On the number of databases and closure operations

Burosch, Gustav; Demetrovics, János; Katona, Gyula O. H.; Kleitman, Daniel J.; Sapozhenko, A. A.

doi:10.1016/0304-3975(91)90359-a

Cited by 21 publications

(6 citation statements)

References 5 publications

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“…, n} is union-free. Balogh and Wagner [19] proved the following natural counting counterpart of Kleitman's theorem, confirming a conjecture of Burosch, Demetrovics, Katona, Kleitman, and Sapozhenko [22]. Theorem 8.7.…”

Section: List Colouringmentioning

confidence: 55%

The Method of Hypergraph Containers

Balogh

Morris

Samotij

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.

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Section: List Colouringmentioning

confidence: 55%

The Method of Hypergraph Containers

Balogh

Morris

Samotij

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…If H corresponds to the class of formulas only expressible with Horn logic and variables V, then the number of logically different hypothesis in H is close to [1,3]:…”

Section: The Lrn * Algorithmmentioning

confidence: 99%

Extracting Rules from Neural Networks with Partial Interpretations

Persia

Ozaki

2022

nldl

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We investigate the problem of extracting rules, expressed in Horn logic, from neural network models.Our work is based on the exact learning model, in which a learner interacts with a teacher (the neural network model) via queries in order to learn an abstract target concept, which in our case is a set of Horn rules. We consider partial interpretations to formulate the queries. These can be understood as a representation of the world where part of the knowledge regarding the truthness of propositions is unknown. We employ Angluin’s algorithm for learning Horn rules via queries and evaluate our strategy empirically.

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“…Even the lack of such a formula has not been proved. In [6], Burosh et al consider the issue of counting Moore families as natural, so they provide an upper bound for that number (see also [13]). An often supported approach to try to obtain such a formula involves counting the number of objects for the first values of n using a systematic procedure.…”

Section: Introductionmentioning

confidence: 99%

Recursive decomposition tree of a Moore co-family and closure algorithm

Colomb

Irlande

Raynaud

et al. 2013

Ann Math Artif Intell

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A collection of sets on a ground set U n (U n = {1, 2, ..., n}) closed under intersection and containing U n is known as a Moore family. The set of Moore families for a fixed n is in bijection with the set of Moore co-families (union-closed families containing the empty set) denoted M n. This paper follows the work initiated in [9] in which we have given an inductive definition of the lattice of Moore co-families. In the present paper we use this definition to define a recursive decomposition tree of any Moore co-family and we design an original algorithm to compute the closure under union of any family. Then we compare performance of this algorithm to performance of Ganter's algorithm and Norris' algorithm.

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On the number of databases and closure operations

Cited by 21 publications

References 5 publications

The Method of Hypergraph Containers

The Method of Hypergraph Containers

Extracting Rules from Neural Networks with Partial Interpretations

Recursive decomposition tree of a Moore co-family and closure algorithm

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