Generalizing Computability Theory to Abstract Algebras

  • J. V. Tucker
  • J. I. ZuckerEmail author


We present a survey of our work over the last four decades on generalizations of computability theory to many-sorted algebras. The following topics are discussed, among others: (1) abstract v concrete models of computation for such algebras; (2) computability and continuity, and the use of many-sorted topological partial algebras, containing the reals; (3) comparisons between various equivalent and distinct models of computability; (4) generalized Church-Turing theses.


Computability and continuity Computability on abstract structures Computability on the reals Generalized church-turing thesis Generalized computability 



We are grateful to Mark Armstrong, Sol Feferman, Diogo Poças and an anonymous referee for very helpful comments on earlier drafts of this chapter. We also thank the editors, Giovanni Sommaruga and Thomas Strahm, for the opportunity to participate in this volume, and for their helpfulness during the preparation of this chapter.

This research was supported by a grant from the Natural Sciences and Engineering Research Council (Canada).


  1. 1.
    M. Armstrong, Notions of semicomputability in topological algebras over the reals. M.Sc. Thesis, Department of Computing and Software, McMaster University, 2015. Archived at Scholar
  2. 2.
    L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, New York, 1998)CrossRefGoogle Scholar
  3. 3.
    V. Brattka, P. Hertling, Topological properties of real number representations. Theor. Comput. Sci. 284(2), 241–257 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II (Interscience, New York, 1953). Translated and revised from the German edition [1937]Google Scholar
  5. 5.
    Y.L. Ershov, S.S. Goncharov, A.S. Nerode, J.B. Remmel (eds.), Handbook of Recursive Mathematics (Elsevier, Amsterdam, 1998). In 2 volumesGoogle Scholar
  6. 6.
    S. Feferman, Theses for computation and recursion on abstract structure, in Turing’s Revolution. The Impact of his Ideas About Computability, ed. by G. Sommaruga, T. Strahm (Birkhäuser/Springer, Basel, 2015)Google Scholar
  7. 7.
    J.E. Fenstad, Computation theories: an axiomatic approach to recursion on general structures, in Logic Conference, Kiel 1974, ed. by G. Muller, A. Oberschelp, K. Potthoff (Springer, Berlin, 1975), pp. 143–168Google Scholar
  8. 8.
    J.E. Fenstad, Recursion Theory: An Axiomatic Approach (Springer, Berlin, 1980)CrossRefzbMATHGoogle Scholar
  9. 9.
    H. Friedman, Algebraic procedures, generalized Turing algorithms, and elementary recursion theory, in Logic Colloquium ‘69, ed. by R.O. Gandy, C.M.E. Yates (North Holland, Amsterdam, 1971), pp. 361–389CrossRefGoogle Scholar
  10. 10.
    A. Fröhlich, J. Shepherdson, Effective procedures in field theory. Philos. Trans. R. Soc. Lond. (A) 248, 407–432 (1956)CrossRefzbMATHGoogle Scholar
  11. 11.
    M.Q. Fu, J.I. Zucker, Models of computability for partial functions on the reals. J. Log. Algebraic Methods Program. 84, 218–237 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    J. Hadamard, in Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover, New York, 1952). Translated from the French edition [1922]Google Scholar
  13. 13.
    J. Hadamard, La Théorie des Équations aux Dérivées Partielles (Éditions Scientifiques, Warsaw, 1964)zbMATHGoogle Scholar
  14. 14.
    N.D. James, J.I. Zucker, A class of contracting stream operators. Comput. J. 56, 15–33 (2013)CrossRefGoogle Scholar
  15. 15.
    S.C. Kleene, Introduction to Metamathematics (North Holland, Amsterdam, 1952)zbMATHGoogle Scholar
  16. 16.
    G. Kreisel, J.L. Krivine, Elements of Mathematical Logic (North Holland, Amsterdam, 1971)zbMATHGoogle Scholar
  17. 17.
    G. Kreisel, D. Lacombe, J. Shoenfield, Partial recursive functions and effective operations, in Constructivity in Mathematics: Proceedings of the Colloqium in Amsterdam, 1957, ed. by A. Heyting (North Holland, Amsterdam, 1959), pp. 290–297Google Scholar
  18. 18.
    A.I. Mal’cev, Constructive algebras I, in The Metamathematics of Algebraic Systems. A.I. Mal’cev, Collected papers: 1936–1967 (North Holland, Amsterdam, 1971), pp. 148–212Google Scholar
  19. 19.
    W. Magnus, A. Karass, D. Solitar, Combinatorial Group Theory (Dover, New York, 1976)zbMATHGoogle Scholar
  20. 20.
