Hostname: page-component-77f85d65b8-lfk5g Total loading time: 0 Render date: 2026-04-21T19:00:28.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Computable elements and functions in effectively enumerable topological spaces

Published online by Cambridge University Press:  23 June 2016

MARGARITA KOROVINA  and
OLEG KUDINOV
MARGARITA KOROVINA
Affiliation:
A.P. Ershov Institute of Informatics systems, NSU, Novosibirsk, Russia Email: rita.korovina@gmail.com
OLEG KUDINOV
Affiliation:
Sobolev Institute of Mathematics, NSU, Novosibirsk, Russia Email: kud@math.nsc.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is a part of the ongoing program of analysing the complexity of various problems in computable analysis in terms of the complexity of the associated index sets. In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective ω–continuous domains and the natural numbers with the discrete topology satisfy this condition. We prove weak and strong analogues of Rice's theorem for computable elements. Then, we construct principal computable numberings of partial majorant-computable real-valued functions and co-effectively closed sets and calculate the complexity of index sets for important problems such as root verification and function equality. For example, we show that, for partial majorant-computable real functions, the equality problem is Π1 1-complete.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ash, C.J. and Knight, J.F. (2000). Computable Structures and the Hyperarithmetical Hierarchy. Studies in Logic and the Foundations of Mathematics, 144, Elsevier.Google Scholar
Becher, V., Heiber, P. and Slaman, T.A. (2014). Normal numbers and the Borel hierarchy. Fundamenta Mathematicae 22, 63–77.Google Scholar
Berger, U. (1993). Total sets and objects in domain theory. Annals of Pure and Applied Logic 60 (2) 91–117.Google Scholar
Brattka, V. (2001). Computable versions of Baire's category theorem. In: MFCS'99. Springer Lecture Notes in Computer Science 2136 224–235.CrossRefGoogle Scholar
Brattka, V. and Gherardi, G. (2009). Borel complexity of topological operations on computable metric spaces. Journal of Logic and Computation 19 (1) 45–76.Google Scholar
Brattka, V. and Gherardi, G. (2011). Weihrauch degrees, omniscience principles and weak computability. Journal of Symbolic Logic 76 (1) 143–176.Google Scholar
Brattka, V. and Weihrauch, K. (1999). Computability on subsets of euclidean space I: Closed and compact sets. Theoretical Computer Science 219 (1–2) 65–93.CrossRefGoogle Scholar
Brodhead, P. and Cenzer, D.A. (2008). Effectively closed sets and enumerations. Archive for Mathematical Logic 46 (7–8) 565–582.CrossRefGoogle Scholar
Calvert, W., Fokina, E., Goncharov, S.S., Knight, J.F., Kudinov, O.V., Morozov, A.S. and Puzarenko, V. (2007). Index sets for classes of high rank structures. Journal of Symbolic Logic 72 (4) 1418–1432.Google Scholar
Calvert, W., Harizanov, V.S., Knight, J.F. and Miller, S. (2006). Index sets of computable structures. Journal Algebra and Logic 45 (5) 306–325.CrossRefGoogle Scholar
Ceitin, G.S. (1971). Mean value theorems in constructive analysis. American Mathematical Society Translations: Series 2 98 11–40.Google Scholar
Cenzer, D.A. and Remmel, J.B. (1998). Index sets for Π0 1 classes. Annals of Pure and Applied Logic 93 (1–3) 3–61.Google Scholar
Cenzer, D.A. and Remmel, J.B. (1999). Index sets in computable analysis. Theoretical Computer Science 219 (1–2) 111–150.CrossRefGoogle Scholar
Delzell, C.N. (1982). A finiteness theorem for open semi-algebraic sets, with application to Hilbert's 17th problem. In: Ordered Fields and Real Algebraic Geometry, Contemp. Math., 8, AMS 79–97.CrossRefGoogle Scholar
Downey, R.G., Hirschfeldt, D.R. and Khoussainov, B. (2003). Uniformity in computable structure theory. Journal of Algebra and Logic 42 (5) 318–332.Google Scholar
Downey, R.G. and Montalban, A. (2008). The isomorphism problem for torsion-free Abelian groups is analytic complete. Journal of Algebra 320 2291–2300.Google Scholar
Ershov, Yu.L. (1973). Theorie der Numerierungen I. Zeitschrift fur mathematische Logik Grundlagen der Mathematik 19 289–388.Google Scholar
