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Studies in Computational Intelligence 819 László T. Kóczy Jesús  Medina-Moreno Elsa Ramírez-Poussa Alexander Šostak   Editors Computational Intelligence and Mathematics for Tackling Complex Problems Studies in Computational Intelligence Volume 819 Series Editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland The series “Studies in Computational Intelligence” (SCI) publishes new developments and advances in the various areas of computational intelligence—quickly and with a high quality The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems, and hybrid intelligent systems Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution, which enable both wide and rapid dissemination of research output The books of this series are submitted to indexing to Web of Science, EI-Compendex, DBLP, SCOPUS, Google Scholar and Springerlink More information about this series at http://www.springer.com/series/7092 László T Kóczy Jesús Medina-Moreno Elsa Ramírez-Poussa Alexander Šostak • • • Editors Computational Intelligence and Mathematics for Tackling Complex Problems 123 Editors László T Kóczy Faculty of Engineering Sciences Széchenyi István University Gỹr, Hungary Budapest University of Technology and Economics Budapest, Hungary Eloísa Ramírez-Poussa Faculty of Economic and Business Sciences Department of Mathematics University of Cádiz Cádiz, Spain Jesús Medina-Moreno Science Faculty Department of Mathematics University of Cádiz Cádiz, Spain Alexander Šostak Institute of Mathematics and Computer Science University of Latvia Riga, Latvia Department of Mathematics University of Latvia Riga, Latvia ISSN 1860-949X ISSN 1860-9503 (electronic) Studies in Computational Intelligence ISBN 978-3-030-16023-4 ISBN 978-3-030-16024-1 (eBook) https://doi.org/10.1007/978-3-030-16024-1 Library of Congress Control Number: 2019934777 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Advances in Computational Intelligence and Mathematics for Tackling Complex Problems Many areas of modern knowledge encounter problems, whose solution is impossible without applying advanced mathematical techniques as well as computational intelligence-based methods Moreover, the interaction and interplay between mathematical techniques and computational technologies are fundamental for the adequate approach of the research of such problems Among the most important mathematical tools for this interaction are fuzzy logic-based mathematical technologies and rough set-based methodologies These technologies and specifically their interplay allow to address different challenges of the present technological age Many areas of modern knowledge encounter the problems the solution of which presupposes the use of both advanced mathematical techniques and computational intelligence-based research methods In this volume, these two research areas, computational intelligence and mathematics, are connected in attractive contributions devoted to the solution of some tempting theoretical and real-world important problems The volume is mainly composed of the extended and reviewed versions of the highest quality papers presented by participants from diverse countries of the world such as Japan, Mexico, Chile and Cuba at the Tenth European Symposium on Computational Intelligence and Mathematics (ESCIM 2018) held in Riga, the capital of Latvia, from October to 10 Besides, the technical programme of the conference included four substantial keynote presentations given by Profs Janusz Kacprzyk from Poland (description of human-centric systems: a crucial role of bipolarity in judgements and intentions), Oscar Castillo from Mexico (nature-inspired optimization of type-2 fuzzy logic controllers), Gabriella Pasi, from Italy (aggregation guided by fuzzy quantifiers in IR and social media analytics) and Alexander Šostak from Latvia (on many-level fuzzy rough approximation systems) In the sequel, we give a brief summary of the contributions contained in this volume v vi Preface The first and the twelfth papers, written by a group of Hungarian authors, are devoted to the development of fuzzy signature-based models In the first paper, a new fuzzy signature modelling packaging decision is developed It is based on logistics expert opinions and aimed to support the decision-making process by choosing the right packaging system specifically aimed for dangerous goods packaging In its turn, the twelfth paper presents a new fuzzy signature-based model for the qualification of residential buildings This model is using a structure of fuzzy signature with variable aggregations, where the definition of aggregation is made by parameters and the values of parameters are changing depending on the specific application The second paper presents a new classifier architecture based on fuzzy fingerprint relevance classifier This classifier is worked out by a group of Portuguese researchers Specifically, this classifier allows to get good results in the process of automatic identifying patient innovation solutions from texts obtained by means of the Web The authors of the third paper, researchers from the Szechenyi Istvan University in Hungary, propose an interesting population-based memetic algorithm, so-called discrete bacterial memetic evolutionary algorithm, appropriate for solving the one-commodity pickup-and-delivery travelling salesman problem The algorithm was tested on benchmark instances, and the results were compared with the state-of-the-art methods in the literature, illustrating the advantages of the proposed algorithm In the fourth paper presented by researchers from University of Debrecen, some interesting possible connections between concepts of roughness and fuzziness are studied It is shown that a rough membership function can be interpreted as a special type of a fuzzy membership