Primitive Recursive Selection Functions over Abstract Algebras

  • J. I. Zucker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)


We generalise to abstract many-sorted algebras the classical proof-theoretic result due to Parsons and Mints that an assertion \({\forall} x {\exists} y {\it P}(x,y)\) (where P is ∑\(^{\rm 0}_{\rm 1}\)), provable in Peano arithmetic with ∑\(^{\rm 0}_{\rm 1}\) induction, has a primitive recursive selection function. This involves a corresponding generalisation to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalisation turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulae in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realisability of existential assertions over such algebras, and to work in an intuitionistic proof system. This investigation gives some insight into the relationship between verifiable specifications and computability on topological data types such as the reals, where the atomic formulae, i.e., equations between terms of type real, are not computable.


Atomic Formula Proof Theory Abstract Algebra Partial Algebra Peano Arithmetic 
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  1. 1.
    Beeson, M.: Foundations of Constructive Mathematics. Springer, Heidelberg (1985)CrossRefMATHGoogle Scholar
  2. 2.
    Feferman, S.: Definedness. Erkenntnis 43, 295–320 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)Google Scholar
  4. 4.
    Kleene, S.C.: On the interpretation of intuitionistic number theory. Journal of Symbolic Logic 10, 109–124 (1945)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kleene, S.C.: Introduction to Metamathematics. North Holland, Amsterdam (1952)MATHGoogle Scholar
  6. 6.
    Kreisel, G.: Some reasons for generalizing recursion theory. In: Gandy, R.O., Yates, C.M.E. (eds.) Logic Colloquium 1969, pp. 139–198. North Holland, Amsterdam (1971)Google Scholar
  7. 7.
    Meinke, K., Tucker, J.V.: Universal algebra. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 1, pp. 189–411. Oxford University Press, Oxford (1992)Google Scholar
  8. 8.
    Mints, G.: Quantifier-free and one-quantifier systems. Journal of Soviet Mathematics 1, 71–84 (1973)CrossRefMATHGoogle Scholar
  9. 9.
    Parsons, C.: On a number theoretic choice scheme II. Journal of Symbolic Logic 36, 587 (1971) (Abstract)CrossRefGoogle Scholar
  10. 10.
    Parsons, C.: On n-quantifier induction. Journal of Symbolic Logic 37, 466–482 (1972)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Takeuti, G.: Proof Theory, 2nd edn. North Holland, Amsterdam (1987)MATHGoogle Scholar
  12. 12.
    Troelstra, A.S. (ed.): Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344. Springer, Heidelberg (1993); Second corrected edition, Institute for Logic, Language and Computation, Technical Notes X-93-05, Amsterdam (1993)Google Scholar
  13. 13.
    Tucker, J.V., Zucker, J.I.: Program Correctness over Abstract Data Types, with Error-State Semantics. CWI Monographs, vol. 6, North Holland, Amsterdam (1988)Google Scholar
  14. 14.
    Tucker, J.V., Zucker, J.I.: Provable computable selection functions on abstract structures. In: Aczel, P., Simmons, H., Wainer, S.S. (eds.) Proof Theory, pp. 277–306. Cambridge University Press, Cambridge (1993)Google Scholar
  15. 15.
    Tucker, J.V., Zucker, J.I.: Computation by ‘while’ programs on topological partial algebras. Theoretical Computer Science 219, 379–420 (1999)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many-sorted algebras. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 5, pp. 317–523. Oxford University Press, Oxford (2000)Google Scholar
  17. 17.
    Tucker, J.V., Zucker, J.I.: Abstract computability and algebraic specification. ACM Transactions on Computational Logic 3, 279–333 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tucker, J.V., Zucker, J.I.: Abstract versus concrete computation on metric partial algebras. ACM Transactions on Computational Logic 5, 611–668 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tucker, J.V., Zucker, J.I.: Computable total functions, algebraic specifications and dynamical systems. Journal of Logic and Algebraic Programming 62, 71–108 (2005)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • J. I. Zucker
    • 1
  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

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