Free dynamics and algebraic semantics

  • Michael A. Arbib
Section B Computation Theory in Category
Part of the Lecture Notes in Computer Science book series (LNCS, volume 56)


Adámek has recently given general criteria for the construction of a free X-dynamics. We specialize this result to the case in which the free dynamics is over the initial object, noting that the result, μ0:A X → A is an isomorphism. This result not only specializes to the theory of minimal fixed points, but provides a new method of constructing solutions to Scott's domain equations which does not require coincidence of limits and colimits. Finally, we show how free dynamics allow us to construct semantics for programming schemes.


Free Algebra Program Scheme Tree Automaton Algebraic Semantic Domain Equation 
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. 1.
    ARBIB, M.A. and MANES, E.G.: Machines in a Category: An Expository Introduction, SIAM Review, 16 (1974), 163–192.Google Scholar
  2. 2.
    ADÁMEK, J.: Free Algebras and Automata Realizations in the Language of Categories, Commentationes Mathematicae Universitatis Carolinae, (Prague), 15 (1974), 589–602.Google Scholar
  3. 3.
    ARBIB, M.A.: Categorical Notes on Scott's Theory of Computation, Proc. Conf. on Categorical and Algebraic Methods in Computer Science and System Theory, Dortmund, November 1976.Google Scholar
  4. 4.
    KNASTER, B.: Un Théoreme sur les Fonctions des Ensembles, Ann. Soc. Polon. Math., 6 (1928), 133–134.Google Scholar
  5. 5.
    TARSKI, A.: A Lattice-Theoretical Fixpoint Theorem and its Applications, Pacific J. Math. 5 (1955), 285–309.Google Scholar
  6. 6.
    SCOTT, D.: Continuous Lattices, in Toposes, Algebraic Geometry and Logic, (F.W. Lawvere, Ed.) Lecture Notes in Mathematics 274, Berlin: Springer-Verlag (1972) 97–136.Google Scholar
  7. 7.
    SMYTH, M.B.: Category-Theoretic Solution of Recursive Domain Equations, Theory of Computation Report No. 14, Dept. of Computer Science, University of Warwick, (1976).Google Scholar
  8. 8.
    BLOOM, S.L. and ELGOT, C.C.: The Existence and Construction of Free Iterative Theories, J. Comp. Syst. Sci. 12 (1976) 305–318.Google Scholar
  9. 9.
    GOGUEN, J.A., THATCHER, J.W., WAGNER, E.G. and WRIGHT, J.B.: Initial Algebra Semantics and Continuous Algebras, J. Assoc. Comp. Mach. 24 (1977) 68–95.Google Scholar
  10. 10.
    MANES, E. Algebraic Theories, Berlin: Springer-Verlag (1976).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • Michael A. Arbib
    • 1
  1. 1.Computer and Information ScienceUniversity of Massachusetts at AmherstUSA

Personalised recommendations