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Wadge hardness in Scott spaces and its effectivization

  • VERÓNICA BECHER (a1) and SERGE GRIGORIEFF (a2)

Abstract

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.

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References

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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.
Becher, V. and Grigorieff, S. (2004) Recursion and topology on 2 ≤ω for possibly infinite computations. Theoretical Computer Science 322 (1) 85136.
Becher, V. and Grigorieff, S. (2005) Random reals and possibly infinite computations. part I: Randomness in ∅'. Journal of Symbolic Logic 70 (3) 891913.
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) 124156.
Becher, V. and Grigorieff, S. Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization. This volume.
Brattka, V. and Hertling, P. (1994) Continuity and computability of relations. Informatif Berichte 164, Fern Universität Gesamthochschule in Hagen, Fachbereich Informatik.
de Brecht, M. (2011) Quasi-Polish spaces. ArXiv:1108.1445v1, 40 pages.
Edalat, A. (1997) Domains for computation in mathematics, physics and exact real aritmetic. Bulletin of Symbolic Logic 3 (4) 401452.
Ershov, Y. (1968) On a hierarchy of sets II. Algebra and Logic 7 (4) 1547.
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) 837850.
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous Lattices and Domains, Cambridge University Press.
Hertling, P. (1996a) Unstetigkeitgrade von Funktionen in der effektiven Analysis, Ph.D. thesis, Fern Universität Gesamthochschule in Hagen.
Hertling, P. (1996b) Topological complexity with continuous operations. Journal of Complexity 12 (4) 315338.
Ikegami, D. (2010) Games in Set Theory and Logic, Ph.D. thesis, University of Amsterdam.
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
Kechris, A. S. (1995) Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer.
Kuratowski, K. (1966) Topology, volume 1, Academic Press.
Marker, D. (2002) Lecture notes on Descriptive Set Theory. On Marker's home page, http://homepages.math.uic.edu/~marker/math512/dst.ps.
Moschovakis, Y. (1979/2009) Descriptive Set Theory, volume 155, American Mathematical Society.
Rogers, H. (1967) Theory of recursive functions and effective computability, McGraw-Hill.
Schlicht, P. (2012) Continuous reducibility and dimension, available at http://www.math.uni-bonn.de/people/schlicht/Continuous%20reducibility%20and%20dimension/crd.pdf
Selivanov, V. L. (2003) Wadge degrees of ω-languages of deterministic Turing machines. Theoretical Informatics and Applications 37 (1) 6783.
Selivanov, V. L. (2003) Extended abstract in STACS 2003 Proceedings. Lecture Notes in Computer Science 2607 97108.
Selivanov, V. L. (2005) Variations on Wadge reducibility. Siberian Advances in Mathematics 15 (3) 4480. (Also in Sixth Int. Workshop on Computability and Complexity in Analysis, Informatik Berichte, Uni-Hagen, 320-8/2004, 145–156.)
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 159171.
Selivanov, V. L. (2005b). Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly 51 (1) 4561.
Selivanov, V. L. (2006) Towards a descriptive set theory for domain-like structures. Theoretical Computer Science 365 (3) 258282.
Selivanov, V. L. (2007) Hierarchies of Δ0 2-measurable k-partitions. Mathematical Logic Quarterly 53 (4–5) 446461.
Selivanov, V. L. (2008) On the difference hierarchy in countably based T 0-spaces. Electronic Notes in Theoretical Computer Science 221 257269.
Tang, A. (1979) Chain properties in . Theoretical Computer Science 9 (2) 153172.
Tang, A. (1981) Wadge reducibility and hausdorff difference hierarchy in . Lectures Notes in Mathematics 871 360371.
Wadge, W. W. (1972) Degrees of complexity of subsets of the Baire space. Notices of the American Mathematical Society A-714.
Weihrauch, K. (2000) Computable Analysis. An Introduction, Springer.

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Wadge hardness in Scott spaces and its effectivization

  • VERÓNICA BECHER (a1) and SERGE GRIGORIEFF (a2)

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