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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References

  • [1] Avigad, J., E. T. Dean, and J. Rute, “Algorithmic randomness, reverse mathematics, and the dominated convergence theorem,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 1854–64.
  • [2] Barmpalias, G., A. R. Day, and A. E. M. Lewis-Pye, “The typical Turing degree,” Proceedings of the London Mathematical Society (3), vol. 109 (2014), pp. 1–39.
  • [3] Bienvenu, L., and L. Patey, “Diagonally non-computable functions and fireworks,” Information and Computation, vol. 253 (2017), pp. 64–77.
  • [4] Brattka, V., “Computable invariance,” Theoretical Computer Science, vol. 210 (1999), pp. 3–20.
  • [5] Brattka, V., “Computable versions of Baire’s category theorem,” pp. 224–35 in Mathematical Foundations of Computer Science, 2001(Mariánské Lázně), edited by J. Sgall, A. Pultr, and P. Kolman, vol. 2136 of Lecture Notes in Computer Science, Springer, Berlin, 2001.
  • [6] Brattka, V., “Effective Borel measurability and reducibility of functions,” Mathematical Logic Quarterly, vol. 51 (2005), pp. 19–44.
  • [7] Brattka, V., “From Hilbert’s 13th problem to the theory of neural networks: constructive aspects of Kolmogorov’s superposition theorem,” pp. 253–80 in Kolmogorov’s Heritage in Mathematics, edited by É. Charpentier, A. Lesne, and N. Nikolski, Springer, Berlin, 2007.
  • [8] Brattka, V., M. de Brecht, and A. Pauly, “Closed choice and a uniform low basis theorem,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 986–1008.
  • [9] Brattka, V., and G. Gherardi, “Borel complexity of topological operations on computable metric spaces,” Journal of Logic and Computation, vol. 19 (2009), pp. 45–76.
  • [10] Brattka, V., and G. Gherardi, “Effective choice and boundedness principles in computable analysis,” Bulletin of Symbolic Logic, vol. 17 (2011), pp. 73–117.
  • [11] Brattka, V., and G. Gherardi, “Weihrauch degrees, omniscience principles and weak computability,” Journal of Symbolic Logic, vol. 76 (2011), pp. 143–76.
  • [12] Brattka, V., G. Gherardi, and R. Hölzl, “Probabilistic computability and choice,” Information and Computation, vol. 242 (2015), pp. 249–86.
  • [13] Brattka, V., G. Gherardi, and A. Marcone, “The Bolzano-Weierstrass theorem is the jump of weak Kőnig’s lemma,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 623–55.
  • [14] Brattka, V., M. Hendtlass, and A. P. Kreuzer, “On the uniform computational content of computability theory,” Theory of Computing Systems, vol. 61 (2017), pp. 1376–426.
  • [15] Brattka, V., and A. Pauly, “On the algebraic structure of Weihrauch degrees,” preprint, arXiv:1604.08348v7 [cs.LO].
  • [16] Brattka, V., and G. Presser, “Computability on subsets of metric spaces,” Theoretical Computer Science, vol. 305 (2003), pp. 43–76.
  • [17] Brown, D. K., and S. G. Simpson, “The Baire category theorem in weak subsystems of second-order arithmetic,” Journal of Symbolic Logic, vol. 58 (1993), pp. 557–78.
  • [18] Dorais, F. G., D. D. Dzhafarov, J. L. Hirst, J. R. Mileti, and P. Shafer, “On uniform relationships between combinatorial problems,” Transactions of the American Mathematical Society, vol. 368 (2016), pp. 1321–59.
  • [19] Downey, R. G., and D. R. Hirschfeldt, Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.
  • [20] Gherardi, G., and A. Marcone, “How incomputable is the separable Hahn-Banach theorem?,” Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 393–425.
  • [21] Hertling, P., “Unstetigkeitsgrade von Funktionen in der effektiven Analysis,” Ph.D. dissertation, Fernuniversität Hagen, Hagen, 1996.
  • [22] Hirschfeldt, D. R., Slicing the Truth: On the Computable and Reverse Mathematics of Combinatorial Principles, vol. 28 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, World Scientific, Singapore, 2015.
  • [23] Hirschfeldt, D. R., R. A. Shore, and T. A. Slaman, “The atomic model theorem and type omitting,” Transactions of the American Mathematical Society, vol. 361 (2009), pp. 5805–37.
  • [24] Kučera, A., “Measure, $\Pi^{0}_{1}$-classes and complete extensions of $\mathrm{PA}$,” pp. 245–59 in Recursion Theory Week (Oberwolfach, 1984), vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985.
  • [25] Kurtz, S. A., “Randomness and genericity in the degrees of unsolvability,” Ph.D. dissertation, University of Illinois at Urbana–Champaign, Urbana, Ill., 1981.
  • [26] Miller, J. S., “Pi-0-1 classes in computable analysis and topology,” Ph.D. dissertation, Cornell University, Ithaca, N.Y., 2002.
  • [27] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, New York, 2009.
  • [28] Nies, A., F. Stephan, and S. A. Terwijn, “Randomness, relativization and Turing degrees,” Journal of Symbolic Logic, vol. 70 (2005), pp. 515–35.
  • [29] Pauly, A., “How incomputable is finding Nash equilibria?,” Journal of Universal Computer Science, vol. 16 (2010), pp. 2686–710.
  • [30] Pauly, A., “On the (semi)lattices induced by continuous reducibilities,” Mathematical Logic Quarterly, vol. 56 (2010), pp. 488–502.
  • [31] Rumyantsev, A., and A. Shen, “Probabilistic constructions of computable objects and a computable version of Lovász local lemma,” Fundamenta Informaticae, vol. 132 (2014), pp. 1–14.
  • [32] Schröder, M., “Admissible representations for continuous computations,” Ph.D. dissertation, Fernuniversität Hagen, Hagen, 2002.
  • [33] Simpson, S. G., “Baire categoricity and $\Sigma^{0}_{1}$-induction,” Notre Dame Journal of Formal Logic, vol. 55 (2014), pp. 75–78.
  • [34] Stein, T. V., “Vergleich nicht konstruktiv lösbarer Probleme in der Analysis,” Ph.D. dissertation, Fernuniversität Hagen, Hagen, 1989.
  • [35] Tavana, N. R., and K. Weihrauch, “Turing machines on represented sets, a model of computation for analysis,” Logical Methods in Computer Science, vol. 7 (2011), no. 2:19.
  • [36] Weihrauch, K., “The degrees of discontinuity of some translators between representations of the real numbers,” technical report TR-92-050, International Computer Science Institute, Berkeley, Calif., 1992, http://www.icsi.berkeley.edu/pubs/techreports/tr-92-050.pdf.
  • [37] Weihrauch, K., “The TTE-interpretation of three hierarchies of omniscience principles,” Informatik Berichte 130, Fernuniversität Hagen, Hagen, 1992.
  • [38] Weihrauch, K., Computable Analysis, Springer, Berlin, 2000.