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Complexity for partial computable functions over computable Polish spaces

Published online by Cambridge University Press:  19 December 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
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Abstract

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Special Issue 3: Mind, Mechanism and Mathematics: Computability Unchained , March 2018 , pp. 429 - 447
Copyright
Copyright © Cambridge University Press 2016 

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References

Ash, C.J. and Knight, J.F. (2000). Computable Structures and the Hyperarithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, Elsevier.Google 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.CrossRefGoogle Scholar
Brodhead, P. and Cenzer, D.A. (2008). Effectively closed sets and enumerations. Archive for Mathematical Logic 46 (7–8) 565–582.Google 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 Computation 72 (4) 1418–1432.Google Scholar
Calvert, W., Harizanov, V.S., Knight, J.F. and Miller, S. (2006). Index sets of computable structures. Journal of Algebra and Logic 45 (5) 306–325.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.Google 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, Contemporary Mathematics, vol. 8, AMS, 79–97.Google 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 (6) 2291–2300.Google Scholar
Ershov, Yu. L. (1973). Theorie der Numerierungen I. Zeitschrift fur Mathematische Logik Grundlagen der Mathematik 19 (19–25) 289–388.Google Scholar
Ershov, Yu. L. (1977). Model 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
Grubba, T. and Weihrauch, K. (2007). On computable metrization. Electronic Notes in Theoretical Computer Science 167 345–364.CrossRefGoogle Scholar
Hodges, W. (1993). Model theory. In: Encyclopedia of Mathematics, Cambridge University Press.Google Scholar
Kechris, A.S. (1995). Classical Descriptive Set Theory, Springer-Verlag.Google Scholar
Kolmogorov, A.N. and Fomin, S.V. (1999). Elements of the Theory of Functions and Functional Analysis, Dover Publications.Google Scholar
Korovina, M.V. (2003). Computational aspects of Σ-definability over the real numbers without the equality test. In: Baaz, M. and Makowsky, J.A. (eds.) CSL'03, Lecture Notes in Computer Science, vol. 2803, Springer, 330–344.Google Scholar
Korovina, M.V. and Kudinov, O.V. (2005). Towards computability of higher type continuous data. In: Proceedings CiE'05, Lecture Notes in Computer Science, vol. 3526, Springer-Verlag, 235–241.Google 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.Google Scholar
Korovina, M.V. and Kudinov, O.V. (2015a). 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. (2015b). Index sets as a measure of continuous constraints complexity. Lecture Notes in Computer Science vol. 8974, Springer, 201–215.Google Scholar
Montalban, A. and Nies, A. (2013). Borel structures: A brief survey. Lecture Notes in Logic 41, 124–134.Google Scholar
Moschovakis, Y.N. (1964). Recursive metric spaces. Fundamenta Mathematicae 55 215–238.Google Scholar
Moschovakis, Y.N. (2009). Descriptive Set Theory. North-Holland.Google Scholar
Rogers, H. (1967). Theory of Recursive Functions and Effective Computability, McGraw-Hill.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
Shoenfield, J.R. (1971). Degrees of Unsolvability, North-Holland Publisher.Google Scholar
Selivanov, V. (1988). Index sets of factor-objects of the Post numbering. Algebra and Logic 27 (3) 215–224.Google Scholar
Selivanov, V. (2015). Towards the effective descriptive set theory. Lecture Notes in Computer Science 9136 324–333.Google Scholar
Selivanov, V. and Schröder, M. (2014). Hyperprojective hierarchy of qcb 0-space. Journal of Computability 4 (1) 1–17.Google Scholar
Spreen, D. (1984). On r.e. inseparability of cpo index sets. In: Logic and Machines: Decision Problems and Complexity, Lecture Notes in Computer Science, vol. 171, Springer, 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.Google Scholar
Weihrauch, K. and Deil, Th. (1980). Berechenbarkeit auf cpo-s. Schriften zur Angew. Math. u. Informatik vol. 63. RWTH Aachen.Google Scholar