Skip Navigation


Journal of Logic and Computation Advance Access originally published online on August 30, 2008
Journal of Logic and Computation 2009 19(1):159-174; doi:10.1093/logcom/exn033
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Korovina, M.
Right arrow Articles by Kudinov, O.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

This article appears in the following Journal of Logic and Computation issue: Special Issue: Logic and Computation in the Real World: CiE 2007 [View the issue table of contents]

Original Articles

The Uniformity Principle for {Sigma}-definability

Margarita Korovina

Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA), School of Mathematics, The University of Manchester, UK and IIS SB RAS, Novosibirsk, Russia
E-mail: Margarita.korovina{at}manchester.ac.uk

Oleg Kudinov

Sobolev Institute of Mathematics, Novosibirsk, Russia
E-mail: kud{at}math.nsc.ru

Received 24 October 2007.

This article is an extended version of the paper published in Korovina and Kudinov (2007, Lecture Notes in Computer Science, Vol. 4497, pp. 416–425). The main goal of this research is to develop logical tools and techniques for effective reasoning about continuous data based on {Sigma}-definability. In this article we invent the Uniformity Principleand prove it for {Sigma}-definability over the real numbers extended by open predicates. Using the Uniformity Principle, we investigate different approaches to enrich the language of {Sigma}-formulas in such a way that simplifies reasoning about computable continuous data without enlarging the class of {Sigma}-definable sets. In order to do reasoning about computability of certain continuous data we have to pick up an appropriate language of a structure representing these continuous data. We formulate several major conditions how to do that in a right direction. We also employ the Uniformity Principleto argue that our logical approach is a good way for formalization of computable continuous data in logical terms.

Keywords: {Sigma}-Definability; Uniformity Principle; effective reasoning about continuous data; continuous data types; computable analysis



References

  1. Barwise J. Admissible Sets and Structures. (1975) Berlin: Springer Verlag.
  2. Brattka V, Weihrauch K. Computability on subsets of euclidean space I: closed and compact sets. Theoretical Computer Science (1999) 219:65–93.[CrossRef][Web of Science]
  3. Ershov Yu L. Definability and Computability. (1996) New-York: Plenum.
  4. Gherardi G. Some Results in Computable Analysis and Effective Borel Measurability. (2006) Siena: University of Siena. PhD thesis.
  5. Korovina M, Kudinov O. The Uniformity Principle for {Sigma}-definability with applications to computable analysis. In. In: Lecture Notes in Computer Science.—Cooper SB, Löwe B, Sorbi A, eds. (2007) 4497. Berlin/Heidelberg: Springer. 416–425. CiE’07.[CrossRef][Web of Science]
  6. Korovina MV, Kudinov OV. Towards computability of higher type continuous data. In. In: Lecture Notes in Computer Science.—Barry Cooper S, Löwe Benedikt, Torenvliet Leen, eds. (2005) 3526. Berlin/Heidelberg: Springer. 235–241. CiE.[Web of Science]
  7. Korovina MV. Computational aspects of sigma-definability over the real numbers without the equality test. In. In: Lecture Notes in Computer Science.—Baaz M, Makowsky JA, eds. (2003) 2803. Berlin/Heidelberg: Springer. 330–344. CSL.[Web of Science]
  8. Korovina MV. Gandy's theorem for abstract structures without the equality test. In. In: Lecture Notes in Computer Science.—Vardi MY, Voronkov A, eds. (2003) 2850. Berlin/Heidelberg: Springer. 290–301. LPAR.
  9. Korovina MV, Kudinov OV. Semantic characterisations of second-order computability over the real numbers. In. In: Lecture Notes in Computer Science.—Fribourg L, ed. (2001) 2142. Berlin/Heidelberg: Springer. 160–172. CSL.[CrossRef]
  10. Korovina MV, Kudinov OV. Characteristic properties of majorant-computability over the reals. In. In: Lecture Notes in Computer Science.—Gottlob G, Grandjean E, Seyr K, eds. (1998) 1584. Berlin/Heidelberg: Springer. 188–203. CSL.
  11. Weihrauch K. Computable Analysis. (2000) Berlin: Springer Verlag.

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Korovina, M.
Right arrow Articles by Kudinov, O.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?