Notre Dame Journal of Formal Logic

On the Uniform Computational Content of the Baire Category Theorem

Vasco Brattka, Matthew Hendtlass, and Alexander P. Kreuzer

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Abstract

We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete (i.e., “large”) metric space cannot be decomposed into countably many nowhere dense (i.e., small) pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets.

Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 4 (2018), 605-636.

Dates
Received: 10 October 2015
Accepted: 21 September 2016
First available in Project Euclid: 13 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1539396028

Digital Object Identifier
doi:10.1215/00294527-2018-0016

Mathematical Reviews number (MathSciNet)
MR3871904

Subjects
Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]

Keywords
computable analysis Weihrauch lattice Baire category genericity reverse mathematics

Citation

Brattka, Vasco; Hendtlass, Matthew; Kreuzer, Alexander P. On the Uniform Computational Content of the Baire Category Theorem. Notre Dame J. Formal Logic 59 (2018), no. 4, 605--636. doi:10.1215/00294527-2018-0016. https://projecteuclid.org/euclid.ndjfl/1539396028


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