function This fact is applied to investigate some interplay between the two theories The main goal of the fifth paper, written by a group of Hungarian scientists, is to find a method that would allow to indicate some characteristic points that can be used for fitting two measured surfaces together In order to get such a method, the authors introduce fuzzy version of a Hough transform that allows to detect straight line segments on the image As different from the classical Hough transform, its fuzzy version is useful in case when the lines are not precisely given or not precisely straight The authors apply this method for finding the same location on two measured versions of a surface The authors of the sixth paper analyse the behaviour of the so-called rescaled algorithm for fuzzy cognitive maps with respect to the existence and uniqueness of fixed points of such maps This problem is important for the use of fuzzy cognitive maps in network-like decision support tools In the seventh paper, the class of group-like uninorms is introduced as a subclass of group-like FLe algebras, where the underlying universe is order isomorphic to the real line interval (0,1) The author, from the University of Pécs in Hungary, presents some methods of construction of such uninorms and besides gives a complete characterization of this class in case of uninorms with a finite number of idempotent elements Preface vii The next paper presents the results of the research work in the field of adjoint triples done in the Department of Mathematics, University of Cádiz, Spain Adjoint triples arise as an important generalization of t-norms and corresponding residuation, since they provide more flexibility and increase the range of possible application The authors analyse how the exchange principle law should be defined on adjoint triples and what conditions the conjunction of an adjoint triple should fulfil in order to guarantee that the corresponding residuated implications satisfy the prescribed property The ninth paper presents the results of the research done by mathematicians from Serbia Basing on the known fact that in every finite poset each element can be presented as a join of completely join-irreducible elements, the authors justify the introduction of a new concept of a poset-valued reciprocal (preference) relations as well as its intuitionistic counterpart In this representation, join-irreducible elements represent pieces of information that reflect the grade of preference in this framework In their previous research, the authors of the tenth paper showed that the category of Chu correspondences between Hilbert contexts is equivalent to the category of propositional systems (the algebraic counterpart of the set of closed subspaces of a Hilbert space) These researchers from Spain and Slovakia extend in the present paper the previously obtained results to the big toy models (in the sense of S Abramsky) introduced as a tool to represent quantum systems in terms of Chu spaces Specifically, the authors obtain a categorical equivalence between big toy models and a suitable subcategory of the category of Hilbert formal contexts and Chu correspondences This result is a new example of interesting structures which are representable in terms of Chu correspondences In the next paper written by Polish scientists, the authors continue their previous research on attribute selection by Hellwig method in case when the data set is expressed via an Atanassov intuitionistic fuzzy set The authors present a novel extension of Hellwig method for the reduction of data, which was primarily proposed for economic data analysis In this method, the authors use three-term representation of fuzzy intuitionistic sets, that is taking into account the degree of membership, non-membership and hesitation The next two papers are written by scientists from different institutions in Poland The thirteenth paper is devoted to the day and night design of a fuzzy system for the classification of blood pressure load The authors analyse the load of 30 patients, which were classified by the fuzzy classifier and indicated a high index of people with a pressure load The executed analysis indicates that for these patients, a cardiovascular event could occur at any time of day and night The authors of the fourteenth paper develop an example of the application of a flower pollination algorithm for the probabilistic neural network learning process Special attention authors pay on the investigation of the inertial parameters of this algorithm In the next paper, written by a group of Japanese scientists and presented by Tsuchiya Takeshi, the authors propose a new search method of various tourist information in one prefecture in Japan This method is using paragraph vector that viii Preface extracts features by a combination of words and word ordering that is included in the content The sixteenth paper is presented by a researcher of the Siemens AG, Corporate Technology in Germany It deals with transform utility values to preference values— the problem actual for decision support and recommender systems Basing on the Łukasiewicz transitivity, the author derives a new transformation and examines its mathematical properties Fuzzy relational equations and inequalities play an important role in many tools of fuzzy modelling and have been extensively studied In the seventeenth paper, the authors from the University of Latvia present a method allowing to convert a system of fuzzy relational constraints with max-t-norm composition to a linear constraint system by adding integer variables A numerical example is given to illustrate this method The eighteenth paper, written by a team composed by researchers from Cuba, Chile and Spain, analyses the main variables—causes and effect—related to the enterprise architecture and the multifactorial