Free dynamics and algebraic semantics
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.
KeywordsFree Algebra Program Scheme Tree Automaton Algebraic Semantic Domain Equation
Unable to display preview. Download preview PDF.
- 1.ARBIB, M.A. and MANES, E.G.: Machines in a Category: An Expository Introduction, SIAM Review, 16 (1974), 163–192.Google Scholar
- 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.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.KNASTER, B.: Un Théoreme sur les Fonctions des Ensembles, Ann. Soc. Polon. Math., 6 (1928), 133–134.Google Scholar
- 5.TARSKI, A.: A Lattice-Theoretical Fixpoint Theorem and its Applications, Pacific J. Math. 5 (1955), 285–309.Google Scholar
- 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.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.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.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.MANES, E. Algebraic Theories, Berlin: Springer-Verlag (1976).Google Scholar