Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T12:33:55.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive predicate structures for continuous data†

Published online by Cambridge University Press:  13 November 2014

MARGARITA KOROVINA  and
OLEG KUDINOV
MARGARITA KOROVINA
Affiliation:
A.P. Ershov Institute of Informatics systems, SB RAS, NSU, Novosibirsk, Russia Email: [email protected]
OLEG KUDINOV
Affiliation:
Sobolev Institute of Mathematics, SB RAS, NSU, Novosibirsk, Russia Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we develop a general framework for continuous data representations using positive predicate structures. We first show that basic principles of Σ-definability which are used to investigate computability, i.e., existence of a universal Σ-predicate and an algorithmic characterization of Σ-definability hold on all predicate structures without equality. Then we introduce positive predicate structures and show connections between these structures and effectively enumerable topological spaces. These links allow us to study computability over continuous data using logical and topological tools.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 8: Computing with Infinite Data: Topological and Logical Foundations Part 2 , December 2015 , pp. 1669 - 1684
Copyright
Copyright © Cambridge University Press 2014 

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.)

Footnotes

†

This research was partially supported by EU project PIRSES-GA-2011-294962, grants DFG/RFBR 1267/14-1, 14-01-91334 and RFBR 13-01-0015_a.

References

Ajtai, M. (1989) First-order definability on finite structures. Annals of Pure and Applied Logic 45 211–225.Google Scholar
Barwise, J. (1975) Admissible Sets and Structures. Springer Verlag.CrossRefGoogle Scholar
Brattka, V. and Presser, G. (2003) Computability on subsets of metric spaces. Theoretical Computer Science 305 (1–3) 43–76.Google Scholar
Ershov, Y. L. (1996) Definability and Computability. Plenum, New York.Google Scholar
Grübba, T., Schröder, M. and Weihrauch, K. (2007) Computable metrization. Mathematical Logic Quaterly 53 (4–5) 595–604.Google Scholar
Hodges, W. (1993) The meaning of specifications: Set-theoretical specification. Semantics of Programming Languages and Model Theory, Algebra, Logic and Applications Series 5 43–68.Google Scholar
Immerman, N. (1999) Descriptive Complexity. Springer Verlag.CrossRefGoogle Scholar
Korovina, M. (2003) Gandy's theorem for abstract structures without the equality test. In: Vardi, Y. M. and Voronkov, A. (eds.) LPAR'03, Springer Lecture Notes in Computer Science 2850 290–301.CrossRefGoogle Scholar
Korovina, M. (2003) Computational aspects of Σ-definability over the real numbers without the equality test. In: Baaz, M. and Makowsky, J. A. (eds.) CSL'03, Springer Lecture Notes in Computer Science 2803 330–344.CrossRefGoogle Scholar
Korovina, M. and Kudinov, O. (2005) Towards computability of higher type continuous data. In: Cooper, S. B., Löwe, B. and Torenvliet, L. (eds) CiE'05, Springer Lecture Notes in Computer Science 3526 235–241.Google Scholar
Korovina, M. and Kudinov, O. (2007) The uniformity principle for Σ-definability with applications to computable analysis. In: Cooper, S. B., Löwe, B. and Sorbi, A. (eds.) CiE'07, Springer Lecture Notes in Computer Science 4497 416–425.Google Scholar
Korovina, M. and Kudinov, O. (2008) Towards Computability over Effectively Enumerable Topological Spaces. Elsevier Electronic Notes in Theoretical Computer Science 221 115–125.Google Scholar
Morozov, A. and Korovina, M. (2008) On Σ-definability without equality over the real numbers. Mathematical Logic Quarterly 54 (5) 535–544.Google Scholar
Moschovakis, Y. N. (1969) Abstract first order computability I, II. Transactions of the American Mathematical Society 138 427–504.Google Scholar
Moschovakis, Y. N. (1974) Elementary induction on abstract structures. Studies in Logic and the Foundations of Mathematics volume 77, North-Holland.Google Scholar
Moschovakis, Y. N. (1976) Recursive metric spaces. Fundamenta Mathematicae 55 215–238.CrossRefGoogle Scholar
Sazonov, V. (2001) Using agents for concurrent querying of web-like databases via hyper-set-theoretic approach. In: Bjørner, D., Broy, M. and Zamulin, A. V. (eds.) PSI'01, Springer Lecture Notes in Computer Science 2244 378–394.CrossRefGoogle Scholar
Weihrauch, K. (1993) Computability on computable metric spaces. Theoretical Computer Science 113 191–210.Google Scholar
Weihrauch, K. (2000) Computable Analysis. Springer Verlag.CrossRefGoogle Scholar