The Uniformity Principle for Sigma-definability

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 Σ-definability. In this article we invent the Uniformity Principleand prove it for Σ-definability over the real numbers extended by open predicates. Using the Uniformity Principle, we investigate different approaches to enrich the language of Σ-formulas in such a way that simplifies reasoning about computable continuous data without enlarging the class of Σ-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.