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

Morozov, Andrei and Korovina, Margarita (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
Uncontrolled Keywords:	CICADA, the real numbers Σ–definability, computably enumerable open sets
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations MSC 2010, the AMS's Mathematics Subject Classification > 68 Computer science
Depositing User:	Dr Margarita Korovina
Date Deposited:	08 Jan 2010
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1383

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