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Why concept lattices are large: extremal theory for generators, concepts, and VC-dimension

Alexandre Albano, Bogdan Chornomaz
2017 International Journal of General Systems  
We make this precise by giving an upper bound for the number of minimal generators (and thereby for the number of concepts) of contexts without contranominal scales larger than a given size.  ...  A unique type of subcontexts is always present in formal contexts with many concepts: the contranominal scales.  ...  Acknowledgements We want to deeply thank Bernhard Ganter for the invaluable feedback and fruitful discussions.  ... 
doi:10.1080/03081079.2017.1354798 fatcat:zvw4ul67cfdltkowiihoxrm6rm

Formal Concept Analysis in knowledge processing: A survey on models and techniques

Jonas Poelmans, Sergei O. Kuznetsov, Dmitry I. Ignatov, Guido Dedene
2013 Expert systems with applications  
This is the first part of a large survey paper in which we analyze recent literature on Formal Concept Analysis (FCA) and some closely related disciplines using FCA.  ...  In this first part, we zoom in on and give an extensive overview of the papers published between 2003 and 2011 on developing FCAbased methods for knowledge processing.  ...  Acknowledgements We are grateful to Paul Elzinga for his help in processing the papers and generating the context files on which the lattices are based.  ... 
doi:10.1016/j.eswa.2013.05.007 fatcat:july7aafprcjrbaely2swksvhi

An algebra of human concept learning

Jacob Feldman
2006 Journal of Mathematical Psychology  
The basic theory of algebraic decomposition is extended in several ways, including algebraic accounts of the typicality of individual objects within concepts, and estimation of the power series from noisy  ...  , amounting to a kind of "theory" of the pattern.  ...  Preparation of this article was supported by NSF SBR-9875175 and NSF SES-0339062.  ... 
doi:10.1016/j.jmp.2006.03.002 fatcat:5lnya7duuzgadflttwoicuxe3e

A Learnable Representation for Syntax Using Residuated Lattices [chapter]

Alexander Clark
2011 Lecture Notes in Computer Science  
We propose a representation for natural language syntax based on the theory of residuated lattices: in particular on the Galois lattice between contexts and substrings, which we call the syntactic concept  ...  In this paper we define the basic algebraic properties of the syntactic concept lattice, together with a representation derived from this lattice and discuss the generative power of the formalism.  ...  Categorial grammars We have shown that this concept lattice is a residuated lattice; the theory of categorial grammars is based largely on the theory of residuation [14] .  ... 
doi:10.1007/978-3-642-20169-1_12 fatcat:jgw4w6nixfedde6hc3cfsqlfbm

Design of class hierarchies based on concept, (Galois) lattices

Robert Godin, Hafedh Mili, Guy W. Mineau, Rokia Missaoui, Amina Arfi, Thuy-Tien Chau
1998 Theory and Practice of Object Systems  
Concept (or Galois) lattices and variant structures are presented as a framework for dealing with the design and maintenance of class hierarchies.  ...  Experiments are presented to better assess the applicability of the approach. Design of class hierarchies based on concept (Galois) lattices  ...  Between these two extremes are a large number of potential designs that can be generated by pruning only a well chosen subset of the empty concepts.  ... 
doi:10.1002/(sici)1096-9942(1998)4:2<117::aid-tapo6>3.0.co;2-q fatcat:osiym4hwuzagjdzhjnid6mmapy

Design of class hierarchies based on concept, (Galois) lattices

Robert Godin, Hafedh Mili, Guy W. Mineau, Rokia Missaoui, Amina Arfi, Thuy‐Tien Chau
1998 Theory and Practice of Object Systems  
Concept (or Galois) lattices and variant structures are presented as a framework for dealing with the design and maintenance of class hierarchies.  ...  Experiments are presented to better assess the applicability of the approach. Design of class hierarchies based on concept (Galois) lattices  ...  Between these two extremes are a large number of potential designs that can be generated by pruning only a well chosen subset of the empty concepts.  ... 
doi:10.1002/(sici)1096-9942(1998)4:2<117::aid-tapo6>3.3.co;2-i fatcat:3zqfgmskfbenpkran7qayvwrt4

L-INDEX: A Conceptual Index of Textual Data on MapReduce Platform

Dunlu Peng, Kai Duan, Lei Xie
2015 International Journal of Grid and Distributed Computing  
We first discuss using FCA to discover the relationships among the data, and then employ Formal Concept Lattice (FCL) to connect the documents semantically with each other.  ...  Section 4 provides the algorithms of generating concepts from massive set of documents on MapReduce platform.  ...  Acknowledgements The authors gratefully acknowledge the support of Shanghai Key Science and Technology Project in Information Technology Field (14511107902), Shanghai Leading Academic Discipline Project  ... 
doi:10.14257/ijgdc.2015.8.2.20 fatcat:c4lfomgyjjdzzfwudghcrpc7oa

Software Conceptual Integrity: Deconstruction, Then Reconstruction [article]

