Abstract
This paper investigates some of the underlying axioms allowing the fixpoint-semantics approach to hold for tree-like recursion schemes. The notions of scheme and interpretation are generalized. The axioms satisfied by "algebraic theories" are shown to be adequate for the definition of the notion of an interpretation. It is also shown that in order to provide the semantics of arbitrary finite recursion schemes, rational algebraic theories are insufficient and it is necessary to introduce a new class of "recursion-closed" algebraic theories. Finally, free recursion-closed algebraic theories are shown to exist.
This research has been partially supported by the National Science Foundation under Grant #MCS77-11360.
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References
Arnold, A., Systemes d'equations dans le magmoide. Ensembles Rationels et algebriques d'arbres, These d'Etat, Universite de Lille, (March 1977).
Arnold, A. and Dauchet, M., Theorie des Magmoides, Publications du Laboratoire de Calcul, Universite de Lille I, (January 1977).
Arnold, A. and Nivat, M., Nondeterministic Recursive Program Schemes. IRIA Technical Report No. 262, Domaine de Voluceau, Le Chesnay, France (November 1977).
Arnold, A. and Nivat, M., Algebraic semantics of nondeterministic recursive program schemes, Technical Report no. 78-4, Universite Paris VII, (Feb. 1978).
Burstall, R.M., and Thatcher, J.W., The algebraic theory of recursive program schemes, Symposium on Category Theory Applied to Computation and Control. Lecture Notes in Computer Science, Vol. 25, Springer-Verlag, New York, 1975, 126–131.
Courcelle, B., A Representation of trees by Languages, Part I and Part II, Theoretical Computer Science. Part I: Vol. 6, pp. 255–279; Part II: Vol. 7, pp. 25–55, (1978).
Courcelle, B., On the Definition of Classes of Interpretations, IRIA Technical Report No. 236, Domaine de Voluceau, Le Chesnay, France (May 1977).
Courcelle, B. and Nivat, M., Algebraic families of interpretations, Proc. 17th IEEE Symp. on Foundations of Comp. Sci., Houston, Texas (October 1976), 137–146.
Eilenberg, S. and Wright, J.B., Automata in general algebras, Inf. and Cont. 11 (1967), 452–470.
Elgot, C.C., Monadic computation and iterative algebraic theories. In H.E. Rose and J.C. Shepherdson (Eds.), Logic Colloquium '73, Studies in Logic, Vol. 80, North-Holland, Amsterdam, 1975, 175–230.
Elgot, C.C., Algebraic Theories and Program Schemes. Symposium on Semantics of Algorithmic Languages, (Ed. E. Engeler), Springer-Verlag (1971), 71–88.
Elgot, C.C., Structured programming with and without GOTO statements. IBM Research Report, RC-5626 (1975); IEEE Transactions on Software Engineering SE1 (1976), 41–53.
Ginali, S., Iterative Algebraic Theories, Infinite Trees and Program Schemata. Ph.D. Thesis, Department of Mathematics, University of Chicago, Chicago, Illinois, June 1976.
Goguen, J.A., Thatcher, J.W., Wagner, E.G., and Wright, J.B. An Introduction to Categories, Algebraic Theories and Algebras. IBM Report RC-5369, Yorktown Heights, New York (1975).
Goguen, J.A., Thatcher, J.W., Wagner, E.G., and Wright, J.B. Rational algebraic theories and fixed-point solutions. Proc. 17th IEEE Symp. on Foundations of Comp. Sci., Houston, Texas (October 1976). 147–158.
Goguen, J.A., Thatcher, J.W., Wagner, E.G., and Wright, J.B. Initial algebra semantics and continuous algebras, JACM 24 (1977), 68–95.
Guessarian, I. Schemas de Programmes Recursif Polyadiques: Equivalence Semantiques et Classes d'Interpretations. These d'Etat, Universite de Paris VII, 1975.
Manes, E.G. Algebraic Theories. Graduate Texts in Mathematics, Vol. 26, Springer-Verlag, New York, 1976.
Mezei, J. and Wright, J.B. Algebraic automata and context-free sets, Inf. and Cont. 11 (1967), 3–29.
Nivat, M. On the interpretation of recursive polyadic program schemes, Symposia Mathematica, Vol. 15, Academic Press, New York, 1975, 255–281.
Nivat, M. and Arnold, A., Calculs infinis, interpretations metriques et plus grands points fixes, Technical Report no. 78–19, Universite Paris VII, (May 1978).
Scott, D. Outline of a Mathematical Theory of Computation. Technical Monograph PRG-2, Oxford University Computing Laboratory, Programming Research Group (1970).
Scott, D. The Lattice of Flow Diagrams. Technical Monograph PRG-3, Oxford University Computing Laboratory, Programming Research Group (1971).
Scott, D. Continuous Lattices. Technical Monograph PRG-7, Oxford University Computing Laboratory, Programming Research Group (1971).
Scott, D. Data types as lattices, SIAM J. Comp. 5 (1976), 522–587.
Thatcher, J.W., Wagner, E.G. and Wright, J.B., Programming languages as mathematical objects, to appear in Mathematical Foundations of Computer Science, '78.
Thatcher, J.W., Wagner, E.G. and Wright, J.B., Free continuous theories, Technical Report RC 6906, IBM T.J. Watson Research Center, Yorktown Heights, New York, (December 1977).
Wagner, E.G. Languages for defining sets in arbitrary algebras: Proceedings, 12th IEEE Symposium on Switching and Automata Theory, East Lansing, Michigan (1971).
Wagner, E.G. An algebraic theory of recursive definitions and recursive languages. Proceedings, 3rd Annual ACM Symposium on Theory of Computing, Shaker Heights, Ohio (1971).
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Gallier, J.H. (1979). Recursion schemes and generalized interpretations. In: Maurer, H.A. (eds) Automata, Languages and Programming. ICALP 1979. Lecture Notes in Computer Science, vol 71. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09510-1_20
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DOI: https://doi.org/10.1007/3-540-09510-1_20
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