Abstract
The order-theoretic enrichment of the algebraic approach to theoretical computer science, and to the theory of programming in particular, can be accomplished rather directly and elegantly, and appears to have a wide range of applications. This paper presents mathematical fundamentals of order-algebraic semantics. The principal new results concern free continuous algebraic theories and minimal conditions for solving general "systems of (recursive) equations" in algebraic theories.
Partial support received from the University of Colorado and Naropa Institute, both of Boulder, Colorado 80302.
This set of authors is herein, and we hope subsquently will be, referred to by the symbols ADJ.
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Bibliography
ADJ (Coauthored by J.A. Goguen, J.W. Thatcher, E.G. Wagner and J.B. Wright.) (1975) "Initial algebra semantics and continuous algebras," IBM Research Report RC 5701, November 3, 1975. To appear, JACM.
ADJ (Coauthored by J.A. Goguen, J.W. Thatcher, E.G. Wagner and J.B. Wright.) (1976) "A junction between computer science and category theory: I, Basic definitions and examples," Part 1, IBM Research Report RC 4526 (September 1973); Part 2, IBM Research Report 5908 (March 1976); Part 3 to appear.
ADJ (Coauthored by J.A. Goguen, J.W. Thatcher, E.G. Wagner and J.B. Wright.) (1976a) "Rational algebraic theories and fixed-point solutions," submitted for presentation; IBM Research Report to appear.
Arden, D.N. (1961) "Delayed-logic and finite-state machines," Course Notes 6.531, E.E.Dept., MIT, Summer, 1961.
Bekic, H. (1969) "Definable operations in general algebra, and the theory of automata and flowcharts," Report IBM Laboratory Vienna, (1969).
Berry, G. (1975) "Bottom up computation of recursive programs," IRIA Research Report 133, September 1975.
Birkhoff, G. (1967) Lattice Theory, Amer. Math. Soc. Colloq. Pub. 25 New York (1948). (revised edition) (1967).
Blikle, A. (1971) "Nets, complete lattices with a composition," Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Math. Astr., et Phys. 19 (1971) 1123–1127.
Blikle, A. (1973) "Equations in nets, complete oriented lattices," CCPAS Report No. 99, Warsaw (1973).
Blikle, A. (1974) "An extended approach to mathematical analysis of programs," CCPAS Report No. 169, Warsaw (1974).
Bloom, S.L. and Elgot, C.C. (1974) "The existence and construction of free iterative theories," IBM Research Report RC-4937 (1974). To appear JCSS.
Burstall, R.M. and Thatcher, J.W. "The algebraic theory of recursive program schemes," Proceedings AAAS Symposium on Category Theory Applied to Computation and Control, Univ. of Mass. Press, Amherst (1974); Lecture Notes in Computer Science 25 (1975) 126–131.
de Bakker, J.W. and Scott, D. (1969) "A theory of programs," unpublished notes, IBM Seminar, Vienna (1969).
de Roever, W.P. (1974) "Operational, mathematical and axiomatized semantics for recursive procedures and data structures," Mathematical Centre Report ID 1/74 (1974).
Eilenberg, S. and Wright, J. (1967) "Automata in general algebras," Information and Control 11 (1967) 52–70.
Elgot, C.C. (1970) "The common algebraic structure of exit-automta and machines," Computing 6 (1970) 349–370.
Elgot, C.C. (1971) "Algebraic theories and program schemes," Symp. on Semantics of Algorithmic Languages, (Ed. E.Engeler), Springer-Verlag (1971) 71–88.
Elgot, C.C. (1972) "Remarks on one-argument program schemes," Formal Semantics of Programming Languages, (Ed. R. Rustin), Prentice-Hall, N.J. (1972) 59–64.
Elgot, C.C. (1973) "Monadic computation and iterative algebraic theories," IBM Research Report RC 4564, October 1973. Proceedings, Logic Colloquium '73, North Holland Publishing Company (1975) 175–230.
Elgot, C.C. (1974) See Bloom and Elgot (1974).
(1975) "Structured programming with and without GO TO statements," IBM Research Report RC 5626 (1975). IEEE Trans.on Software Eng. SE-2 (1976) 41–54.
Englefriet, J. (1974) "Simple program schemes and formal languages," Lecture Notes in Computer Science 20, Springer-Verlag (1974).
Engelfriet, J. and Schmidt, E.M. (1975) "IO and OI," Datalogisk Afdeling Report DAIMI PB-417, Aarhus University, Denmark, July, 1975.
Ginsburg, S. (1966) The Mathematical Theory of Context-Free Languages, McGraw-Hill, N.Y.(1966).
Ginali, S. (1976) Ph.D. Dissertation, University of Chicago, forthcoming.
Goguen, J.A. (1974) "On homomorphisms, correctness, termination, unfoldments, and equivalence of flow diagram programs," J. of Comp. and Sys. Sci. 8 (1974) 333–365.
Herrlich, H. and Strecker, C.E. (1973) Category Theory, Allyn and Bacon (1973)
Karp, R.M. (1959) "Some applications of logical syntax to digital computer programming," Harvard University Thesis (1959).
Lawvere, F.W. (1963) "Functional semantics of algebraic theories," Proceedings, Nat'l Acad. Sci. 50 (1963) 869–872.
Mac Lane, S. (1971a) Category Theory for the Working Mathematician, Springer-Verlag (1971).
Mezei, J. and Wright, J.B. (1967) "Algebraic automata and context-free sets," Information and Control 11 (1967) 3–29.
Mitchell, B. (1965) Theory of Categories, Academic Press, New York (1965).
Pareigis, B. (1970) Categories and Functors, Academic Press, New York (1970).
Park, D.M.R. (1969) "Fixpoint induction and proofs of program properties," Machine Intelligence 5 (Eds. B.Meltzer and D.Michie) Edinburgh Univ.Press (1969) 59–78.
Scott, D. (1970) "Outline of a mathematical theory of computation," Proceedings, 4th Princeton Conf. on Inform. Science and Systems (1970).
Wagner, E.G. (1971) "Languages for defining sets in arbitrary algebras," Proceedings, 12th IEEE Symp. on Switching and Automata Th., E.Lansing, Mich. (1971).
Wagner, E.G. (1971a) "An algebraic theory of recursive definitions and recursive languages," Proceedings, 3rd ACM Symp. on Theory of Comp., Shaker Hghts, Ohio (1971).
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Wagner, E.G., Wright, J.B., Goguen, J.A., Thatcher, J.W. (1976). Some fundamentals of order-algebraic semantics. In: Mazurkiewicz, A. (eds) Mathematical Foundations of Computer Science 1976. MFCS 1976. Lecture Notes in Computer Science, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-07854-1_169
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DOI: https://doi.org/10.1007/3-540-07854-1_169
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