elements impregnated with uncertainty that affects it The knowledge given by experts is translated into dependence rules, which also have been analysed from a fuzzy point of view using the fuzzy relation equation theory For the use of time series, it is often crucial to obtain, as much detailed information as possible, from these series The team of researchers from the University of Castilla–La Mancha, from Spain, presents in the nineteenth paper a technique for obtaining linguistic description from time series using a representation called fuzzy piecewise linear segments It is shown how to obtain the information of a modelled series using this representation and the necessary steps to generate the description by using templates The twentieth paper is written by Lithuanian and Hungarian researchers, and it presents a new approach for the evaluation of management questionnaires It combines expert knowledge about the fuzzy signature structure with the hybrid minimization of squared errors among leaves and reconstructed values at the leaves It is shown that this method is more advanced than the mere use of expert knowledge or expert knowledge enhanced with statistical analysis Fuzzy partitions in many cases are the core of the first step of fuzzification procedure They are defined in different ways, but usually by taking into account aspects of the whole universe On the other hand, the twenty-first paper, written by mathematicians from Malaga University, presents a method to define fuzzy partition for elements in the universe holding certain fuzzy attribute Specifically, the presented technique for the construction of fuzzy partitions according to a fuzzy context is based on fuzzy transforms The next two papers are written by Latvian mathematicians In the twenty-second paper, a special construction of a general aggregation operator is proposed This construction allows to aggregate fuzzy sets taking into account the distance between elements of the universe Specifically, the authors describe how this construction could be applied for the risk assessment in the case when a strong fuzzy metric is used to characterize the similarity of objects under evaluation Preface ix In the twenty-third paper, a many-level approach to fuzzy rough approximation for fuzzy sets is developed It is based on the many-level rough approximation operators introduced in the paper Basic properties of such operators are studied Besides, the measure of this approximation is defined and studied This measure in some sense describes the quality of the obtained approximation The last paper in this volume presents the research of mathematicians from University of Cádiz, Spain Here, the philosophy of rough set theory is applied in order to reduce formal context in the environment of formal concept analysis Specifically, a reduction mechanism based on the consideration of bireducts is proposed, and some properties of the reduced contexts are studied Finally, we would like to finish this preface showing our acknowledgement to the authors members of the programme committee and reviewers, since without their effort and interest, this special issue would not have been possible We also acknowledge the support received from the University of Cádiz, the Hungarian Fuzzy Association, the Szechenyi Istvan University, the Institute of Mathematics and CS, University of Latvia and the State Research Agency (AEI) and the European Regional Development Fund (FEDER) research project TIN2016-76653-P Finally, a word of thanks is also due to EasyChair, for the facilities provided in the submission/acceptance of the papers, and in the preparation of this book Gyõr/Budapest, Hungary Cádiz, Spain Cádiz, Spain Riga, Latvia October 2018 László T Kóczy Jesús Medina-Moreno Elsa Ramírez-Poussa Alexander Šostak 186 A Šostak et al M-Level Rough Approximation of an L-Fuzzy Set Given an M-level L-fuzzy preoder R : X × X × M → L, we define the upper M-level rough approximation operator u R : L X × M → L X and the lower Mlevel rough approximation operator l R : L X × M → L X by u R (A)(x)(α) = y ((R(y, x, α)) ∗ A(y)) and l R (A)(x)(α) = y ((R(x, y, α)) → A(y)) Theorem M-rough approximation operators satisfy the following properties: (1u) u R (a X , α) = a X ∀α ∈ M where a X : X → L is constant with value a; (2u) A ≤ u R (A, α) ∀A ∈ L X , ∀α ∈ M; X (3u) u R i Ai , α = i u R (Ai , α) ∀{Ai | i ∈ I } ⊆ L ∀α ∈ M; X (4u) u R (u R (A, α), α) = u R (A, α) ∀A ∈ L ∀α ∈ M; (1l) l R (a X , α) = a X ∀α ∈ M; (2l) A ≥ l R (A, α) ∀A ∈ L X ∀α ∈ M; X (3l) l R i Ai , α = i l R (Ai , α) ∀{Ai | i ∈ I } ⊆ L ∀α ∈ M; X (4l) l R (l R (A, α), α) = l R (A, α) ∀A ∈ L ∀α ∈ M Theorem Let A be an L-fuzzy subset of a set X endowed with an M-level L-fuzzy preoder R and α ∈ M Then u R (A, α) is the extensional hull and l R (A, α) is the extensional kernel of A in the L-fuzzy preodered space (X, R α ) where R α is the restriction of R to [0, 1] × [0, 1] × {α} Theorem For every L-fuzzy set A and every α ∈ M, it holds u R (l R (A, α), α) = l R (A, α) and l R (u R (A, α), α) = u R (A, α) Definition The pair (u R , l R ), where R : X × X × M → L is an M-level L-fuzzy preoder on X and u R , l R : L X × M → L X are upper and lower M-level rough approximation operators generated by R, is called an M-level rough approximative pair, and the corresponding triple (X, u R , l R ) an M-level rough approximation space Definition Let (X, u R X , l R X ) and (Y, u RY , l RY ) be M-level rough approximation spaces A mapping f : X → Y is called continuous if (1) f (u R X (A)) ≤ u RY ( f (A)) for every A ∈ L X , and (2) f −1 (l RY (B)) ≤ l R X ( f −1 (B)) for every B ∈ L Y By MRAS we denote the category of M-level rough approximation spaces and their continuous mappings Theorem If f : (X, R X ) → (Y, RY ) is a monotone mapping (that is R X (x, x , α) ≤ RY ( f (x), f (x ), α) ∀x, x ∈ X, α ∈ M), then the mapping f : (X, u R X , l R X ) → (Y, u RY , l RY ) is continuous Hence by assigning the space (X, u R X , l R X ) to an M-level L-fuzzy preodered set (X, R) and interpreting a monotone mapping f : (X, R X ) → (Y, RY ) as a mapping f : (X, u R X , l R X ) → (Y, u RY , l RY ) we get an embedding functor from the category of M-level L-fuzzy preodered sets into the category MRAS Remark The results presented in this section is a many-level generalization of the corresponding results of the theory developed in [3] On the Measure of Many-Level Fuzzy Rough Approximation for L-Fuzzy Sets 187 Measure of M-Level L-Fuzzy Rough Approximation 4.1 Measure of Inclusion of L-Fuzzy Sets Definition By setting A → B = x∈X (A(x) → B(x)) where A, B ∈ L X and →: L × L → L is the residuum corresponding to the operation ∗ by Galois connection, we obtain a mapping →: L X × L X → L We call A → B by the measure of inclusion of the L-fuzzy set A into the L-fuzzy set B The following properties of the mapping →: L X × L X → L can be found in the recent works of different authors; X X (1) i Ai → B = i (Ai → B) ∀{Ai | i ∈ I } ⊆ L , B ∈ L , X X (2) A → ( i Bi ) = i (A → Bi ) ∀A ∈ L , {Bi | i ∈ I } ⊆ L ; (3) A → B = L whenever A ≤ B; (4) X → A = x A(x) ∀A ∈ L X ; (5) (A → B) ≤ (A ∗ C → B ∗ C) ∀A, B, C ∈ L X ; (6) (A → B) ∗ (B → C) ≤ (A → C) ∀A, B, C ∈ L X ; X (7) i Ai → i Bi ≥ i (Ai → Bi ) ∀{Ai : i ∈ I }, {Bi : i ∈ I } ⊆ L ; X → ≥ A B (A → B ) ∀{A : i ∈ I }, {B : i ∈ I } ⊆ L (8) i i i i i i i i i 4.2 Measure of M-Level L-Fuzzy Rough Approximation Let (X, R) be an M-level L-fuzzy preodered set Given an L-fuzzy set A ∈ L X , we define the measure U(A, ·) : M → L of its upper M-level L-fuzzy rough approximation by U R (A, α) = u R (A, α) → A and the measure L(A, ·) : M → L of its lower M-level L-fuzzy rough approximation by L R (A, α) = A → l R (A, α) If R is symmetric then U R (A, α) = L R (A, α) for every L-fuzzy set A In this case we call R R (A, ·) := U R (A, ·) = L R (A, ·) by the measure of M-level L-fuzzy rough approximation of an L-fuzzy set A The above defined measures of lower and upper M-level rough approximation of L-fuzzy sets give rise to the M-level operators of upper and lower L-fuzzy rough approximation U R : L X × M → L and L R : L X × M → L and the operator of Mlevel rough approximation R R : L X × M → L if R is symmetric In the next theorem we collect the main properties of these operators Theorem U R (a X , α) = L ∀α ∈ M; L R (a X , α) = L ∀α ∈ M; U R (u R (A, α), α) = L ∀A ∈ L X , ∀α ∈ M; L R (l R (A), α), α) = L ∀A ∈ L X ∀α ∈ M; U R ( i Ai , α) ≥ i U R (Ai , α) ∀{Ai | i ∈ I } ⊆ L X ∀α ∈ M; U R ( i Ai , α) ≥ i U R (Ai , α) ∀{Ai | i ∈ I } ⊆ L X , ∀α ∈ M; L R ( i Ai , α) ≥ i L R (Ai , α) ∀{Ai | i ∈ I } ⊆ L X , ∀α ∈ M; L R ( i Ai , α) ≥ i L R (Ai , α) ∀{Ai | i ∈ I } ⊆ L X , ∀α ∈ M; 188 A Šostak et al U R (a X ∗ A, α) ≥ U R (A, α) for all A ∈ L X and all constants a X ; 10 L R (a X → A, α) ≥ L R (A, α) for all A ∈ L X and all constants a X Remark We know that u R (A, α) is the extensional hull and l R (A, α) is the extensional kernel of the L-fuzzy set A in the L-fuzzy preodered set (X, R α ) This observation allows to interpret U R (A, ·) as the M-level measure of upper extensionality, L R (A, ·) as the M-level measure of lower extensionality and in case R is symmetric, to interpret R R (A, ·) as the measure of M-extensionality of an L-fuzzy set A in the M-level L-fuzzy preodered set (X, R) 4.3 On the Category of M-Level L-Fuzzy Rough Approximation Spaces Given an M-level L-fuzzy preodered set (X, R), we call the quadruple (X, R, U R , L R ) by an M-level L-fuzzy rough approximation space Definition We call a mapping of M-level L-fuzzy rough approximation spaces f : (X, U R X , L R X ) → (Y, U RY , L RY ) continuous if (1con) U R X ( f −1 (B)) ≥ UY (B) ∀B ∈ L Y ; (2con) L R X ( f −1 (B)) ≥ LY (B) ∀B ∈ L Y Let MLRAS be the category whose objects are M-level L-fuzzy rough approximation spaces and whose morphisms are continuous mappings Theorem Let R X : X × X × M → L and RY : Y × Y × M → L be M-level Lfuzzy preoders on sets X and Y respectively and let f : (X, R X ) → (Y, RY ) be a monotone mapping Then the mapping f : (X, U R X , L R X ) → (Y, U RY , L RY ) is continuous Thus assigning the M-level L-fuzzy rough approximation space (X, U R X , L R X ) to an M-level L-fuzzy preoder space (X, R) and interpreting monotone mappings f : (X, R X ) → (Y, RY ) as mappings f : (X, U R X , L R X ) → (Y, U RY , L RY ), we obtain an embedding functor from the category of M-level L-fuzzy preodered sets into the category MLRAS 4.4 Ditopological Interpretation of M-Level L-Fuzzy Rough Approximation Spaces Let (X, L R , U R ) be an M-level L-fuzzy rough approximation space Properties (1), (5) and (6) of Theorem characterize L R : L X × M → L as an M-level enriched Lfuzzy topology on the set X [6] In its turn, properties (2), (7) and (8) characterize U R : L X × M → L as an M-level enriched L-fuzzy co-topology Now from Theorem we get the following: On the Measure of Many-Level Fuzzy Rough Approximation for L-Fuzzy Sets 189 Theorem Let (X, R) be an M-level L-fuzzy preodered set Then the triple (X, L X , U X ) is an enriched M-level L-fuzzy ditopology.1 Theorem By assigning the M-level L-fuzzy ditopological space (X, L R X , U R X ) to an M-level L-fuzzy-preodered set (X, R) and interpreting monotone mappings f : (X, R X ) → (Y, RY ) as mappings of the corresponding M-level L-fuzzy ditopological spaces f : (X, L R X , U R X ) → (Y, L RY , U RY ), we get an embedding functor from the category of M-level L-fuzzy preodered sets into the category of M-level enriched L-fuzzy ditopological spaces 4.5 Examples of Measures for Many-Level Fuzzy Rough Approximation of L-Fuzzy Sets The case of Łukasiewicz t-norm Let ∗ L be the Łukasiewicz t-norm on the interval L = [0, 1], and → L : L × L → L be the corresponding residuum Then, given an M-level L-fuzzy relation R on a set X , A ∈ L X and α ∈ M, we have: U R (A, α) = x x (2− A(x)+ A(x )− R(x, x , α)) L R (A, α) = x x (2− A(x)+ A(x )− R(x , x, α)) In particular, if R : X × X × M → [0, 1] is global, then R R (A, M ) = L and R R (A, M ) = − inf x,x | A(x) − A(x ) | for all A ∈ L X The case of the minimum t-norm Let ∗ = ∧ be the minimum t-norm on the unit interval L = [0, 1], and →: L × L be the corresponding residuum, Then U R (A, α) = inf x,x (A(x ) ∧ R(x, x , α) → A(x)), L R (A, α) = inf x,x (A(x ) ∧ R(x , x, α) → A(x)) In particular, if R is global, then R R (A, α) = L ∀A ∈ L X , α ∈ M The case of the product t-norm Let ∗ = · be the product t-norm on the unit interval [0,1] and →: L × L be the corresponding residuum, Then U(A, α) = inf x,x (A(x ) · R(x, x , α) → A(x)), L(A, α) = inf x,x (A(x ) · R(x , x, α) → A(x)) In particular in case R is global, A(x) R R (A, M ) = L and R R (A, M ) = inf x,x ∈X A(x ) Conclusions Basing on the research done in our papers [3, 4, 8, 9], we initiate here the many level approach to fuzzy rough approximation for L-fuzzy sets, introduce the measure of the quality of this approximation, illustrate it with examples and sketch its topological interpretation As the main perspectives for the further work, we see both developing its theoretical aspects and applications to problems of practical nature We speak here about a ditopology [1] and not a topology since the degrees of openness and closedness of L-fuzzy sets in our case may be unrelated 190 A Šostak et al Concerning the theoretical issues, as first, we plan to develop further the qualitative approach to the theory of many-level fuzzy rough approximation for L-fuzzy sets in the framework of category theory An investigation of the relations between the many-level approach to rough approximation and the theory of multigranual rough sets is one of the prospectives for the future work [10] As possible applications of our approach to practical problems, as the first one we see image processing The idea of this application was sketched by an example in the Introduction Besides, we guess that our approach could be helpful when studying the problems of decision making in fuzzy environment Acknowledgements The author is thankful to the referees for reading the paper carefully and making some remarks that allowed to improve the exposition References Brown, L.M., Ertürk, R., Dost, S.: ¸ Ditopological texture spaces and fuzzy topology, I Basic Concepts Fuzzy Sets Syst 110, 227–236 (2000) Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets Intern J Gen Syst 17, 191–209 (1990) E¸lkins, A., Šostak, A., U¸ljane, I.: On a category of extensional fuzzy rough approximation operators In: Communication in Computer Information Science, vol 611 (2016) Han, S.-E., Šostak, A.: On the measure of M-rough approximation of L-fuzzy sets Soft Comput 22, 2843–2855 (2018) Höhle, U.: M-valued sets and sheaves over integral commutative cl-monoids, Chapter In: Rodabaugh, S.E., Klement, E.P., Höhle, U (eds.) Applications of Category Theory to Fuzzy Sets, pp 33–73 Kluwer Acad Publ (1992) Höhle, U., Šostak, A.: Axiomatic foundations of fixed-based fuzzy topology, In: Höhle, U., Rodabaugh, S (eds.) Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, pp 123–272 Kluwer Acad Publ (1999) Pawlak, Z.: Rough sets Intern J Comput Inform Sci 11, 341–356 (1982) Šostak, A., E¸lkins, A.: LM-valued equalities, LM-rough approximation operators and MLgraded ditopologies Hacettepe J Math Stat 46, 15–32 (2017) Šostak A., U¸ljane, I.: Bornological structures on many-valued sets RadHAZU Matematiˇcke Znanosti 21, 145–170 (2017) 10 Yao, Y., She, Y.: Rough set models in multigranual spaces Inform Sci 327, 40–56 (2016) Bireducts in Formal Concept Analysis M José Benítez-Caballero, Jesús Medina-Moreno and Eloísa Ramírez-Poussa Abstract In this paper we apply the philosophy of Rough Set Theory to reduce formal context in the environment of Formal Concept Analysis Specifically, we propose a reduction mechanism based on the consideration of bireducts and we also study several properties of the reduced contexts Keywords Formal concept analysis · Rough set theory · Bireduct · Size reduction Introduction In the last years, the use of databases in order to store knowledge has increased and the necessity to extract information from them is a research topic increasingly important For that reason, some mathematical tools have been studied Two of them are Rough Set Theory (RST) and Formal Concept Analysis (FCA), which deal with databases composed of objects and attributes related between them One of the main goals in both theories is the reduction of the size of the database keeping the main knowledge In order to reduce the attributes of databases, reducts (minimal subsets of attributes) have been studied in many works within both theories [2, 5–7, 10, 12] In [3], we studied the effects of applying the philosophy of reduction given in RST to remove attributes of databases corresponding to FCA en- Partially supported by the State Research Agency (AEI) and the European Regional Development Fund (FEDER) project TIN2016-76653-P M J Benítez-Caballero (B) · J Medina-Moreno · E Ramírez-Poussa Department of Mathematics, University of Cádiz, Cádiz, Spain e-mail: mariajose.benitez@uca.es J Medina-Moreno e-mail: jesus.medina@uca.es E Ramírez-Poussa e-mail: eloisa.ramirez@uca.es © Springer Nature Switzerland AG 2020 L T Kóczy et al (eds.), Computational Intelligence and Mathematics for Tackling Complex Problems, Studies in Computational Intelligence 819, https://doi.org/10.1007/978-3-030-16024-1_24 191 192 M J Benítez-Caballero et al vironments Specifically, we consider the reducts of RST to reduce formal contexts given in FCA theory A new point of view of the notion of reduction arose within