On the Uniform Computational Content of the Baire Category Theorem

TL;DR

This paper analyzes the computational content of various forms of the Baire Category Theorem within the Weihrauch lattice, revealing differences in their computational complexity and logical interpretations.

Contribution

It distinguishes between logical versions and input representations of the theorem, showing their different computational behaviors and connections to genericity.

Findings

The two logical versions are not Weihrauch equivalent.

Input information type affects computational complexity via the jump operation.

Connections to genericity and comeager sets are established.

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 how the sequence of closed sets is "given". Essentially, we can distinguish between positive and negative information on closed sets. We discuss all the 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.