Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T12:42:57.268Z Has data issue: false hasContentIssue false
  • English
  • Français

The logic of recursive equations

Published online by Cambridge University Press:  12 March 2014

A. J. C. Hurkens ,
Monica McArthur ,
Yiannis N. Moschovakis ,
Lawrence S. Moss  and
Glen T. Whitney
A. J. C. Hurkens
Affiliation:
Klaasstokseweg 7, 5443 NS HAPS, The Netherlands, E-mail: hurkens@sci.kun.nl
Monica McArthur
Affiliation:
Department of Mathematics, Indiana University, Bloomington IN 47405, USA, E-mail: mcarthur@iu-math.math.indiana.edu
Yiannis N. Moschovakis
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90024, USA, E-mail: ynm@math.ucla.edu
Lawrence S. Moss
Affiliation:
Departments of Mathematics and Computer Science, Indiana University, Bloomington IN 47405, USA, E-mail: lmoss@ucs.indiana.edu
Glen T. Whitney
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, E-mail: glen@rentec.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in “propositional” fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The “pure,” propositional fragment FLR0 turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 451 - 478
Copyright
Copyright © Association for Symbolic Logic 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bloom, Stephen L. and Ésik, Zoltán, Iteration theories: The equational logic of iterative processes, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin, 1993.Google Scholar
[2] Bloom, Stephen L. and Ésik, Zoltán, Solving polynomial fixed-point equations, Mathematical foundations of computer science '94, Lecture Notes in Computer Science, no. 841, Springer-Verlag, Berlin, 1994.Google Scholar
[3] Bloom, Stephen L. and Ésik, Zoltán, Some quasi-varieties of iteration theories, Mathematical foundations of programming semantics '93, Lecture Notes in Computer Science, no. 802, Springer-Verlag, Berlin, 1994.Google Scholar
[4] Courcelle, B., Kahn, G., and Vuillemin, J., Algorithmes a'equivalence et de reduction a des expressions minimales dans une classe d'equations recursives simples, Automata, languages, and programming, 2nd colloquium (Loeckx, J., editor), Lecture Notes in Computer Science, no. 14, Springer-Verlag, Berlin, 1974.Google Scholar
[5] de Bakker, J. W., Recursive procedures, Mathematical Centre Tracts 24, Mathematisch Centrum Amsterdam, Amsterdam, 1971.Google Scholar
[6] de Barker, J. W. and de Roever, W. P., A calculus for recursive program schemes, Automata, languages, and programming (Nivat, M., editor), North-Holland, Amsterdam, 1973.Google Scholar
[7] Elgot, C. C., Monadic computation and iterative algebraic theories, Logic colloquium 1973 (Shepherdson, J. C., editor), Studies in Logic, no. 80, North-Holland, Amsterdam, 1975.Google Scholar
[8] Ésik, Z. and Bernátsky, L., Scott induction and equational proofs, Mathematical foundations of programming semantics '95, Electronic Notes in Theoretical Computer Science, no. 1, 1995.Google Scholar
[9] Ésik, Zoltán, Completeness of park induction, Theoretical Computer Science , to appear.Google Scholar
[10] Ésik, Zoltán, Identities in iterative and rational algebraic theories, Comptational Linguistics and Computer Languages, vol. 14 (1980), pp. 183–207.Google Scholar
[11] Ivanov, L. L., Algebraic recursion theory, Ellis Horwood Limited, Chichester, 1986.Google Scholar
[12] Ivanov, L. L., Categorial generalization of algebraic recursion theory, Journal of Pure and Applied Algebra, vol. 101 (1995), pp. 91–128.Google Scholar
[13] Kahn, G., The semantics of a simple language for parallel programming, Information processing 74 (Rosenfeld, J. L., editor), North-Holland, Amsterdam, 1974.Google Scholar
[14] Lawvere, F. W., Functorial semantics of algebraic theories, Proceedings of the National Academy of Sciences of the United States of America, vol. 50 (1963), pp. 869–872.Google Scholar
[15] Manna, Z. and Vuillemin, J., Fixpoint approach to the theory of computation, Automata, languages, and programming (Nivat, M., editor), North-Holland, Amsterdam, 1973.Google Scholar
[16] Milner, R., Communication and concurrency, Prentice-Hall, 1989.Google Scholar
[17] Moschovakis, Y. N., The logic of functional recursion, to appear in Proceedings of the 10th International Congress of Logic, Methodology and the Philosophy of Science, held in Florence, 1995.Google Scholar
[18] Moschovakis, Y. N., The formal language of recursion, this Journal, vol. 54 (1989), pp. 1216–1252.Google Scholar
[19] Moschovakis, Y. N., A model of concurrency with fair merge and full recursion, Information and Computation, vol. 93 (1991), pp. 114–171.Google Scholar
[20] Moschovakis, Y. N. and Whitney, G. T., Powerdomains, powerstructures, and fairness, Proceedings of computer science logic '94, Kasimierz, Poland (Pacholski, L. and Tiuryn, J., editors), Lecture Notes in Computer Science, no. 933, Springer-Verlag, Berlin, 1995, pp. 382–396.Google Scholar
[21] Skordev, D. G., Computability in combinatory spaces, Kluwer Academic Publishers, Dordrecht, 1992.Google Scholar
[22] Stoy, J. E., Denotational semantics: The Scott-Strachey approach to programming language theory, MIT Press, Cambridge, MA, 1977.Google Scholar
[23] Whitney, G. T., Recursion structures for non-determinism and concurrency, Ph.D. thesis , University of California, Los Angeles, 1994.Google Scholar