Polynomial Time Categoricity and Linear Orderings

  • Jeffrey B. Remmel
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)


Let ℒ = < {ci}i∈S, {Ri}i∈T, {fi}i∈U> be a recursive language, i.e. assume that S, T and U are initial segments of the natural numbers N = (0, 1, 2, ,…), ci is a constant symbol for each i ∈ S, and there are partial recursive functions t and u such that for all i ∈ T, Ri is a t(i)-ary relation symbol and for all j ∈ U, f is a u(i)-ary function symbol. A structure A = <A, {ci A}i∈S, {Ri A}i∈T, {fi A}i∈U> is said to be a recursive structure if A, the universe of A, is a recursive subset of N, Ri A is a recursive relation for each i ∈ T, and fi A is a partial recursive function from Au(i) into A for each i ∈ U. Two recursive structures A and A′ over ℒ are recursively isomorphic, denoted by A ≈ r A′, if there is a partial recursive function f which maps A onto A’ which is an ℒ-isomorphism from A onto A′. We say that a recursive structure A over ℒ is recursively categorical if any recursive structure A′ over ℒ which is isomorphic to A is recursively isomorphic to A. The notion of a recursively categorical structure was first defined by Mal’cev [M] and, in the Russian literature, such structures are called autostable. Recursively categorical structures have been widely studied in the literature of recursive algebra and recursive model theory. For example, general semantic conditions for when a decidable model is recursively categorical were given by Nurtazin [Nu] and similar results were found by Ash and Nerode [AN] for models in which one can effectively decide all Σ1-formulas. Recursively categorical structures for various theories have been classified: Boolean algebras independently by Goncharov [Go] and La Roche [L], linear orderings independently by Dzgoev [GO] and Remmel [R], Abelian p-groups by Smith [S], and decidable dense two-dimensional partial orderings by Manaster and Remmel.


Polynomial Time Binary Representation Order Type Recursive Structure Partial Recursive Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    Ash, C.J., Private communication, December 1992.Google Scholar
  2. [AN]
    Ash, C.J. and A. Nerode [1981], Intrinsically recursive relations. Effective Aspects of Algebra (ed. J.N. Crossley), Upside Down A Book Co., Yarra Glen, Victoria, Australia, 26–41.Google Scholar
  3. [CR1]
    Cenzer, D. and J.B. Remmel [1991], Polynomial-time versus recursive models. Annals of Pure and Applied Logic, 54, 17–58.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [CR2]
    Cenzer, D. and J.B. Remmel [1992], Polynomial-time Abelian groups. Annals of Pure and Applied Logic, 56, 313–363.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [CR3]
    Cenzer, D. and J.B. Remmel, Feasibly categorical Abelian groups. Preprint.Google Scholar
  6. [GD]
    Goncharov, S.S. and V.D. Dzgoev [1980], Autostability of Models. Algebra i. Logika, 19, 28–37.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Go]
    Goncharov, S.S. [1975], Some properties of the constructivization of Boolean algebras. Sibirsk Mat. Ž, 16, 264–278.MathSciNetzbMATHGoogle Scholar
  8. [Gr]
    Grigorieff, S. [1990], Every recursive linear ordering has a copy in DTIME(n). J. of Symb. Logic 55, 260–276.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [HU]
    Hopcroft, J.E. and J.D. Ullman [1969], Formal Languages and Their Relations to Automata. Addison Wesley.Google Scholar
  10. [L]
    La Roche, P.E. [1977], Recursively presented Boolean algebras. Not. Amer. Math. Soc, 24, A–552.Google Scholar
  11. [M]
    Mal’cev, A.I. [1962], On recursive Abelian groups. Soviet Math. 3, 1431–1432.MathSciNetGoogle Scholar
  12. [NR1]
    Nerode, A. and J.B. Remmel [1987], Complexity theoretic algebra I, vector space over finite fields. Proceedings of Structure in Complexity Theory, 2nd Annual Conference, Computer Science Press of the IEEE, 218–239.Google Scholar
  13. [NR2]
    Nerode, A. and J.B. Remmel [1989], Complexity Theoretic Algebra II, the free Boolean algebra. Ann. Pure and Appl. Logic, 44, 71–99.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [NR3]
    Nerode, A. and J.B. Remmel [1990], Complexity Theoretic Algebra: vector space bases. Feasible Mathematics (Eds. S. Buss & P. Scott), Progress in Computer Science and Applied Logic, vol. 9, Birkhäuser, 293–319.Google Scholar
  15. [P]
    Plotkin, J.M., Who put the “back” in back-and-forth?, (this volume).Google Scholar
  16. [R1]
    Remmel, J.B. [1981], Recursively categorical linear orderings. Proc. Amer. Math. Soc, 83, 387–391.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [R2]
    Remmel, J.B. [1990], When is every recursive linear ordering of type μ recursively isomorphic to a p-time linear order over the binary representation of the natural numbers?, Feasible Mathematics (Eds. S. Buss and P. Scott), Progress in Computer Science, and Applied Logic, vol. 9, Birkhäuser, 321–341.Google Scholar
  18. [Ro]
    Rogers, H.J. [1967], Theory of Recursive Functions and Effective Computability. McGraw-Hill.Google Scholar
  19. [S]
    Smith, R. [1981], Two theorems on autostability in p-groups. Logic Year 1979-80 (Storrs, Conn.), Lecture Notes in Math Springer-Verlag, 302–311.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Jeffrey B. Remmel
    • 1
  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations