Comparing Classes of Finite Structures

Calvert, Wesley; Cummins, Desmond F.; Knight, Julia F.; Miller, Sara

doi:10.1023/b:allo.0000048827.30718.2c

Cited by 76 publications

(71 citation statements)

References 17 publications

(19 reference statements)

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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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Section: Introductionmentioning

confidence: 99%

Index sets of computable structures

et al. 2006

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“…Moreover, the transformations work perfectly well for structures with universe an arbitrary subset of ω. Therefore, these classes are also "on top" under the relation ≤ tc in [4]. We have shown that for the class K of trees, the relation I(E, K) (the set of pairs of indices for computable members of K that are isomorphic) lies "on top" under the relation ≤ F F on Σ • undirected graphs,…”

Section: Abelian P-groupsmentioning

confidence: 97%

“…In the next section we will explain the relationship between F Freducibility and the notion of tc-reducibility introduced in [4] to compare the classes of countable structures.…”

Section: σ 1 1 Sets and Relationsmentioning

confidence: 99%

“…We write K ≤ tc K if there is a Turing computable operator Φ = ϕ e taking the atomic diagram of each A ∈ K to the atomic diagram of some B ∈ K , such that Φ is 1 − 1 on isomorphism types. This notion was introduced in [4]. If I(K) and I(K ) are hyperarithmetical, and K ≤ tc K , then I( ∼ =, K) ≤ F F I( ∼ =, K ).…”

Section: Abelian P-groupsmentioning

confidence: 99%

“…They showed that the class of undirected graphs, the class of fields of any fixed characteristic, the class of 2-step nilpotent groups, and the class of linear orderings all lie "on top" in this setting. In [4], it was observed that the Borel transformations are all effective. Moreover, the transformations work perfectly well for structures with universe an arbitrary subset of ω.…”

Section: Abelian P-groupsmentioning

confidence: 99%

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Isomorphism relations on computable structures

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Friedman²,

Harizanov³

et al. 2012

J. symb. log.

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On Σ¹₁ equivalence relations over the natural numbers

Fokina

Friedman

2011

Mathematical Logic Qtrly

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We study the structure of Σ 1 1 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly Σ 1 1 i.e., Σ 1 1 but not Δ 1 1 equivalence classes. We also show the existence of incomparable Σ 1 1 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ 1 1 equivalence relations (under both reducibilities) and show that existence of infinitely many properly Σ 1 1 equivalence classes that are complete as Σ 1 1 sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ 1 1 equivalence relations.

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Comparing Classes of Finite Structures

Cited by 76 publications

References 17 publications

Index sets of computable structures

Index sets of computable structures

Isomorphism relations on computable structures

On Σ¹₁ equivalence relations over the natural numbers

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Comparing Classes of Finite Structures

Cited by 76 publications

References 17 publications

Index sets of computable structures

Index sets of computable structures

Isomorphism relations on computable structures

On Σ11 equivalence relations over the natural numbers

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On Σ¹₁ equivalence relations over the natural numbers