On the complexity of categoricity in computable structures

White, Walker

doi:10.1002/malq.200310066

Cited by 17 publications

(11 citation statements)

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“…Therefore, since computable categoricity and relative computable categoricity coincide for trees of finite height, there is a Σ 0 3 predicate which expresses "T is computably categorical". Theories are known to exist in which the property of computable categoricity is strictly more complex than Σ 0 3 ; we refer the reader to [35] for details.…”

Section: Theorem 18mentioning

confidence: 99%

Computable categoricity of trees of finite height

2005

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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 Σ 0 3 -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 ∆ 0 n+1 -categorical but not ∆ 0 n -categorical.

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Section: Theorem 18mentioning

confidence: 99%

Computable categoricity of trees of finite height

2005

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“…The index set for A is the set I(A) of all indices for computable (isomorphic) copies of A. For a class K of structures, closed under isomorphism, the index set is the set I(K) of all indices for computable members of K. There is quite a lot of work on index sets [14], [6], [3], [2], [5], [8], [20], [21], [7], etc. Our work is very much in the spirit of Louise Hay, and Hay together with Doug Miller (see [16]).…”

Section: Introductionmentioning

confidence: 99%

Index sets of computable structures

et al. 2006

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The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Q-vector spaces, Archimedean real closed ordered fields, reduced Abelian p-groups of length less than ω 2 , and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be m-completen , for various n. In each case, the calculation involves finding an "optimal" sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Π n , d-Σ n , or Σn) yields a bound on the complexity of the index set. When we show m-completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.

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“…If we can show that I(A) is m-complete d-Σ 0 2 , then this Scott sentence is optimal. Scott sentences and index sets have been studied for a number of different kinds of structures [20], [10], [7], [6], [5], [13], [31], [30], [11], [8], [9], [22]. In [9] and [22], as in the current paper, the main goal was to find optimal Scott sentences.…”

Section: Introductionmentioning

confidence: 99%

Scott sentences for certain groups

Knight

Saraph

2017

Arch. Math. Logic

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We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable Σ3 Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable d-Σ2" (the conjunction of a computable Σ2 sentence and a computable Π2 sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. 1 Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of Q. We show that for some of these groups, the computable Σ3 Scott sentence is best possible, while for others, there is a computable d-Σ2 Scott sentence.

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On the complexity of categoricity in computable structures

Abstract: We investigate the computational complexity the class of Γ-categorical computable structures. We show that hyperarithmetic categoricity is Π 1 1 -complete, while computable categoricity is Π 0 4 -hard.

Cited by 17 publications

References 17 publications

Computable categoricity of trees of finite height

Computable categoricity of trees of finite height

Index sets of computable structures

Scott sentences for certain groups

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