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## 2010.6: Remarks on \$\Sigma\$-definability without the equality test over the Reals

2010.6: Andrei Morozov and Margarita Korovina (2008) Remarks on \$\Sigma\$-definability without the equality test over the Reals. Electronic Notes in Theoretical Computer Science, 202. pp. 305-313. ISSN 1571-0661

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## Abstract

In this paper we study properties of Σ–definability over the reals without the equality test which is one of the main concepts in the logical approach to computability over continuous data. It has been shown that a set \$B\subset R^n\$ is Σ–definable without the equality test if and only if \$ B \$ is c.e. open. If we allow the equality test, the structur\$e of a Σ–definable subset of \$R^n\$ can be rather complicated. The next natural question to consider is the following. Is there an effective procedure producing a set which is a maximal c.e. open subset of a given Σ–definable with the equality subset of \$R^n\$? It this paper we give the negative answer to this question.

Item Type: Article CICADA, the real numbers Σ–definability, computably enumerable open sets MSC 2000 > 03 Mathematical logic and foundationsMSC 2000 > 68 Computer science 2010.6 Dr Margarita Korovina 08 January 2010