Algebraically closed theories

  • Eric Badouel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 379)


The present work takes place in the study of infinitary behaviours for CCS-like communicating processes. A problem in that area arises from the fact that most of the abstraction morphisms we are interested in don't commute with least fixed points. In order to offer an alternative to least fixed point semantics we present an axiomatization of the notion of fixed point calculus within the formalism of algebraic theories. Such a calculus fixes one solution for each equation resulting from the interpretation of a set of recursive definitions in a way consistent with the free interpretation of the equations. This leads us to the notion of algebraically closed theory which stands for an algebraic theory equipped with a fixed point calculus. The rational theories by ADJ appear to be a special case of algebraically closed theories when least solutions are always chosen.


Operator Symbol Rational Expression Rational Theory Algebraic Theory Recursive Definition 
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  1. [ADJ76]
    J.W. Thatcher, E.G. Wagner and J.B. Wright. Rational algebraic theories and fixed-point solutions. Proceeding 17th IEEE Symposium on foundations of computing, 1976.Google Scholar
  2. [Bad89]
    Eric Badouel. Algebraically closed theories. full version of the paper, to appear in INRIA report, 1989.Google Scholar
  3. [CKV74]
    B. Courcelle, G. Kahn and J. Vuillemin. Algorithmes d'équivalence et de réduction à des expressions minimales dans une classe d'équations récursives simples. Proc. 2nd ICALP, LNCS 14, 1974.Google Scholar
  4. [DG87]
    Ph. Darondeau and B. Gamatié. Modelling infinitary behaviours of communicating systems. INRIA report n 749, 1987.Google Scholar
  5. [Elg75]
    Calvin C. Elgot. Monadic computation and iterative algebraic theories. in Proceeding of logic colloquium Bristol, 1973 pp 175–230. North-Holland, Amsterdam, 1975.Google Scholar
  6. [Eyt84]
    Michel Eytan. La sémantique algébrique ADJ est une sémantique fonctorielle. Séminaire du LITP, janvier, 1984.Google Scholar
  7. [Law63]
    F. W. Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Scien. USA, 50(5), 1963.Google Scholar
  8. [Mil80]
    Robin Milner. A calculus of communicating systems. Springer Verlag LNCS, n 92, 1980.Google Scholar
  9. [MR77]
    M. Makkai and G.E. Reyes. First order categorical logic. Lecture notes in mathematics vol 611, Springer-Berlin, 1977.Google Scholar
  10. [Tiu77]
    Jerzy Tiuryn. Fixed points and algebras with infinitely long expressions part I. regular algebras in Proceedings MFCS 1977. Lecture note in computer science, 53, 1977.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Eric Badouel
    • 1
  1. 1.Irisa, Campus de BeaulieuRennes CedexFrance

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