Generic copies of countable structures

Ash, C. J.; Knight, Julia F.; Manasse, Mark S.; Slaman, Theodore A.

doi:10.1016/0168-0072(89)90015-8

Cited by 143 publications

(103 citation statements)

References 4 publications

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“…In [3] Ash, Knight, Mannasse, and Slaman proved that this condition is actually equivalent, with no extra hypotheses needed, to the stronger notion of relative computable categoricity. In fact, they proved a more general result which applies to relative α-categoricity for any α < ω CK 1 .…”

Section: Relatively Computably Categorical Treesmentioning

confidence: 99%

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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: Relatively Computably Categorical Treesmentioning

confidence: 99%

“…Again, throughout this paper, we will deal only with finitary formulas. Theorem 2.2 (Ash-Knight-Mannasse-Slaman [3]) Let L be a computable language, and let A be a computable L-structure. Then A is relatively computably categorical iff A has a formally Σ 0 1 Scott family.…”

Section: Relatively Computably Categorical Treesmentioning

confidence: 99%

Computable categoricity of trees of finite height

2005

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“…These papers include Ash [1986Ash [ ], [1986a, [1987]; Ash and Knight [1990], [1994], [1995], Barker [1988] and Chisholm [1990a]. Related results when the notions are relativized to the degree of noncomputable models can be found in Ash, Knight, Manasse and Slaman [1989], Ash, Knight and Slaman [1993] and Chisholm [1990]. Here the results are proven by forcing arguments and the extra decidability hypotheses are not needed.…”

Section: Theorem 711mentioning

confidence: 69%

“…Another interesting survey is Millar [1999] in The Handbook of Computability Theory (Griffor [1999]). Two books in progress on the subject are Ash and Knight [1999] and Harizanov [2000]. These are all good sources for material and references.…”

Section: Basic Notionsmentioning

confidence: 99%

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Effective Model Theory: The Number of Models and Their Complexity

Khoussainov¹,

Shore²

1999

Models and Computability

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Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

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Intrinsic Reducibilities

McNicholl

2000

MLQ - Math. Log. Quart.

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Generic copies of countable structures

Cited by 143 publications

References 4 publications

Computable categoricity of trees of finite height

Computable categoricity of trees of finite height

Effective Model Theory: The Number of Models and Their Complexity

Intrinsic Reducibilities

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