RST, with the introduction of bireducts [1, 4, 11] which purposes to reduce the set of objects as well as the set of attributes In this paper, continuing with the idea presented in [3], we analyze the results of carrying out the reduction of a formal context considered in FCA theory by means of the bireducts given in RST The paper is organized as follows: we recall some basic notions and results of FCA and RST in Sect In Sect 3, we present the reduction mechanism considering bireducts and its relationship with FCA Finally, conclusions and future works are presented in Sect Preliminaries In this section, we will recall the basic notions of FCA and RST needed to understand this work In Formal Concept Analysis (FCA) one of the most basic notions is the definition of context, which is a representation of the data of a knowledge system by means of a triple (A, B, R), that is, two non empty sets and a crisp relation between them This relationship, defined as R : A × B → {0, 1}, where A is the set of attributes and B the set of objects, takes the value if a and b are related, for each a ∈ A and b ∈ B, written as R(a, b) = or a Rb If they are not related the relation values is From a context, two concept-forming operators are defined ↑ : B → A , ↓ : A → B , for each X ⊆ B and Y ⊆ A, as follows: X ↑ = {a ∈ A | for all b ∈ X, a Rb} = {a ∈ A | if b ∈ X, then a Rb} Y ↓ = {b ∈ B | for all a ∈ Y, a Rb} = {b ∈ B | if a ∈ Y, then a Rb} (1) (2) Let us consider a pair of subsets X ⊆ B and Y ⊆ A in the context (A, B, R), we say that this pair (X, Y ) is a concept, if the equalities X ↑ = Y and Y ↓ = X are satisfied We also can define the extent (Ext) of the concept (X, Y ) as the subset of objects X and the intent (Int) as the subset of attributes Y The set of all the concepts, denoted as C(A, B, R) with the inclusion ordering on the left argument has the structure of a complete lattice [9] Also, we can consider the opposite inclusion ordering on the right argument For all pair of concepts (X , Y1 ), (X , Y2 ) ∈ C(A, B, R), we define the meet (∧) and join (∨) operators as: (X , Y1 ) ∧ (X , Y2 ) = (X ∧ X , (Y1 ∨ Y2 )↓↑ ) (X , Y1 ) ∨ (X , Y2 ) = ((X ∨ X )↑↓ , Y1 ∧ Y2 ) Bireducts in Formal Concept Analysis 193 Additionally, considering an object b ∈ B, we can define the concept generated by b as the concept (b↑↓ , b↑ ) It is called object-concept We can assert that an objectconcept is a concept for sure because the pair (↑ , ↓ ) is a Galois connection [8] A subcontext of a context (A, B, R) can be obtained from the reduction of the set of attributes and/or a set of object In this case, the subcontext is the triple (Y, Z , R ∩ (Y × Z )), with Y ⊆ A and Z ⊆ B In this subcontext, the concept-forming operators Y ×Z are defined analogously to Eqs (1) and (2) and are denoted as ↓ and ↑Y ×Z The notion of irreducible element of a lattice is recalled next Definition Given a lattice (L , ), such that ∧, ∨ are the meet and the join operators, and an element x ∈ L verifying If L has a top element , then x = If x = y ∧ z, then x = y or x = z, for all y, z ∈ L we call x meet-irreducible (∧-irreducible) element of L A join-irreducible (∨-irreducible) element of L is defined dually Now, we are going to recall the notions needed from Rough Set Theory (RST) First, the notion of information system and indiscernibility relation is presented Definition An information system (U, A) is a tuple, where U = {1, , n} and A = {a1 , , am } are finite, non-empty sets of objects and attributes, respectively Each a in A corresponds to a mapping a¯ : U → Va , where Va is the value set of a over U Definition Given an information system (U, A) and D ⊆ A, the D-indiscernibility relation, I D is defined as the equivalence relation ¯ = a( ¯ j)} I D = {(i, j) ∈ U × U | for all a ∈ D, a(i) (3) where each class can be written as [k] D = {i ∈ U | (k, i) ∈ I B } I D produces a partition on U denoted by U/I D = {[k] D | k ∈ U } When D = {a}, i.e., D is a singleton, we will write Ia instead of I{a} Given two objects i, j ∈ U and a subset D ⊆ A, if the set {a ∈ D | a(i) ¯ = a( ¯ j)} = ∅, we say that D discerns the objects i and j, or equivalently, that the objects i, j ∈ U are discernible by D The main idea of a bireduct is to reduce the sets of attributes and objects, with the goal of preventing the occurrence of incompatibilities and eliminating existing noise in the original data The following definition formalizes this idea Definition Given an information system (U, A), we consider a pair (X, D), where X ∈ U is a subset of objects and D ∈ A is a subset of attributes We say that (X, D) is an information bireduct if and only if every pair of objects i, j ∈ X are discernible by D and the following properties hold: 194 M J Benítez-Caballero et al – There is no subset C D such that C discerns every pair of objects of X – There is no subset of objects X Y such that D discern every pair of objects of Y In order to make easier the computation of bireducts, we consider discernibility matrices and discernibility functions The details of this definition and the considered result are in [4] Definition Given an information system (U, A), its discernibility matrix is an equivalence matrix with order |U | × |U |, denoted as MA , in which the element ¯ = MA (i, j) for each pair of objects (i, j) is defined as: MA (i, j) = {a ∈ A | a(i) a( ¯ j)}, and the discernibility function of (U, A) is defined as: i∨j∨ bir τU,A = {a ∈ A | a(i) ¯ = a( ¯ j)} i, j∈U The next theorem shows a mechanism to obtain bireducts Theorem ([4]) Given a boolean information system (U, A) An arbitrary pair of sets (X, D), where X ⊆ U , D ⊆ A, is a bireduct of the information system if and only if the cube a∈D a ∧ i ∈X / i is a cube in the restricted disjunctive normal form bir (RDNF) of τU,A Bireducts applications in FCA This paper proposes to use the reduction given