Index sets of computable structures

Calvert, Wesley; Harizanov, Valentina S.; Knight, Julia F.; Miller, Sara

doi:10.1007/s10469-006-0029-0

Cited by 37 publications

(32 citation statements)

References 18 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are many results using the strategy above to find the complexities of optimal Scott sentences of structures. For example, Knight et al [CHKM06], [CHK + 12] determined the complexities of optimal Scott sentences for finitely generated free abelian groups, reduced abelian groups, free groups, and many other structures.…”

Section: Introductionmentioning

confidence: 99%

On optimal Scott sentences of finitely generated algebraic structures

Harrison-Trainor

2018

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Scott showed that for every countable structure A, there is a sentence of the infinitary logic Lω 1 ω , called a Scott sentence for A, whose models are exactly the isomorphic copies of A. Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity "describing" the structure. Knight et al. have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely generated structure always has a Σ 0 3 Scott sentence. We give a characterization of the finitely generated structures for whom the Σ 0 3 Scott sentence is optimal. One application of this result is to give a construction of a finitely generated group where the Σ 0 3 Scott sentence is optimal.We use this hierarchy to measure the complexity of a sentence. The sentence given above describing the infinite-dimensional Q-vector space is a Π 0 3 sentence, and it turns out that this is the best possible; there is no d-Σ 0 2 description of this vector space. There is a d-Σ 0 2 description of any finite-dimensional Q-vector space, and so these structures are "simpler" than the infinite-dimensional vectors space.

show abstract

Section: Introductionmentioning

confidence: 99%

On optimal Scott sentences of finitely generated algebraic structures

Harrison-Trainor

2018

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…A common method to characterize the complexity of an object is to describe the degree of its index set [5][6][7][8]10,13,17]. In the present paper, we carry out this computation for the class of objects which are machine-learnable in a particular model.…”

Section: Introductionmentioning

confidence: 99%

PAC learning, VC dimension, and the arithmetic hierarchy

Calvert

2015

Arch. Math. Logic

View full text Add to dashboard Cite

We compute that the index set of PAC-learnable concept classes is mcomplete Σ 0 3 within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable Π 0 1 classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension.

show abstract

Index sets of computable structures

Cited by 37 publications

References 18 publications

On optimal Scott sentences of finitely generated algebraic structures

On optimal Scott sentences of finitely generated algebraic structures

Scott sentences for certain groups

PAC learning, VC dimension, and the arithmetic hierarchy

Contact Info

Product

Resources

About