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From an Implicational System to its Corresponding D-basis

Estrella Rodríguez Lorenzo, Kira V. Adaricheva, Pablo Cordero, Manuel Enciso, Angel Mora
2015 International Conference on Concept Lattices and their Applications  
An open problem is to obtain it from an arbitrary implicational system, so it is our aim in this paper.  ...  We introduce a method to compute the D-basis by means of minimal generators calculated using the Simplification Logic for implications.  ...  confusion.From an implicational system to its corresponding D-basis From an implicational system to its corresponding D-basis  ... 
dblp:conf/cla/LorenzoACEM15 fatcat:rnynufch5vff7n67bgqd4udg2m

Bases via Minimal Generators

Pablo Cordero, Manuel Enciso, Angel Mora Bonilla, Manuel Ojeda-Aciego
2013 International Joint Conference on Artificial Intelligence  
In this work we introduce a method to provide a Duquenne-Guigues basis corresponding to the minimal generators and their closed sets from a context.  ...  One outstanding problem in formal concept analysis and other areas is the study of the equivalences between a given set of implications and its corresponding basis (notice that there exists a wide range  ...  The first step is to use MinGen Algorithm introduced in [2] to compute all minimal generators corresponding to an arbitrary set of implications.  ... 
dblp:conf/ijcai/CorderoEBO13 fatcat:fqqvx46prvgj7l573lgt7imbxe

Computing Left-Minimal Direct Basis of implications

Pablo Cordero, Manuel Enciso, Angel Mora, Manuel Ojeda-Aciego
2013 International Conference on Concept Lattices and their Applications  
We propose in this paper, an automated method to calculate a left-minimal direct basis from the set of all implications built between a closed set and its corresponding minimal generators.  ...  these basis from an arbitrary set of implications have been introduced.  ...  , ac → abcd}, which corresponds to a Duquenne-Guigues basis.  ... 
dblp:conf/cla/CorderoEMO13 fatcat:w2ioehqgofan5btyqktu2ktnuy

Direct and Binary Direct Bases for One-set Updates of a Closure System [article]

Kira Adaricheva, Taylor Ninesling
2018 arXiv   pre-print
Our goal is to obtain the same type of implicational basis for the new closure system as was given for original closure system and to make the basis update an optimal process.  ...  We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the D-basis introduced in Adaricheva, Nation, Rand [Disc.Appl.Math  ...  We thank Sergey Obiedkov for pointing to the important publication of Marcel Wild [18] , and we thank Justin Cabot-Miller for his support in producing valuable test cases in the implementation phase.  ... 
arXiv:1810.08684v1 fatcat:xmid3rqszrertnmkxbbpl33zhi

The multiple facets of the canonical direct unit implicational basis

K. Bertet, B. Monjardet
2010 Theoretical Computer Science  
An efficient generation of a full implicational system (or a closure system) can be performed from equivalent implicational systems and in particular from the bases for such systems, for example, the so-called  ...  Formalization of dependencies leads to the notion of so-called full implicational systems (or full family of functional dependencies) which is in one-to-one correspondence with the other significant notions  ...  Acknowledgements We would like to thank the referees and the editor for their helpful comments. We also thank Emily Tanimura and Claude Del Vigna for their help on form and substance.  ... 
doi:10.1016/j.tcs.2009.12.021 fatcat:evbudiiu5bgb3kvcf5loo4lbnq

Ordered direct implicational basis of a finite closure system

K. Adaricheva, J.B. Nation, R. Rand
2013 Discrete Applied Mathematics  
One can extract the D-basis from any direct unit basis S in time polynomial in the size of S, and it takes only linear time of the cardinality of the D-basis to put it into a proper order.  ...  A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration.  ...  We are grateful to several people who read the draft of this paper at different stages of its preparation and made valuable comments.  ... 
doi:10.1016/j.dam.2012.08.031 fatcat:ue5cpvsmarhyxl7rz3ykcneqpi

On implicational bases of closure systems with unique critical sets

K. Adaricheva, J.B. Nation
2014 Discrete Applied Mathematics  
A further refinement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis.  ...  Thus, closure systems without D-cycles can be effectively recognized. While E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.  ...  The first draft of the paper was written during the first author's visit to University of Hawai'i, supported by AWM-NSF Travel Mentor Grant.  ... 
doi:10.1016/j.dam.2013.08.033 fatcat:nnwsgtg4uvaavcfepxbqxnsfr4

The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey

Nathalie Caspard, Bernard Monjardet
2003 Discrete Applied Mathematics  
In particular we show that this lattice is atomistic and lower bounded and that there exists a canonical basis for the representation of any closure system by "implicational" closure systems.  ...  Since the lattices of closure operators and of full implicational systems are anti-isomorphic with the lattice of closure systems they have the dual properties. ?  ...  Janowitz for their remarks that have lead to signiÿcant improvements.  ... 
doi:10.1016/s0166-218x(02)00209-3 fatcat:bn7ilv6rz5gqjmv2vuko6blmzi

