Vapnik-Chervonenkis density in some theories without the independence property, I

Aschenbrenner, Matthias; Dolich, Alf; Haskell, Deirdre; Macpherson, Dugald; Starchenko, Sergei

doi:10.1090/tran/6659

Cited by 45 publications

(109 citation statements)

References 90 publications

(139 reference statements)

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“…For the sake of completeness, before stating the main result of the section we briefly recall the definition of vc-density. We refer the reader to [3] for a detailed exposition.…”

Section: 2mentioning

confidence: 99%

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The Dp-Rank of Abelian Groups

Halevi¹,

Palacín²

2019

J. symb. log.

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“…For the sake of completeness, before stating the main result of the section we briefly recall the definition of vc-density. We refer the reader to [3] for a detailed exposition.…”

Section: 2mentioning

confidence: 99%

“…The vc-density has been studied for quite some time in combinatorics, computational geometry, statistics and machine learning. In model theory it relates to the classical problem of counting types and was furthermore studied extensively in many specific cases in [3,2].…”

Section: Introductionmentioning

confidence: 99%

The Dp-Rank of Abelian Groups

Halevi¹,

Palacín²

2019

J. symb. log.

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“…At the intersection of CLT and classical model theory is the concept of VC dimension (Aschenbrenner, Dolich, Haskell, Macpherson, & Starchenko, 2011, 2013Kearns and Vazirani, 1994;Laskowski, 1992), which we mentioned briefly. It could be possible to reconcile the results of these two disciplines in E-and F-logics, to the benefit of all involved.…”

Section: Classical Model-theory Techniquesmentioning

confidence: 99%

Computability of validity and satisfiability in probability logics over finite and countable models

Yang

2015

Journal of Applied Non-Classical Logics

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The -logic (which is called E-logic in this paper) of Terwijn is a variant of first-order logic (FOL) with the same syntax in which the models are equipped with probability measures and the ∀x quantifier is interpreted as 'there exists a set A of measure ≥ 1 − such that for each x ∈ A, …'. Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational ∈ (0, 1), respectively 1 1 -complete and 1 1 -hard, and ii) for = 0, respectively decidable and 0 1 -complete. The adjective 'general' here means 'uniformly over all languages'. We extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability and validity with respect to finite models in E-logic are, i) for rational ∈ (0, 1), respectively 0 1 -complete and 0 1 -complete, and ii) for = 0, respectively decidable and 0 1 -complete. Although partial results toward the countable case are also achieved, the computability of E-logic over countable models still remains largely unsolved. In addition, most of the results here and of Kuyper and Terwijn do not apply to individual languages with a finite number of unary predicates. Reducing this requirement continues to be a major point of research. On the positive side, we derive the decidability of the corresponding problems for monadic relational languages -equality-and function-free languages with finitely-many unary and arbitrarily-many nullary predicates. This result holds for all three of the unrestricted, countable, and finitemodel cases. Applications in computational learning theory (CLT), weighted graphs, and artificial neural networks (ANNs) are discussed in the context of these decidability and undecidability results.

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“…A number of results were proved about the inexistance of such expansions. In [3] it was shown that (Z, +, 0, 1, <) has no proper dp-minimal expansions. This was later significantly strengthened in [7] by the following: Fact 1.1 ([7,Corollary 2.20]).…”

Section: Introductionmentioning

confidence: 99%