Abstract
In this article the general form recursive equations (GFRE) are considered. A necessary and sufficient condition for these equations to have a solution in the family of partial recursive functions is found. We show that there exists such a GFRE that, in contrast with usual case, it has a non-computable solution but has no solution in the class P of partial recursive functions. The problem of solution existence to GFRE is shown to be Σ 03 -complete and Σ 11 -complete in the classes P and the class of total recursive functions, respectively.
This research was supported byAUA/ERC grant
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
H. P. Barendregt The Lambda Calculus. Its Syntax and Semantics. North-Holland Publ. Comp., Amsterdam, 1981.
D. B. Benson and I. Guessarian Algebraic solution to recursion schemata. Journal of Computer and System Sciences, 35:365–400, 1987.
N. Dershowitz and J.-P. Jouannaud Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 6, pages 243–320. Elsevier-MIT Press, Amsterdam, New York, Oxford, Tokyo, 1990.
J. P. Gallier and W. Snyder Designing unification procedures using transformations: A survey. Bulletin of the European Association for Theoretical Computer Science, 40:273–326, February 1990.
G. Huet and D. C. Oppen Equations and rewriting rules: A survey. In Formal Language Theory: Perspectives and Open Problems, pages 349–405. Academic Press, New York, 1980.
A. J. Kfoury, J. Tiuryn, and P. Urzyczyn Computational consequences and partial solutions of a generalized unification problem. In Proc. Fourth Annual Symposium on Logic in Computer Science, pages 98–105, Washington, 1989.
S. C. Kleene Introduction to Metamathematics. D. Van Nostrand Co., Inc., New York, Toronto, 1952.
S. C. Kleene Extension of an effectively generated class of functions by enumeration. Colloquium Mathematicum, VI:67–78, 1958.
G. Kreisel Interpretation of analysis by means of constructive functionals of finite types. In A. Heyting, editor, Constructivity in Mathematics, Studies in Logic, pages 101–128. North-Holland Publ. Co., Amsterdam, 1959.
H. B. Marandjian On recursive equations. In COLOG-88, Papers presented at the Int. Conf.on Computer Logic, Part II, pages 159–161, Tallinn, 1988.
H. B. Marandjian Selected Topics in Recursive Function Theory in Computer Science. DTH, Lyngby, 1990.
V. P. Orevkov Complexity of Proofs and Their Transformations in Axiomatic Theories, volume 128 of Transl. of Math. Monographs. American Mathematical Society, New York, 1993.
R. Parikh, A. Chandra, J. Halpern, and A. R. Meyer Equations between regular terms and an application to process logic. SIAM J. Comput., 14(4):935–942, 1985.
T. Pietrzykowski A complete mechanization of second-order type theory. JACM, 20(2):333–364, 1973.
A. Robinson. Equational logic of partial functions under Kleene equality: a complete and an incomplete set of rules. JSL, LIV(2):354–362, 1989.
H. Rogers. Jr. Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Company, New York, 1967.
L. Rudak. A completeness theorem for weak equational logic. Algebra Universalis, 16:331–337, 1983.
D. Scott Data types as lattices. SIAM J. Comput., 5(3):522–587, 1976.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Marandjian, H.B. (1995). General form recursive equations I. In: Pacholski, L., Tiuryn, J. (eds) Computer Science Logic. CSL 1994. Lecture Notes in Computer Science, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022278
Download citation
DOI: https://doi.org/10.1007/BFb0022278
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60017-6
Online ISBN: 978-3-540-49404-1
eBook Packages: Springer Book Archive