Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Category Theory and Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 283))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Burstall R.M. (1980) Electronic Category Theory. Proc. Ninth Annual Symposium on the Mathematical Foundations of Computer Science. Rydzyua, Poland.

    Google Scholar 

  • 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 

  • 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 

  • Colmerauer A. et al. (1973) Etude et realisation d'un système PROLOG. Convention de Research IRIA-Sesori No. 77030.

    Google Scholar 

  • Eriksson L.H. (1984) Synthesis of a Unification Algorithm in a Logic Programming Calculus. Journal of Logic Programming, 1,1.

    Google Scholar 

  • Fay M. (1979) First-Order Unification in an Equational Theory. 4th Workshop on Automated Deduction, Texas.

    Google Scholar 

  • Goldblatt R. (1979) Topoi: The Categorial Analysis of Logic. Studies in Logic and the Foundations of Mathematics, Vol. 98, North-Holland.

    Google Scholar 

  • Gordon M.J.C., Milner R. and Wadsworth C.P. (1979) Edinburgh LCF. LNCS Lecture Notes in Computer Science, Springer-Verlag. 78.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • Huet G. (1980) Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems. J. ACM 27,4 pp 797–821.

    Google Scholar 

  • 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 

  • Hullot J.-M. (1980) Canonical Forms and Unification. 5th Conference on Automated Deduction, Les Arcs, France. LNCS 87, Springer-Verlag.

    Google Scholar 

  • 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 

  • Kelly G.M. (1982) Basic Concepts of Enriched Category Theory. Cambridge University Press.

    Google Scholar 

  • 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 

  • Kirchner C. (1987) Methods and Tools for Equational Unification. Internal report 87-R-008, Centre de Recherche en Informatique de Nancy.

    Google Scholar 

  • Kleisli H. (1965) Every Standard Construction is Induced by a Pair of Adjoint Functors. Proc. Am. Maths. Soc. 16. pp. 544–546.

    Google Scholar 

  • Lawvere F.W. (1963) Functorial Semantics of Algebraic Theories. Proc. Nat. Acad. of Sciences. 50. pp 869–872.

    Google Scholar 

  • 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 

  • 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 

  • Mac Lane S. (1971) Categories for the Working Mathematician. Springer-Verlag, New York.

    Google Scholar 

  • MacQueen D. (1984) Modules for Standard ML. Proc. ACM Conf. on LISP and Functional Prog. Languages.

    Google Scholar 

  • Manna Z. and Waldinger R. (1980) Deductive Synthesis of The Unification Algorithm. S.R.I. Research Report.

    Google Scholar 

  • Martelli A. and Montanari U. (1982) An Efficient Unification Algorithm. ACM Trans. on Prog. Languages and Systems, 4, 2.

    Google Scholar 

  • Milner R. (1978) A theory of type polymorphism in programming. J. Comp. Sys. Sci. 17, 3. pp. 348–375.

    Google Scholar 

  • Milner R. (1984) A Proposal for Standard ML. Proc. ACM Symp. on LISP and Functional Programming.

    Google Scholar 

  • Paterson M.S. and Wegman M.N. (1978) Linear Unification. J. Comp. Sys. Sci. 16, 2. pp. 158–167.

    Google Scholar 

  • Paulson L.C. Verifying the Unification Algorithm in LCF. Science of Computer Programming 5.

    Google Scholar 

  • Robinson J.A. (1965) A machine-oriented logic based on the resolution principle. J. ACM 12,1. pp. 23–41.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Rydeheard D.E. and Burstall R.M. (1988) Computational Category Theory. To appear, Prentice-Hall.

    Google Scholar 

  • Seely R.A.G. (1983) Hyperdoctrines, natural deduction and the Beck condition. Zeit. fur Math. Logik. 29 pp 505–542.

    Google Scholar 

  • 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 

  • 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 

  • Smyth M.B. and Plotkin G.D. (1977) The Category-Theoretic Solution of Recursive Domain Equations. Proc. Foundations of Computer Science.

    Google Scholar 

  • Street R. (1976) Limits indexed by Category-Valued 2-Functors. J. Pure and Applied Algebra 8. pp 149–181.

    Google Scholar 

  • Taylor P. (1987) Recursive Domains, Indexed Categories and Polymorphism. PhD. Thesis, Dept. Pure Math. and Math. Statistics, University of Cambridge.

    Google Scholar 

  • 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 

  • 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 

Download references

Author information

Authors and Affiliations

  1. University of Manchester, UK

    D. E. Rydeheard & J. G. Stell

Authors
  1. D. E. Rydeheard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. G. Stell
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

David H. Pitt Axel Poigné David E. Rydeheard

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rydeheard, D.E., Stell, J.G. (1987). Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms. In: Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds) Category Theory and Computer Science. Lecture Notes in Computer Science, vol 283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18508-9_23

Download citation

  • DOI: https://doi.org/10.1007/3-540-18508-9_23

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18508-6

  • Online ISBN: 978-3-540-48006-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation