Index Sets as a Measure of Continuous Constraint Complexity

  • Margarita KorovinaEmail author
  • Oleg Kudinov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8974)


We develop the theory of index sets in the context of computable analysis considering classes of effectively open sets and computable real-valued functions. First, we construct principal computable numberings for effectively open sets and computable real-valued functions. Then, we calculate the complexity of index sets for important problems such as root realisability, set equivalence and inclusion, function equivalence which naturally arise in continuous constraint solving. Using developed techniques we give a direct proof of the generalised Rice-Shapiro theorem for effectively open sets of Euclidean spaces and present an analogue of Rice’s theorem for computable real-valued functions. We illustrate how index sets can be used to estimate complexity of continuous constraints in the settings of the Kleene-Mostowski arithmetical hierarchy.


  1. 1.
    Brattka, V., Weihrauch, K.: Computability on subsets of euclidean space I: closed and compact sets. Theor. Comput. Sci. 219(1–2), 65–93 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Benhamou, F., Goualard, F., Languénou, E., Christie, M.: Interval constraint solving for camera control and motion planning. ACM Trans. Comput. Log. 5(4), 732–767 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Collins, P.: Continuity and computability of reachable sets. Theor. Comput. Sci. 341(1–3), 162–195 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Calvert, W., Fokina, E., Goncharov, S.S., Knight, J.F., Kudinov, O.V., Morozov, A.S., Puzarenko, V.: Index sets for classes of high rank structures. J. Symb. Log. 72(4), 1418–1432 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Calvert, W., Harizanov, V.S., Knight, J.F., Miller, S.: Index sets of computable structures. J. Algebra Log. 45(5), 306–325 (2006)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ershov, Y.L.: Model \(\mathbb{C}\) of partial continuous functionals. In: Proceedings of the Logic Colloquium 76, pp 455–467. North-Holland, Amsterdam (1977)Google Scholar
  7. 7.
    Ershov, Y.L.: Numbering Theorey. Nauka, Moscow (1977). (in Russian) Google Scholar
  8. 8.
    Korovina, M.V., Kudinov, O.V.: Positive predicate structures for continuous data. J. Math. Struct. Comput. Sci (2015, To appear). doi: 10.1017/S0960129513000315
  9. 9.
    Korovina, M., Kudinov, O.: \(\varSigma _{K}\)-constraints for hybrid systems. In: Pnueli, A., Virbitskaite, I., Voronkov, A. (eds.) PSI 2009. LNCS, vol. 5947, pp. 230–241. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Korovina, M.V., Kudinov, O.V.: The uniformity principle for sigma-definability. J. Log. Comput. 19(1), 159–174 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Korovina, M.V., Kudinov, O.V.: Towards computability over effectively enumerable topological spaces. Electr. Notes Theor. Comput. Sci. 221, 115–125 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Korovina, M.V., Kudinov, O.V.: Towards computability of higher type continuous data. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 235–241. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  13. 13.
    Korovina, M.V.: Computational aspects of \(\varSigma \)-definability over the real numbers without the equality test. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 330–344. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  14. 14.
    Korovina, M.V.: Gandy’s theorem for abstract structures without the equality test. In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 290–301. Springer, Heidelberg (2003)Google Scholar
  15. 15.
    Nerode, A., Kohn, W.: Models for hybrid systems: automata, topologies, controllability, observability. In: Grossman, R.L., Ravn, A.P., Rischel, H., Nerode, A. (eds.) HS 1991 and HS 1992. LNCS, vol. 736, pp. 317–357. Springer, Heidelberg (1993)Google Scholar
  16. 16.
    Ratschan, S., She, Z.: Constraints for continuous reachability in the verification of hybrid systems. In: Calmet, J., Ida, T., Wang, D. (eds.) AISC 2006. LNCS (LNAI), vol. 4120, pp. 196–210. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  17. 17.
    Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  18. 18.
    Shoenfield, J.R.: Degrees of Unsolvability. North-Holland Publ., New York (1971) zbMATHGoogle Scholar
  19. 19.
    Spreen, D.: On r.e. inseparability of CPO index sets. In: Börger, E., Rödding, D., Hasenjaeger, G. (eds.) Rekursive Kombinatorik 1983. LNCS, vol. 171, pp. 103–117. Springer, Heidelberg (1984) Google Scholar
  20. 20.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.A.P. Ershov Institute of Informatics SystemsNovosibirskRussia
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations