Algebra and Logic

, Volume 43, Issue 6, pp 365–373 | Cite as

Complexity of Categorical Theories with Computable Models

  • S. S. Goncharov
  • B. Khoussainov

Abstract

M. Lerman and J. Scmerl specified some sufficient conditions for computable models of countably categorical arithmetical theories to exist. More precisely, it was shown that if T is a countably categorical arithmetical theory, and the set of its sentences beginning with an existential quantifier and having at most n+1 alternations of quantifiers is Σn+10for any n, then T has a computable model. J. Night improved this result by allowing certain uniformity and omitting the requirement that T is arithmetical. However, all of the known examples of theories of0-categorical computable models had low level of algorithmic complexity, and whether there are theories that would satisfy the above conditions for sufficiently large n was unknown. This paper will include such examples.

computable model countably categorical theory 

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Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • S. S. Goncharov
    • 1
  • B. Khoussainov
    • 2
  1. 1.Institute of Mathematics SB RASAkademika Koptyuga Prospekt, 4NovosibirskRussia
  2. 2.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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