Iaakov Exman
2018 arXiv   pre-print
The tension stems from the Modularity Matrix linking two very different kinds of entities - system concepts and abstract mathematical constructs - as seldom linked before.  ...  Conceptualization and Modularization, iteratively applied during software system design; second, to show that the algebraic Linear Software Models already provide the hitherto lacking formalization of  ...  Acknowledgments The author wishes to thank Alessio Plebe from the University of Messina, Italy, and Gonzalo Genova from the University Carlos III of Madrid, Spain, for their incisive and helpful comments  ... 
arXiv:1811.04315v1 fatcat:ox3baelpvjet7ngky3zroqtazy

Formal Concept Analysis and Information Retrieval – A Survey [chapter]

Victor Codocedo, Amedeo Napoli
2015 Lecture Notes in Computer Science  
One of the first models to be proposed as a document index for retrieval purposes was a lattice structure, decades before the introduction of Formal Concept Analysis.  ...  We try to be as exhaustive as possible by reviewing the last 25 years of research as chronicles of the domain, yet we are also concise in relating works by its theoretical foundations.  ...  explaining the meaning of the axis or why the documents and the query are located in the space as they are is out of the scope of this paper.  ... 
doi:10.1007/978-3-319-19545-2_4 fatcat:ibbabj5khbecrj3zgvmalpb6ja

Shaking the causal tree: On the faithfulness and minimality assumptions beyond pairwise interactions [article]

Tiago Martinelli, Diogo O. Soares-Pinto, Francisco A. Rodrigues
2022 arXiv   pre-print
We also connect this result with the intrinsic violation of faithfulness and elucidate the importance of the concept of causal minimality.  ...  Built upon the concept of causal faithfulness, the so-called causal discovery algorithms propose the breakdown of mutual information (MI) and conditional mutual information (CMI) into sets of variables  ...  Acknowledgments The authors would like to thanks prof. D. Mašulović for fruitful discussion about connections of Ramsey Theory and spurious correlations. Also, to P.  ... 
arXiv:2203.10665v2 fatcat:k5gk6m44znh3fh47zwgf72a3v4

Conceptual knowledge discovery--a human-centered approach

Joachim Hereth Correia, Gerd Stumme, Rudolf Wille, Uta Wille
2003 Applied Artificial Intelligence  
In this framework, Formal Concept Analysis is considered as a theory for concepts, while Conceptual Graphs [So84] are used as building blocks for a theory for judgments and conclusions.  ...  sets A and B are called the extent and the intent of the formal concept (A, B).  ... 
doi:10.1080/713827122 fatcat:hlnaloupd5ao3dcmfs7vdpuj5m

Revisiting the GreCon Algorithm for Boolean Matrix Factorization

Martin Trnecka, Roman Vyjidacek
2020 International Conference on Concept Lattices and their Applications  
We show that the reasons for not using GreCon algorithm are no longer truth.  ...  We revise the algorithm and on various experiments we demonstrate that the revised version is competitive to current BMF algorithms in term of running time.  ...  JG 2020 003 of the Palacký University Olomouc. Support by Grant No. IGA PrF 2020 030 of IGA of Palacký University is also acknowledged.  ... 
dblp:conf/cla/TrneckaV20 fatcat:slyko6sahffetmg5bi4wrdeuga

A short introduction to the quantum formalism[s] [article]

Francois David
2012 arXiv   pre-print
These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics).  ...  The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation.  ...  They rely on the concept of atoms, or minimal proposition, which are the analog for propositions of the concept of minimal projectors on of pure states in the algebraic formalism. 4.2.5.a -Atoms An element  ... 
arXiv:1211.5627v1 fatcat:rdzoeubuprcsngtr2cgegzzrma

A guide to mathematical quasicrystals [article]

Michael Baake
1999 arXiv   pre-print
The emphasis is on proper generalizations of concepts and ideas from classical crystallography.  ...  Various discrete patterns with non-crystallographic symmetries are described in detail, and some of their magic properties are introduced.  ...  Minimal embedding and further examples In spite of the warnings given, the projection method, if used properly, is an extremely powerful tool, e.g. for practically indexing a diffraction pattern.  ... 
arXiv:math-ph/9901014v1 fatcat:2bxg7xnenzcshntfnzv7xdjdyi

Latticial structures in data analysis

Vincent Duquenne
1999 Theoretical Computer Science  
In general, lattice theory has helped to simplify, unify and generalize many aspects of mathematics, and it has suggested many interesting new problems.  ...  I]) Diagrams prove nothing, but bring outstanding features readily to the eye; they are therefore no substitute for such critical tests as may be applied to the data, but are valuable in suggesting such  ...  Thanks are due to the team URA 1294 for its nice welcome, to Nanou Duquenne and John Mohr for their editorial advice, and to the GLAD users for pointing out bugs and discussing the topics of Lattice Analysis  ... 
doi:10.1016/s0304-3975(98)00279-5 fatcat:t45v75io6baarah5a65jfgjo4e
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