The Uniformity Principle for Σ-Definability with Applications to Computable Analysis

Authors:
Margarita Korovina;Oleg Kudinov
Affiliations:
Institute of Informatics Systems, pr. Lavrenteva 6, 630090, Novosibirsk, Russia;Sobolev Institute of Mathematics, pr. Koptuga 4, 630090, Novosibirsk, Russia
Venue:
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Year:
2007

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Abstract

In this paper we prove the Uniformity Principle for Σ---definability over the real numbers extended by open predicates. Using this principle we show that if we have a ΣK-formula, i.e. a formula with quantifier alternations where universal quantifiers are bounded by computable compact sets, then we can eliminate all universal quantifiers obtaining a Σ-formula equivalent to the initial one. We also illustrate how the Uniformity Principle can be employed for reasoning about computability over continuous data in an elegant way.