    J. Moldestad, V. Stoltenberg-Hansen, J.V. Tucker, Finite algorithmic procedures and computation theories. Math. Scand. 46, 77–94 (1980)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Y.N. Moschovakis, Axioms for computation theories—first draft, in Logic Colloquium ‘69 ed. by R.O. Gandy, C.E.M. Yates (North Holland, Amsterdam, 1971), pp. 199–255Google Scholar
  22. 22.
    M. Rabin, Computable algebra, general theory and the theory of computable fields. Trans. Am. Math. Soc. 95, 341–360 (1960)MathSciNetzbMATHGoogle Scholar
  23. 23.
    J.C. Shepherdson, Algebraic procedures, generalized Turing algorithms, and elementary recursion theory, in Harvey Friedman’s Research on the Foundations of Mathematics, ed. by L.A. Harrington, M.D. Morley, A. Ščedrov, S.G. Simpson (North Holland, Amsterdam, 1985), pp. 285–308CrossRefGoogle Scholar
  24. 24.
    V. Stoltenberg-Hansen, J.V. Tucker, Computable rings and fields, in Handbook of Computability Theory, ed. by E. Griffor (Elsevier, Amsterdam, 1999)Google Scholar
  25. 25.
    B.C. Thompson, J.V. Tucker, J.I. Zucker, Unifying computers and dynamical systems using the theory of synchronous concurrent algorithms. Appl. Math. Comput. 215, 1386–1403 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    G.S. Tseitin, Algebraic operators in constructive complete separable metric spaces. Dokl. Akad. Nauk SSSR 128, 49–52 (1959). In RussianMathSciNetGoogle Scholar
  27. 27.
    G.S. Tseitin, Algebraic operators in constructive metric spaces. Tr. Mat. Inst. Steklov 67, 295–361 (1962); In Russian. Translated in AMS Translations (2) 64:1–80. MR 27#2406Google Scholar
  28. 28.
    J.V. Tucker, Computing in algebraic systems, in Recursion Theory, Its Generalisations and Applications. London Mathematical Society Lecture Note Series, vol. 45, ed. by F.R. Drake, S.S. Wainer (Cambridge University Press, Cambridge, 1980), pp. 215–235CrossRefGoogle Scholar
  29. 29.
    J.V. Tucker, J.I. Zucker, Program Correctness Over Abstract Data Types, with Error-State Semantics. CWI Monographs, vol. 6 (North Holland, Amsterdam, 1988)Google Scholar
  30. 30.
    J.V. Tucker, J.I. Zucker, Computation by ‘while’ programs on topological partial algebras. Theor. Comput. Sci. 219, 379–420 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    J.V. Tucker, J.I. Zucker, Computable functions and semicomputable sets on many-sorted algebras, in Handbook of Logic in Computer Science, vol. 5, ed. by S. Abramsky, D. Gabbay, T. Maibaum (Oxford University Press, Oxford, 2000), pp. 317–523Google Scholar
  32. 32.
    J.V. Tucker, J.I. Zucker, Origins of our theory of computation on abstract data types at the Mathematical Centre, Amsterdam, 1979–1980, in Liber Amicorum: Jaco de Bakker, ed. by F. de Boer, M. van der Heijden, P. Klint, J.J.M.M. Rutten (Centrum Wiskunde & Informatica, Amsterdam, 2002), pp. 197–221Google Scholar
  33. 33.
    J.V. Tucker, J.I. Zucker, Abstract versus concrete computation on metric partial algebras. ACM Trans. Comput. Log. 5, 611–668 (2004)CrossRefMathSciNetGoogle Scholar
  34. 34.
    J.V. Tucker, J.I. Zucker, Computable total functions, algebraic specifications and dynamical systems. J. Log. Algebraic Program. 62, 71–108 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    J.V. Tucker, J.I. Zucker, Computability of analog networks. Theor. Comput. Sci. 371, 115–146 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    J.V. Tucker, J.I. Zucker, Continuity of operators on continuous and discrete time streams. Theor. Comput. Sci. 412, 3378–3403 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    J.V. Tucker, J.I. Zucker, Computability of operators on continuous and discrete time streams. Computability 3, 9–44 (2014)MathSciNetzbMATHGoogle Scholar
  38. 38.
    A.M. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 42, 230–265 (1936). With correction, ibid., 43, 544–546 (1937). Reprinted in The Undecidable, ed. by M. Davis (Raven Press, New York, 1965)Google Scholar
  39. 39.
    K. Weihrauch, Computable Analysis: An Introduction (Springer, Berlin, 2000)CrossRefGoogle Scholar
  40. 40.
    B. Xie, M.Q. Fu, J. Zucker, Characterizations of semicomputable sets of real numbers. J. Log. Algebraic Methods Program. 84, 124–154 (2015)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceSwansea UniversityWalesUK
  2. 2.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

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