by a bireduct in RST to reduce the set of attributes and the set of objects of a context in the FCA framework From this reduction a subcontext of the original one will be obtained preserving the philosophy in RST, that is, a consistent pair of objects and attributes is obtained, from which the objects of the subcontext that were discernible in the original context continue been discernible, and superfluous attributes, that are not necessary to preserve this property, are removed from the original context Hence, this section will present several properties of this reduction and how the proposed mechanism is applied to an example Hereon, we will consider a finite set of objects and a finite set of attributes First of all, we present how to define an information system from a given context Definition Let (A, B, R) be a context, a context information system is defined as the pair (B, A) where the mappings a¯ : B → {0, 1}, are defined as a(b) ¯ = R(a, b), for all a ∈ A, b ∈ B Clearly, a context information system is a particular case of information system where the valued set Va of each attribute a ∈ A is the boolean set {0, 1} Now, we detail the proposed mechanism First of all, we consider a context (A, B, R) Then, we build the corresponding context information system, according to Definition 6, Bireducts in Formal Concept Analysis 195 and we compute the bireducts of this environment, applying Theorem Finally, we reduce the original context considering some of the obtained bireducts The following results show how the extension operators on the original context and on the subcontext obtained from a bireduct are related Proposition Let (A, B, R) be a context, (B, A) the corresponding context information system and (X, D) a bireduct of (B, A) If two objects k, j ∈ X fulfill that k ↑ = j ↑ , then the inequality k ↑ X ×D = j ↑ X ×D is satisfied Moreover, the (strict) inequality between object-concepts is preserved Proposition Given a context (A, B, R) and its corresponding context information system (B, A) If the pair (X, D) is a bireduct of (B, A), for every two objects j ↑ , we have that the inequality k ↑ X ×D j ↑ X ×D holds k, j ∈ X satisfying that k ↑ As a consequence, the ordering among the object-concepts is practically maintained carrying out this kind of reduction In [3], it was proved that there is no new join-irreducible elements in the reduced concept lattice considering a consistent set of RST framework That is, if an object does not generate a join-irreducible concept in the original context, then it cannot generate a join-irreducible concept in the reduced one In the following example, as we can expect, we will see that this statement is not necessarily satisfied if we consider bireducts for reducing a formal context Example In this example, the proposed reduction mechanism based on the computation of bireducts will be applied The considered formal context (A, B, R) was presented in [3], and consist of a group of cultivated fields (objects), the set A = {high temperature (ht), humidity (hh), windy (wa), fertilizer (f), pesticide (p)} represents the attributes and a relation R given by the following table: R ht 0 0 1 hh 0 1 0 wa 1 1 1 f 1 1 p 1 1 0 The concept lattice associated with this context is displayed on the left of Fig The right side of this figure shows the extent and intent of each concept of the concept lattice The discernibility matrix1 corresponding to the associated information system is displayed below: Since the discernibility relationship is symmetric, the discernibility matrix is a symmetric matrix 196 M J Benítez-Caballero et al Ci C9 C8 C10 C6 C11 C7 C4 C5 C12 C1 C2 C3 C0 C13 10 11 12 13 Generated by object {} {ht, hh, wa, f,p} {1} {ht, wa, f,p} {4} {hh, wa, f,p} {7} {ht, hh, wa, f} {1, 4, 5} {wa, f,p} {1, 7} {ht, wa, f} {1, 2, 4, 5} {wa, p} {1, 4, 5, 7} {wa, f} {1, 2, 4, 5, 6, 7} {wa} {1, 2, 3, 4, 5, 6, 7} {} {1, 3, 4, 5, 7} {f} {1, 6, 7} {ht, wa} {4, 7} {hh, wa, f} {3, 4, 7} {hh, f} Extent Intent Fig Concept lattice and fuzzy concepts of the context of Example ⎞ ∅ ⎟ ⎜ {ht, f} ∅ ⎟ ⎜ ⎟ ⎜{ht,hh,wa,p} {hh,wa,f,p} ∅ ⎟ ⎜ ⎟ ⎜ {ht,hh} {hh,f} {wa,p} ∅ ⎟ ⎜ ⎟ ⎜ {ht} {f} {hh,wa,p} {hh} ∅ ⎟ ⎜ ⎠ ⎝ {f,p} {ht,p} {ht,hh,wa,f} {ht,hh,f,p} {ht,f,p} ∅ {hh,p} {ht,hh,f,p} {ht,wa} {ht,p} {ht,hh,p} {hh,f} ∅ ⎛ We can build the discernibility function considering the matrix above: bir τU, A = {1 ∨ ∨ ht ∨ f} ∧ {1 ∨ ∨ ht ∨ hh ∨ wa ∨ p} ∧ {1 ∨ ∨ ht ∨ hh} ∧ {1 ∨ ∨ ht} ∧ {1 ∨ ∨ f ∨ p} ∧ {1 ∨ ∨ hh ∨ p} ∧ {2 ∨ ∨ hh ∨ wa ∨ f ∨ p} ∧ {2 ∨ ∨ hh ∨ f} ∧ {2 ∨ ∨ f} ∧ {2 ∨ ∨ ht ∨ p} ∧ {2 ∨ ∨ ht ∨ hh ∨ f ∨ p} ∧ {3 ∨ ∨ wa ∨ p} ∧ {3 ∨ ∨ hh ∨ wa ∨ p} ∧ {3 ∨ ∨ ht ∨ hh ∨ wa ∨ f} ∧ {3 ∨ ∨ ht ∨ wa} ∧ {4 ∨ ∨ hh} ∧ {4 ∨ ∨ ht ∨ hh ∨ f ∨ p} ∧ {4 ∨ ∨ ht ∨ p} ∧ {5 ∨ ∨ ht ∨ f ∨ p} ∧ {5 ∨ ∨ ht ∨ hh ∨ p} ∧ {6 ∨ ∨ hh ∨ f} We use the usual laws of classical logic in order to obtain the restricted disjunctive normal form and, from that, the bireducts Some of them are the following: (X , D1 ) = ({1, 3, 4, 5, 6, 7}, {ht, hh, p}) (X , D2 ) = ({1, 2, 3, 5, 6, 7}, {ht, f, p}) Bireducts in Formal Concept Analysis 197 ({1, 2, 3, 5, 6, 7}, {}) ({1, 3, 4, 5, 6, 7}, {}) (Ext(C4 ), {p}) (Ext(C11 ), {ht}) (Ext(C13 ), {hh}) ({1, 2, 5}, {p}) (Ext(C1 ), {ht, p}) (Ext(C2 ), {hh, p}) (Ext(C3 ), {ht, hh}) ({1, 5}, {f, p}) ({1, 3, 5, 7}, {f }) (Ext(C11 ), {ht}) (Ext(C5 ), {ht, f }) (Ext(C1 ), {ht, f, p}) (Ext(C0 ), {ht, hh, p}) Fig Reduced concept lattice from bireducts (X , D1 ) (left) and (X , D2 ) (right) The reduced concept lattices considering bireducts (X , D1 ) and (X , D2 ) are shown in Fig Given this figure, it is easy to check that if we take into consideration the bireduct (X , D1 ), all the extents of the concepts from the reduced concept lattice are equal to the extents of the original one, except for the top element We have denoted Ext(C) to the extent of the concept C in Fig In addition, there