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Index sets for computable structures
 Algebra and Logic
"... 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, Qvector spaces, Archimedean real closed ordered fields, reduced Abelian ..."
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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, Qvector spaces, Archimedean real closed ordered fields, reduced Abelian pgroups of length less than ω 2, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be mcomplete Π 0 n, dΣ 0 n,orΣ 0 n, 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 Πn, dΣn, or Σn) yields a bound on the complexity of the index set. When we show mcompleteness 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.
ABSTRACT
, 2006
"... The Central District 4H Leader Board Training was designed to bring adult 4H leaders and youth 4H members who serve on their county 4H Leader Boards together to investigate ways in which they could form a stronger youthadult partnership on their boards. This program worked through five phases: recru ..."
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The Central District 4H Leader Board Training was designed to bring adult 4H leaders and youth 4H members who serve on their county 4H Leader Boards together to investigate ways in which they could form a stronger youthadult partnership on their boards. This program worked through five phases: recruitment, selfassessment, training, county followup actions, and evaluation and reflection processes. Through followup surveys, a variety of changes in practice for the groups was documented and explored, including policy change, training, meeting formats, and discussion and reflection opportunities. This evaluative paper provides an overview of the methods used to strengthen youthadult partnerships on the boards. Evaluation showed positive shortterm and mediumterm effects for all participating counties and particularly significant impact for counties at a growth stage of development of their youth/adult partnerships. It has been written for possible adaptation and use in other counties, districts, and states. 2 SITUATION Youthadult partnership in groups concerned with organizational development
POLISH GROUP ACTIONS AND COMPUTABILITY
, 903
"... ABSTRACT. Let G be a closed subgroup of S ∞ and X be a Polish Gspace with a countable basis A of clopen sets. Each x ∈ X defines a characteristic function τx on A by τx(A) = 1 ⇔ x ∈ A. We consider computable complexity of τx and some related questions. 1. ..."
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ABSTRACT. Let G be a closed subgroup of S ∞ and X be a Polish Gspace with a countable basis A of clopen sets. Each x ∈ X defines a characteristic function τx on A by τx(A) = 1 ⇔ x ∈ A. We consider computable complexity of τx and some related questions. 1.
POLISH GROUP ACTIONS AND COMPUTABILITY
, 903
"... ABSTRACT 1 2 Let G be a closed subgroup of S ∞ and X be a Polish Gspace with a countable basis A of clopen sets. Each x ∈ X defines a characteristic function τx on A by τx(A) = 1 ⇔ x ∈ A. We consider computable complexity of τx and some related questions. 1. ..."
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ABSTRACT 1 2 Let G be a closed subgroup of S ∞ and X be a Polish Gspace with a countable basis A of clopen sets. Each x ∈ X defines a characteristic function τx on A by τx(A) = 1 ⇔ x ∈ A. We consider computable complexity of τx and some related questions. 1.