Ershov, Yu.L. (1977). Model $\mathbb{C}$ of partial continuous functionals. In: Logic Colloquium, 76, North-Holland 455–467.Google Scholar
Ershov, Yu.L. (1999). Theory of numberings. In: Griffor, E.R. (ed.) Handbook of Computability Theory, Elsevier Science B.V., 473–503.Google Scholar
Frolov, A., Harizanov, V., Kalimullin, I., Kudinov, O. and Miller, R. (2012). Spectra of high n and nonlow n degrees. Electronic Notes in Theoretical Computer Science 22 (4) 755–777.Google Scholar
Gierz, G., HeinrichAAAAHofmann, K., Keime, K., Lawson, J.D. and Mislove, M.W. (2003). Continuous Lattices and Domain. Encyclopedia of Mathemtics and its Applications 93, Cambridge University Press.Google Scholar
Grubba, T. and Weihrauch, K. (2007). On computable metrization. Electronic Notes in Theoretical Computer Science 167, 345–364.Google Scholar
Grubba, T. and Weihrauch, K. (2009). Elementary computable topology. Journal of UCS 15 (6) 1381–1422.Google Scholar
Korovina, M.V. (2003a). Gandy's theorem for abstract structures without the equality test. In: Vardi, M.Y. and Voronkov, A. (eds.) LPAR'03. Springer Lecture Notes in Computer Science 2850 290–301.Google Scholar
Korovina, M.V. (2003b). Computational aspects of Σ-definability over the real numbers without the equality test. In: Baaz, M. and Makowsky, J.A. (eds.) CSL'03. Springer Lecture Notes in Computer Science, 2803 330–344.Google Scholar
Korovina, M.V. and Kudinov, O.V. (2015). Positive predicate structures for continuous data. Journal of Mathematical Structures in Computer Science 25 (8), 1669–1684.Google Scholar
Korovina, M.V. and Kudinov, O.V. (2005). Towards computability of higher type continuous data. In: Proceedings of the CiE'05. Springer-Verlag Lecture Notes in Computer Science 3526 235–241.CrossRefGoogle Scholar
Korovina, M.V. and Kudinov, O.V. (2008). Towards computability over effectively enumerable topological spaces. Electronic Notes in Theoretical Computer Science 221 115–125.Google Scholar
Korovina, M.V. and Kudinov, O.V. (2009). The Uniformity Principle for Sigma-definability. Journal of Logic and Computation 19 (1) 159–174.CrossRefGoogle Scholar
Martin-Löf, P. (1970). Notes on Constructive Mathematics. Almqvist & Wiksell.Google Scholar
Montalban, A. and Nies, A. (2013). Borel structures: A brief survey. Lecture Notes in Logic 41 124–134.Google Scholar
Morozov, A.S. and Korovina, M.V. (2008). On Σ-definability without equality over the real numbers. Mathematical Logic Quarterly 54 (5) 535–544.Google Scholar
Moschovakis, Y.N. (1964). Recursive metric spaces. Fundamenta Mathematicae 55 215–238.Google Scholar
Rogers, H. (1967). Theory of Recursive Functions and Effective Computability, McGraw-Hill.Google Scholar
Selivanov, V. and Schröder, M. (2014) Hyperprojective hierarchy of qcb 0-Space. Journal Computability 4 (1) 1–17, North-Holland.Google Scholar
Shoenfield, J.R. (1971). Degrees of Unsolvability, North-Holland Publ.Google Scholar
Soare, R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer Science and Business Media.Google Scholar
Spreen, D. (1984). On r.e. inseparability of cpo index sets. Logic and Machines: Decision Problems and Complexity, Lecture Notes in Computer Science 171 103–117.Google Scholar
Spreen, D. (1995). On some decision problems in programming. Information and Computation 122 (1) 120–139.Google Scholar
Spreen, D. (1998). On effective topological spaces. Journal of Symbolic Logic 63 (1) 185–221.Google Scholar
Weihrauch, K. (1993) Computability on computable metric Spaces. Theoretical Computer Science 113 (1) 191–210.Google Scholar
Weihrauch, K. (2000). Computable Analysis, Springer Verlag.CrossRefGoogle Scholar
Weihrauch, K. and Deil, Th. (1980) Berechenbarkeit auf cpo-s. Schriften zur Angew. Math. u. Informatik 63. RWTH Aachen.Google Scholar
Welch, L.V. (1984). A hierarchy of families of recursively enumerable degrees. Journal of Symbolic Logic 49 (4) 1160–1170.CrossRefGoogle Scholar
Yates, C.E.M. (1965). Three theorems of the degrees of recursively enumerable sets. Duke Mathematical Journal 32 (3) 461–468.Google Scholar
Yates, C.E.M. (1969). On the degrees of index sets II. Transactions of the American Mathematical Society 135 249–266.Google Scholar