LinCbO: fast algorithm for computation of the Duquenne-Guigues basis [article]

Radek Janostik, Jan Konecny, Petr Krajča
2021 arXiv   pre-print
Our experimental evaluation shows that it is the most efficient algorithm for computation of the Duquenne-Guigues basis.  ...  This combination enables us to reuse attribute counters used in LinClosure and speed up the computation.  ...  It uses the attribute lists to find a subset N 1 Ď N of implications whose left-hand side has an attribute not occurring in D (line 10). It uses the rest N z N 1 of implications to extend D.  ... 
arXiv:2011.04928v2 fatcat:hmliwwjk2fctfjph4zal4jwcya

Computing Minimal Generators from Implications: a Logic-guided Approach

Pablo Cordero, Manuel Enciso, Angel Mora, Manuel Ojeda-Aciego
2012 International Conference on Concept Lattices and their Applications  
The main problem is to enumerate all the minimal generators from a set of implications.  ...  The most widely accepted is the Duquenne-Guigues basis, strongly based on the notion of pseudo-intents. In this work we propose the minimal generators as an element to remove redundancy in the basis.  ...  Implications define a closure system, and for this reason, a closure system can be replaced by their equivalent implications.  ... 
dblp:conf/cla/CorderoEMO12 fatcat:b4njyuvrf5cc3jve2utywpbixa

Decision implication canonical basis: a logical perspective

Yanhui Zhai, Deyu Li, Kaishe Qu
2015 Journal of computer and system sciences (Print)  
In this paper, to deduce an analogous basis for decision implications, we introduce the notion of decision premise and form the so-called decision implication canonical basis.  ...  Due to its special role on logical deduction and practical applications of attribute implications, canonical basis has attracted much attention and been widely studied in Formal Concept Analysis.  ...  Acknowledgments The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.  ... 
doi:10.1016/j.jcss.2014.06.001 fatcat:w5kyweubazarpjtenbdy2ozrse

Formation of the D-basis from implicational systems using Simplification logic

Estrella Rodríguez-Lorenzo, Kira Adaricheva, Pablo Cordero, Manuel Enciso, Angel Mora
2017 International Journal of General Systems  
Due to its solid mathematical foundations, Formal Concept Analysis (FCA) has become an emergent topic in the area of data analysis and knowledge discovering.  ...  The aim of these transformations is to turn the original set of implications into an equivalent one fulfilling some desired properties.  ...  If the application of [SiEq] renders an implication A → / 0, it is removed from the output.  ... 
doi:10.1080/03081079.2017.1349632 fatcat:vc7hrznpsbdwxgfmdjql6tbysu

An Inference System for Exhaustive Generation of Mixed and Purely Negative Implications from Purely Positive Ones

Rokia Missaoui, Lhouari Nourine, Yoan Renaud
2010 International Conference on Concept Lattices and their Applications  
To that end, we define a sound and complete inference system which includes a characterization of implications whose left-hand side is a key in the context K| K representing the apposition of the context  ...  K and its complementary K.  ...  All the authors would like to thank the anonymous referees for their helpful suggestions and are grateful to Sebastian Rudolph and Léonard Kwuida for fruitful discussions about empty implication sets.  ... 
dblp:conf/cla/MissaouiNR10 fatcat:m5hurnejjbba3mpltend2o37na

Optimum basis of finite convex geometry [article]

Kira Adaricheva
2016 arXiv   pre-print
We show that the F-basis introduced in [1] for UC-systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises  ...  Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be open. [1] K.  ...  We are grateful to Laurent Beaudau who pointed to the flaw in original argument about optimum basis of geometries of convex sub-posets.  ... 
arXiv:1205.3236v5 fatcat:t2aay3p52nbotgpsjspjmtkflq

Optimum basis of finite convex geometry

K. Adaricheva
2017 Discrete Applied Mathematics  
Convex geometry, anti-exchange closure operator, affine convex geometry, system of implications, canonical basis, optimum basis, minimum basis, acyclic Horn Boolean functions, minimum CNF-representation  ...  The right sides of non-binary implications can also be optimized, when the convex geometry either satisfies the Carousel property, or does not have D-cycles. The latter generalizes a result of P.L.  ...  We are grateful to Laurent Beaudau who pointed to the flaw in original argument about optimum basis of geometries of convex sub-posets.  ... 
doi:10.1016/j.dam.2017.06.009 fatcat:vaayhbyoejhavjuzdelgxhldk4
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