Journal of Symbolic Logic

Computable categoricity of trees of finite height

Steffen Lempp, Charles McCoy, Russell Miller, and Reed Solomon

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ03-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n≥ 1 in ω, there exists a computable tree of finite height which is δ0n+1-categorical but not δ0n-categorical.

Article information

J. Symbolic Logic Volume 70, Issue 1 (2005), 151-215.

First available in Project Euclid: 1 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Lempp, Steffen; McCoy, Charles; Miller, Russell; Solomon, Reed. Computable categoricity of trees of finite height. J. Symbolic Logic 70 (2005), no. 1, 151--215. doi:10.2178/jsl/1107298515.

Export citation


  • C. J. Ash Categoricity in hyperarithmetical degrees, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 1--14.
  • C. J. Ash and J. F. Knight Computable structures and the hyperarithmetic hierarchy, Elsivier Science, Amsterdam,2000.
  • C. J. Ash, J. F. Knight, M. Mannasse, and T. Slaman Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195--205.
  • J. Chisholm On intrisically $1$-computable trees, unpublished manuscript.
  • J. N. Crossley, A. B. Manaster, and M. F. Moses Recursive categoricity and recursive stability, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 191--204.
  • R. G. Downey On presentations of algebraic structures, Complexity, logic, and recursion theory (A. Sorbi, editor), Dekker, New York,1997, pp. 157--205.
  • R. G. Downey and C. G. Jockusch Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871--880.
  • S. S. Goncharov Autostability and computable families of constructivizations, Algebra and Logic, vol. 14 (1975), pp. 647--680 (Russian), 392--409 (English translation).
  • S. S. Goncharov and V. D. Dzgoev Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 45--58 (Russian), 28--37 (English translation).
  • S. S. Goncharov, S. Lempp, and R. Solomon The computable dimension of ordered abelian groups, Advances in Mathematics, vol. 175 (2003), pp. 102--143.
  • S. S. Goncharov, A. V. Molokov, and N. S. Romanovskii Nilpotent groups of finite algorithmic dimension, Siberian Mathematics Journal, vol. 30 (1989), pp. 63--68.
  • D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko Degree spectra and computable dimension in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71--113.
  • B. Khoussainov and R. A. Shore Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153--193.
  • J. B. Kruskal Well quasi-ordering, the tree theorem, and Vázsonyi's conjecture, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 210--225.
  • O. V. Kudinov An integral domain with finite algorithmic dimension, unpublished manuscript.
  • P. LaRoche Recursively presented Boolean algebras, Notices of the American Mathematical Society, vol. 24 (1977), pp. A--552, research announcement.
  • G. Metakides and A. Nerode Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289--320.
  • R. G. Miller The $\Delta^0_2$ spectrum of a linear order, Journal of Symbolic Logic, vol. 66 (2001), pp. 470--486.
  • C. St. J. A. Nash-Williams On well-quasi-ordering finite trees, Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 833--835.
  • A. T. Nurtazin Strong and weak constructivizations and enumerable families, Algebra and Logic, vol. 13 (1974), pp. 177--184.
  • J. B. Remmel Recursive isomorphism types of recursive Boolean algebras, Journal of Symbolic Logic, vol. 46 (1981), pp. 572--594.
  • S. G. Simpson Nonprovability of certain combinatorial properties of finite trees, Harvey Friedman's research on the foundations of mathematics (L. A. Harrington, M. D. Morley, A. Scedrov, and S. G. Simpson, editors), North-Holland, Amsterdam,1985, pp. 87--117.
  • T. A. Slaman Relative to any nonrecursive set, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2117--2122.
  • R. I. Soare Recursively enumerable sets and degrees, Springer-Verlag, New York,1987.
  • S. Wehner Enumerations, countable structures, and Turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131--2139.
  • W. White On the complexity of categoricity in computable structures, Mathematical Logic Quarterly, vol. 49 (2003), no. 6, pp. 603--614.