Proc. Symposium on Logic in Computer Science. June 22–25, 1987, Ithaca N.Y. Publ. I.E.E.E.Google Scholar
  10. Herbrand J. (1930) Recherches sur la théorie de la démonstration. Thèse, U. de Paris. In: Ecrits logique de Jacques Herbrand, PUF Paris (1968).Google Scholar
  11. Herold A. (1986) Combination of Unification Algorithms. In Proceedings 8th Conference on Automated Deduction, Oxford. pp 450–469. LNCS Lecture Notes in Computer Science, Springer-Verlag. 230.Google Scholar
  12. Hewitt C. (1972) Description and Theoretical Analysis (Using Schemata) of PLANNER: A Language for Proving Theorems and Manipulating Models in a Robot. Ph.D. Dept. Maths. M.I.T. Cambridge. Mass.Google Scholar
  13. Huet G. (1976) Résolution d'équations dans les languages d'ordre 1,2,..., ω. Thèse d'etat, Specialité Maths. University of Paris VII.Google Scholar
  14. Huet G. (1980) Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems. J. ACM 27,4 pp 797–821.Google Scholar
  15. Huet G. and Oppen D.C (1980) Equations and Rewrite Rules: A Survey. In ‘Formal Languages: Perspectives and Open Problems'. Ed. R. Book, Academic Press.Google Scholar
  16. Hullot J.-M. (1980) Canonical Forms and Unification. 5th Conference on Automated Deduction, Les Arcs, France. LNCS 87, Springer-Verlag.Google Scholar
  17. Jouannaud J.-P., Kirchner C. and Kirchner H. (1983) Incremental Construction of Unification Algorithms in Equational Theories. 10th ICALP, LNCS Lecture Notes in Computer Science, Springer-Verlag. 154, Springer-Verlag.Google Scholar
  18. Kelly G.M. (1982) Basic Concepts of Enriched Category Theory. Cambridge University Press.Google Scholar
  19. Kelly G.M. and Street R. (1974) Review of the Elements of 2-Categories. Proc. Category Seminar, Sydney 1972/73. LNM Lecture Notes in Mathematics, Springer-Verlag. 420. Springer-Verlag.Google Scholar
  20. Kirchner C. (1987) Methods and Tools for Equational Unification. Internal report 87-R-008, Centre de Recherche en Informatique de Nancy.Google Scholar
  21. Kleisli H. (1965) Every Standard Construction is Induced by a Pair of Adjoint Functors. Proc. Am. Maths. Soc. 16. pp. 544–546.Google Scholar
  22. Lawvere F.W. (1963) Functorial Semantics of Algebraic Theories. Proc. Nat. Acad. of Sciences. 50. pp 869–872.Google Scholar
  23. Lawvere F.W. (1970) Equality in hyperdoctrines and the comprehension schema as an adjoint functor. In: A. Heller (ed.), Proc. New York Symposium on Applications of Categorical Logic. Amer. Math. Soc. pp 1–14.Google Scholar
  24. Levi G. and Sirovich F. (1975) Proving Program Properties, Symbolic Evaluation and Logical Procedural Semantics. In LNCS Lecture Notes in Computer Science, Springer-Verlag. 32. Math. Foundations of Computer Science. Springer-Verlag.Google Scholar
  25. Mac Lane S. (1971) Categories for the Working Mathematician. Springer-Verlag, New York.Google Scholar
  26. MacQueen D. (1984) Modules for Standard ML. Proc. ACM Conf. on LISP and Functional Prog. Languages.Google Scholar
  27. Manna Z. and Waldinger R. (1980) Deductive Synthesis of The Unification Algorithm. S.R.I. Research Report.Google Scholar
  28. Martelli A. and Montanari U. (1982) An Efficient Unification Algorithm. ACM Trans. on Prog. Languages and Systems, 4, 2.Google Scholar
  29. Milner R. (1978) A theory of type polymorphism in programming. J. Comp. Sys. Sci. 17, 3. pp. 348–375.Google Scholar
  30. Milner R. (1984) A Proposal for Standard ML. Proc. ACM Symp. on LISP and Functional Programming.Google Scholar
  31. Paterson M.S. and Wegman M.N. (1978) Linear Unification. J. Comp. Sys. Sci. 16, 2. pp. 158–167.Google Scholar
  32. Paulson L.C. Verifying the Unification Algorithm in LCF. Science of Computer Programming 5.Google Scholar
  33. Robinson J.A. (1965) A machine-oriented logic based on the resolution principle. J. ACM 12,1. pp. 23–41.Google Scholar
  34. Robinson J.A. and Wos L.T. (1969) Paramodulation and Theorem Proving in First-Order Theories with Equality. Machine Intelligence 4. American Elsevier. pp. 135–150.Google Scholar
  35. Rydeheard D.E. and Burstall R.M. (1985) The Unification of Terms: A Category-Theoretic Algorithm. Internal Report, Universities of Manchester and Edinburgh.Google Scholar
  36. Rydeheard D.E. and Burstall R.M. (1986) A Categorical Unification Algorithms. Proc. Summer Conf. on Category Theory and Computer Programming 1985. LNCS Lecture Notes in Computer Science, Springer-Verlag. 240.Google Scholar
  37. Rydeheard D.E. and Burstall R.M. (1988) Computational Category Theory. To appear, Prentice-Hall.Google Scholar
  38. Seely R.A.G. (1983) Hyperdoctrines, natural deduction and the Beck condition. Zeit. fur Math. Logik. 29 pp 505–542.Google Scholar
  39. Seely R.A.G. (1987) Modelling Computations: A 2-Categorical Framework. Proc. Symposium on Logic in Computer Science. June 22–25, 1987, Ithaca N.Y. Publ. I.E.E.E.Google Scholar
  40. Siekmann J.H. (1984) Universal Unification. In the 7th Internal. Conf. on Automated Deduction. LNCS Lecture Notes in Computer Science, Springer-Verlag. 170.Google Scholar
  41. Smyth M.B. and Plotkin G.D. (1977) The Category-Theoretic Solution of Recursive Domain Equations. Proc. Foundations of Computer Science.Google Scholar
  42. Street R. (1976) Limits indexed by Category-Valued 2-Functors. J. Pure and Applied Algebra 8. pp 149–181.Google Scholar
  43. Taylor P. (1987) Recursive Domains, Indexed Categories and Polymorphism. PhD. Thesis, Dept. Pure Math. and Math. Statistics, University of Cambridge.Google Scholar
  44. Tiden E. (1986) Unification in combinations of collapse-free theories with disjoint sets of function symbols. In Proceedings 8th Conference on Automated Deduction, Oxford. pp 431–449. LNCS Lecture Notes in Computer Science, Springer-Verlag. 230.Google Scholar
  45. Yelick K. (1985) Combining Unification Algorithms for Confined Regular Equational Theories. In Proc. First Intern. Conf. On Rewriting Techniques and Applications. Dijon, France. LNCS Lecture Notes in Computer Science, Springer-Verlag. 202 pp 365–380. Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • D. E. Rydeheard
    • 1
  • J. G. Stell
    • 1
  1. 1.University of ManchesterUK

Personalised recommendations