Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-20T02:56:26.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Wadge hardness in Scott spaces and its effectivization

Published online by Cambridge University Press:  14 November 2014

VERÓNICA BECHER  and
SERGE GRIGORIEFF
VERÓNICA BECHER
Affiliation:
FCEyN, Universidad de Buenos Aires & CONICET, Argentina Email: vbecher@dc.uba.ar
SERGE GRIGORIEFF
Affiliation:
LIAFA, CNRS & Université Paris Diderot - Paris 7, France Email: seg@liafa.univ-paris-diderot.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove some results on the Wadge order on the space of sets of natural numbers endowed with Scott topology, and more generally, on omega-continuous domains. Using alternating decreasing chains we characterize the property of Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated differences of open sets). A similar characterization holds for Wadge one-to-one and finite-to-one completeness. We consider the same questions for the effectivization of the Wadge relation. We also show that for the space of sets of natural numbers endowed with the Scott topology, in each class of the Hausdorff difference hierarchy there are two strictly increasing chains of Wadge degrees of sets properly in that class. The length of these chains is the rank of the considered class, and each element in one chain is incomparable with all the elements in the other chain.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 7: Computing with Infinite Data: Topological and Logical Foundations Part 1 , October 2015 , pp. 1520 - 1545
Copyright
Copyright © Cambridge University Press 2014 

References

Abramsky, S. and Jung, A. (1994) Domain theory. In: Gabbay, A and Maibaum (eds.) Handbook of Logic in Computer Science, volume 3, Oxford University Press.Google Scholar
Becher, V. and Grigorieff, S. (2004) Recursion and topology on 2 ≤ω for possibly infinite computations. Theoretical Computer Science 322 (1) 85–136.Google Scholar
Becher, V. and Grigorieff, S. (2005) Random reals and possibly infinite computations. part I: Randomness in ∅'. Journal of Symbolic Logic 70 (3) 891–913.Google Scholar
Becher, V. and Grigorieff, S. (2009) From index sets to randomness in ∅n. Random reals and possibly infinite computations, part II. Journal of Symbolic Logic 74 (1) 124–156.Google Scholar
Becher, V. and Grigorieff, S. Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization. This volume.Google Scholar
Brattka, V. and Hertling, P. (1994) Continuity and computability of relations. Informatif Berichte 164, Fern Universität Gesamthochschule in Hagen, Fachbereich Informatik.Google Scholar
de Brecht, M. (2011) Quasi-Polish spaces. ArXiv:1108.1445v1, 40 pages.Google Scholar
Edalat, A. (1997) Domains for computation in mathematics, physics and exact real aritmetic. Bulletin of Symbolic Logic 3 (4) 401–452.Google Scholar
Ershov, Y. (1968) On a hierarchy of sets II. Algebra and Logic 7 (4) 15–47.Google Scholar
Fokina, E. B., Friedman, S.-D. and Tornquist, A. (2010) The effective theory of Borel equivalence relations. Annals of Pure and Applied Logic 161 (7) 837–850.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous Lattices and Domains, Cambridge University Press.CrossRefGoogle Scholar
Hertling, P. (1996a) Unstetigkeitgrade von Funktionen in der effektiven Analysis, Ph.D. thesis, Fern Universität Gesamthochschule in Hagen.Google Scholar
Hertling, P. (1996b) Topological complexity with continuous operations. Journal of Complexity 12 (4) 315–338.CrossRefGoogle Scholar
Ikegami, D. (2010) Games in Set Theory and Logic, Ph.D. thesis, University of Amsterdam.Google Scholar
Ikegami, D., Schlicht, P. and Tanaka, H. (2012) Continuous reducibility for the real line, preprint, available at http://www.math.uni-bonn.de/people/schlicht/Ikegami%20Schlicht%20Tanaka%202012%20linespread.pdf Google Scholar
Kechris, A. S. (1995) Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer.Google Scholar
Kuratowski, K. (1966) Topology, volume 1, Academic Press.Google Scholar
Marker, D. (2002) Lecture notes on Descriptive Set Theory. On Marker's home page, http://homepages.math.uic.edu/~marker/math512/dst.ps.Google Scholar
Moschovakis, Y. (1979/2009) Descriptive Set Theory, volume 155, American Mathematical Society.Google Scholar
Rogers, H. (1967) Theory of recursive functions and effective computability, McGraw-Hill.Google Scholar
Schlicht, P. (2012) Continuous reducibility and dimension, available at http://www.math.uni-bonn.de/people/schlicht/Continuous%20reducibility%20and%20dimension/crd.pdf Google Scholar
Selivanov, V. L. (2003) Wadge degrees of ω-languages of deterministic Turing machines. Theoretical Informatics and Applications 37 (1) 67–83.Google Scholar
Selivanov, V. L. (2003) Extended abstract in STACS 2003 Proceedings. Lecture Notes in Computer Science 2607 97–108.Google Scholar
Selivanov, V. L. (2005) Variations on Wadge reducibility. Siberian Advances in Mathematics 15 (3) 44–80. (Also in Sixth Int. Workshop on Computability and Complexity in Analysis, Informatik Berichte, Uni-Hagen, 320-8/2004, 145–156.)Google Scholar
Selivanov, V. L. (2005a). Variations on Wadge reducibility. In: Proceedings of the 6th Workshop on Computability and Complexity in Analysis (CCA 2004). Electronic Notes in Theoretical Computer Science 120 159–171.Google Scholar
Selivanov, V. L. (2005b). Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly 51 (1) 45–61.Google Scholar
Selivanov, V. L. (2006) Towards a descriptive set theory for domain-like structures. Theoretical Computer Science 365 (3) 258–282.CrossRefGoogle Scholar
Selivanov, V. L. (2007) Hierarchies of Δ0 2-measurable k-partitions. Mathematical Logic Quarterly 53 (4–5) 446–461.Google Scholar
Selivanov, V. L. (2008) On the difference hierarchy in countably based T 0-spaces. Electronic Notes in Theoretical Computer Science 221 257–269.Google Scholar
Tang, A. (1979) Chain properties in Pω . Theoretical Computer Science 9 (2) 153–172.Google Scholar
Tang, A. (1981) Wadge reducibility and hausdorff difference hierarchy in Pω . Lectures Notes in Mathematics 871 360–371.Google Scholar
Wadge, W. W. (1972) Degrees of complexity of subsets of the Baire space. Notices of the American Mathematical Society A-714.Google Scholar
Weihrauch, K. (2000) Computable Analysis. An Introduction, Springer.Google Scholar