is no new join-irreducible elements The set of join-irreducible elements is listened below and they are generated by objects 1, and 72 : (1↑1 ↓ , 1↑1 ) = ({1}, {ht, p}) (4↑1 ↓ , 4↑1 ) = ({4}, {hh, p}) (7↑1 ↓ , 7↑1 ) = ({7}, {ht, hh}) On the other hand, if we consider the bireduct (X , D2 ), we can see that the extents are equal or are contained into the extents of concepts of the original concept lattice In this case, a new join-irreducible element appears when the reduction is applied Note that the concept generated by object is not a join-irreducible concept in the original concept lattice, because it is the supremum of the concepts generated by object and object The object is erased when the context is reduced considering the second bireduct Therefore, the concept generated by the object in the reduced concept lattice is a join-irreducible element The others join-irreducible concepts in the reduced concept lattice are preserved, that is, the concept generated by 2, and are also join-irreducible elements of the original concept lattice Conclusions and future works In this paper we have continued the idea presented in [3] considering the bireducts given in RST in order to reduce formal contexts in FCA Moreover, we have analyzed the behavior of such kind of reduction, obtaining interesting properties simplicity, we will write (↑1 , ↓ ) instead of (↑ X ×D1 , ↓ operators defined on the subcontext (X , D1 , R|X ×D1 ) For X ×D1 ) to denote the concept-forming 198 M J Benítez-Caballero et al In the future, a more in-depth study of the reduction mechanism introduced in this paper will be carried out and it will be applied to real examples References ´ ezak, D.: Reducing information systems considering similarity rela1 Benítez, M., Medina, J., Sl¸ tions In: Kacprzyk, J., Koczy, L., Medina, J (eds.) 7th European Symposium on Computational Intelligence and Mathematices (ESCIM 2015), pp 257–263 (2015) Benítez-Caballero, M.J., Medina, J., Ramírez-Poussa, E.: Attribute reduction in rough set theory and formal concept analysis In: Polkowski, L., Yao, Y., Artiemjew, P., Ciucci, D., Liu, D., ´ ezak, D., Zielosko, B (eds.) 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Advances in Fuzzy Logic and Technology, pp 177–186 Springer International Publishing, Cham (2017) ´ ezak, D.: Bireducts with tolerance Benítez-Caballero, M.J., Medina, J., Ramírez-Poussa, E., Sl¸ relations Inf Sci 435, 26–39 (2018) Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Attribute reduction in multi-adjoint concept lattices Inf Sci 294, 41–56 (2015) Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Attribute and size reduction mechanisms in multi-adjoint concept lattices J Comput Appl Math 318, 388–402 (2017) Computational and Mathematical Methods in Science and Engineering (CMMSE-2015) Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Characterizing reducts in multi-adjoint concept lattices Inf Sci 422, 364–376 (2018) Denecke, K., Erné, M., Wismath, S.L (eds.): Galois Connections and Applications Kluwer Academic Publishers, Dordrecht, The Netherlands (2004) Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundation Springer (1999) 10 Shao, M.-W., Li, K.-W.: Attribute reduction in generalized one-sided formal contexts Inf Sci 378, 317–327 (2017) ´ ezak, D.: Recent advances in decision bireducts: complexity, heuristics and 11 Stawicki, S., Sl¸ streams Lect Notes in Comput Sci 8171, 200–212 (2013) 12 Yao, Y., Zhang, X.: Class-specific attribute reducts in rough set theory Inf Sci 418–419, 601–618 (2017) Author Index A Alfonso-Robaina, D., 139 Almeida, João N., Asmuss, Svetlana, 131, 175 Azevedo, Salomé, B Bưrưcz, Péter, Benítez-Caballero, José, 191 Bujnowski, Paweł, 81 Bukovics, Á., 91 C Carvalho, Joao P., Cornejo, M Eugenia, 59 Csajbók, Zoltán Ernő, 23 D Díaz-Gómez, Sergio, 167 Díaz-Moreno, J C., 139 Djukić, Marija, 67 E Elkins, Aleksandrs, 183 F Fogarasi, G., 1, 91 Földesi, Péter, 15 G Guzmán, Juan Carlos, 99 H Harmati, István Á., 43 Hirose, Hiroo, 115 J Jenei, Sándor, 51 Jimenez-Linares, Luis, 149 K Kacprzyk, Janusz, 81 Kóczy, László T., 1, 15, 35, 43, 91, 157 Kưdmưn, József, 23 Kowalski, Piotr A., 107 Koyanagi, Keiichi, 115 Krídlo, O., 75 L Lama, Reinis, 131 M Madrid, Nicolás, 167 Malleuve-Martınez, A., 139 Medina-Moreno, Jesús, 59, 139, 191 Melin, Patricia, 99 Miyosawa, Tadashi, 115 Moreno-Garcia, Antonio, 149 Moreno-Garcia, Juan, 149 N Nagy, Szilvia, 35 O Ojeda-Aciego, M., 75 Orlovs, Pavels, 175 P Prado-Arechiga, German, 99 Purvinis, Ojaras, 157 © Springer Nature Switzerland AG 2020 L T Kóczy et al (eds.), Computational Intelligence and Mathematics for Tackling Complex Problems, Studies in Computational Intelligence 819, https://doi.org/10.1007/978-3-030-16024-1 199 200 Author Index R Ramírez-Poussa, Eloísa, 59, 191 Rodriguez-Benitez, Luis, 149 Rubio-Manzano, C., 139 Runkler, Thomas A., 123 Tsuchiya, Takeshi, 115 Tüű-Szabó, Boldizsár, 15 S Sarkadi-Nagy, Balázs, 35 Sawano, Hiroaki, 115 Solecki, Levente, 35 Šostak, Alexander, 183 Susnienė, Dalia, 157 Sziová, Brigita, 35 Szmidt, Eulalia, 81 V Vöröskői, Kata, T Tepavčević, Andreja, 67 U Uljane, Ingrida, 183 W Wadas, Konrad, 107 Y Yamada, Tetsuyasu, 115 ... Springer Nature Switzerland AG 2020 L T Kóczy et al (eds. ), Computational Intelligence and Mathematics for Tackling Complex Problems, Studies in Computational Intelligence 81 9, https://doi.org/10.1007/978-3-030-16024-1_1... inherent difficulties relate to a crucial role related to inherent characteristics of all human-centric problems, i.e a need to take into account affects, judgments, attitudes, evaluations and intentions... variables—causes and effect—related to the enterprise architecture and the multifactorial elements impregnated with uncertainty that affects it The knowledge given by experts is translated into dependence
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