Abstract
Characteristic properties of majorant-computable real-valued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorant-computability proposed in [13]. A model-theoretical characterization of majorant-computability real-valued functions and their domains is investigated. A theorem which connects the graph of a majorant-computable function with validity of a finite formula on the set of hereditarily finite sets on \({\bar{\mathbb{R} }}\), \({\rm\bf HF}({\bar{\mathbb{R} }})\) (where \({\bar{\mathbb{R} }}\) is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the definition of majorant-computability and the notions of computability earlier proposed by Blum et al., Edalat, Sünderhauf, Pour-El and Richards, Stoltenberg-Hansen and Tucker is given. Examples of majorant-computable real-valued functions are presented.
This research was supported in part by the RFBR (grant N 96-15-96878) and by the Siberian Division of RAS (a grant for young researchers, 1997)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barwise, J.: Admissible sets and structures. Springer, Berlin (1975)
Bochnak, J., Coster, M., Roy, M.-F.: Géometrie algébrique réelle. Springer, Heidelberg (1987)
Bitsadze, A.V.: The bases of analytic function theory of complex variables, Nauka, Moskow (1972)
Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the reals: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.) 21(1), 1–46 (1989)
van den Dries, L.: Remarks on Tarski’s problem concerning (IR,+, ∗, exp). In: Proc. Logic Colloquium 1982, pp. 240–266 (1984)
Edalat, A., Sünderhauf, P.: A domain-theoretic approach to computability on the real line. Theoretical Computer Science (to appear)
Ershov, Y.L.: Definability and computability. Plenum, New York (1996)
Freedman, H., Ko, K.: Computational complexity of real functions. Theoret. Comput. Sci. 20, 323–352 (1982)
Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)
Jacobson, N.: Basic Algebra, vol. II. Freedman and Company, New York (1995)
Korovina, M.: Generalized computability of real functions. Siberian Advance of Mathematics 2(4), 1–18 (1992)
Korovina, M.: About an universal recursive function and abstract machines on the list superstructure over the reals. Vychislitel’nye Sistemy, Novosibirsk 156, 3–20 (1996)
Korovina, M., Kudinov, O.: A New Approach to Computability over the Reals. SibAM 8(3), 59–73 (1998)
Montague, R.: Recursion theory as a branch of model theory. In: Proc. of the third international congr. on Logic, Methodology and the Philos. of Sc., Amsterdam, pp. 63–86 (1968)
Moschovakis, Y.N.: Abstract first order computability. Trans. Amer. Math. Soc. 138, 427–464 (1969)
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1988)
Robinson, A.: Nonstandard Analysis. Studies in Logic and foundation of Mathematics. North-Holland, Amsterdam (1966)
Scott, D.: Outline of a mathematical theory of computation. In: 4th Annual Princeton Conference on Information Sciences and Systems, pp. 169–176 (1970)
Stoltenberg-Hansen, V., Tucker, J.V.: Effective algebras. Handbook of Logic in computer Science, vol. 4, pp. 375–526. Clarendon Press, Oxford (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Korovina, M.V., Kudinov, O.V. (1999). Characteristic Properties of Majorant-Computability Over the Reals. In: Gottlob, G., Grandjean, E., Seyr, K. (eds) Computer Science Logic. CSL 1998. Lecture Notes in Computer Science, vol 1584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10703163_14
Download citation
DOI: https://doi.org/10.1007/10703163_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65922-8
Online ISBN: 978-3-540-48855-2
eBook Packages: Springer Book Archive