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Complexity for partial computable functions over computable Polish spaces


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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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.
Brattka, V. and Gherardi, G. (2009). Borel complexity of topological operations on computable metric spaces. Journal of Logic and Computation 19 (1) 4576.
Brodhead, P. and Cenzer, D.A. (2008). Effectively closed sets and enumerations. Archive for Mathematical Logic 46 (7–8) 565582.
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) 14181432.
Calvert, W., Harizanov, V.S., Knight, J.F. and Miller, S. (2006). Index sets of computable structures. Journal of Algebra and Logic 45 (5) 306325.
Cenzer, D.A. and Remmel, J.B. (1998). Index sets for Π0 1 classes. Annals of Pure and Applied Logic 93 (1–3) 361.
Cenzer, D.A. and Remmel, J.B. (1999). Index sets in computable analysis. Theoretical Computer Science 219 (1–2) 111150.
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, 7997.
Downey, R.G., Hirschfeldt, D.R. and Khoussainov, B. (2003). Uniformity in computable structure theory. Journal of Algebra and Logic 42 (5) 318332.
Downey, R.G. and Montalban, A. (2008). The isomorphism problem for torsion-free Abelian groups is analytic complete. Journal of Algebra 320 (6) 22912300.
Ershov, Yu. L. (1973). Theorie der Numerierungen I. Zeitschrift fur Mathematische Logik Grundlagen der Mathematik 19 (19–25) 289388.
Ershov, Yu. L. (1977). Model of partial continuous functionals. In: Logic Colloquium 76, North-Holland, 455467.
Ershov, Yu. L. (1999). Theory of numberings. In: Griffor, E.R. (ed.) Handbook of Computability Theory, Elsevier Science B.V., 473503.
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) 755777.
Grubba, T. and Weihrauch, K. (2007). On computable metrization. Electronic Notes in Theoretical Computer Science 167 345364.
Hodges, W. (1993). Model theory. In: Encyclopedia of Mathematics, Cambridge University Press.
Kechris, A.S. (1995). Classical Descriptive Set Theory, Springer-Verlag.
Kolmogorov, A.N. and Fomin, S.V. (1999). Elements of the Theory of Functions and Functional Analysis, Dover Publications.
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, 330344.
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.
Korovina, M.V. and Kudinov, O.V. (2008). Towards computability over effectively enumerable topological spaces. Electronic Notes in Theoretical Computer Science 221 115125.
Korovina, M.V. and Kudinov, O.V. (2009). The uniformity principle for sigma-definability. Journal of Logic and Computation 19 (1) 159174.
Korovina, M.V. and Kudinov, O.V. (2015a). Positive predicate structures for continuous data. Journal of Mathematical Structures in Computer Science 25 (8) 16691684.
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, 201215.
Montalban, A. and Nies, A. (2013). Borel structures: A brief survey. Lecture Notes in Logic 41, 124134.
Moschovakis, Y.N. (1964). Recursive metric spaces. Fundamenta Mathematicae 55 215238.
Moschovakis, Y.N. (2009). Descriptive Set Theory. North-Holland.
Rogers, H. (1967). Theory of Recursive Functions and Effective Computability, McGraw-Hill.
Soare, R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer Science and Business Media.
Shoenfield, J.R. (1971). Degrees of Unsolvability, North-Holland Publisher.
Selivanov, V. (1988). Index sets of factor-objects of the Post numbering. Algebra and Logic 27 (3) 215224.
Selivanov, V. (2015). Towards the effective descriptive set theory. Lecture Notes in Computer Science 9136 324333.
Selivanov, V. and Schröder, M. (2014). Hyperprojective hierarchy of qcb 0-space. Journal of Computability 4 (1) 117.
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, 103117.
Spreen, D. (1995). On some decision problems in programming. Information and Computation 122 (1) 120139.
Spreen, D. (1998). On effective topological spaces. Journal of Symbolic Logic 63 (1) 185221.
Weihrauch, K. (1993). Computability on computable metric spaces. Theoretical Computer Science 113 (1) 191210.
Weihrauch, K. (2000). Computable Analysis, Springer-Verlag.
Weihrauch, K. and Deil, Th. (1980). Berechenbarkeit auf cpo-s. Schriften zur Angew. Math. u. Informatik vol. 63. RWTH Aachen.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
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