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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
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© 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.


  Abstract

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


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