Journal paper
Journal paper accepted
30/05/22/11:27
N. Madrid, C. Cornelis. Kitainik axioms do not characterize the class of inclusion measures based on contrapositive fuzzy implications. Information Sciences, 2022. To appear.
ABSTRACT In this short communication, we refute the conjecture by Fodor and Yager that the class of inclusion measures proposed by Kitainik coincides with that of inclusion measures based on contrapositive fuzzy implications. In particular, we show that the conjecture only holds when the considered universe of discourse is finite.
ABSTRACT In this short communication, we refute the conjecture by Fodor and Yager that the class of inclusion measures proposed by Kitainik coincides with that of inclusion measures based on contrapositive fuzzy implications. In particular, we show that the conjecture only holds when the considered universe of discourse is finite.
Journal paper accepted
14/02/22/10:08
F. Pérez-Gámez, D. López-Rodríguez, P. Cordero, Á. Mora, M. Ojeda-Aciego. Simplifying Implications with Positive and Negative Attributes: a Logic-based Approach. Mathematics 2022. To appear
ABSTRACT Concepts and implications are two facets of the knowledge contained within a binary relation between objects and attributes. Simplification Logic (SL) has proved to be valuable for the study of attribute implications in a concept lattice, a topic of interest in the more general framework of Formal Concept Analysis (FCA). Specifically, SL has become the kernel of automated methods to remove redundancy, or obtain different types of bases of implications. Although originally FCA uses only the positive information contained in the dataset, negative information (explicitly stating that an attribute does not hold) has been proposed by several authors, but without an adequate set of equivalence-preserving rules for simplification. In this work, we propose a mixed simplification logic and a method to automatically remove redundancy in implications, which will serve as a foundational standpoint for automated reasoning methods for this extended framework.
ABSTRACT Concepts and implications are two facets of the knowledge contained within a binary relation between objects and attributes. Simplification Logic (SL) has proved to be valuable for the study of attribute implications in a concept lattice, a topic of interest in the more general framework of Formal Concept Analysis (FCA). Specifically, SL has become the kernel of automated methods to remove redundancy, or obtain different types of bases of implications. Although originally FCA uses only the positive information contained in the dataset, negative information (explicitly stating that an attribute does not hold) has been proposed by several authors, but without an adequate set of equivalence-preserving rules for simplification. In this work, we propose a mixed simplification logic and a method to automatically remove redundancy in implications, which will serve as a foundational standpoint for automated reasoning methods for this extended framework.
Journal paper accepted
10/02/22/08:27
P. Cordero, M. Enciso, D. López-Rodríguez, Á. Mora. fcaR, Formal Concept Analysis with R. R Journal, 2022. To appear
ABSTRACT Formal concept analysis (FCA) is a solid mathematical framework to manage information based on logic and lattice theory. It defines two explicit representations of the knowledge present in a dataset as concepts and implications. This paper describes an R package called fcaR that implements FCA's core notions and techniques. Additionally, it implements the extension of FCA to fuzzy datasets and a simplification logic to develop automated reasoning tools. This package is the first to implement FCA techniques in R. Therefore, emphasis has been put on defining classes and methods that could be reusable and extensible by the community. Furthermore, the package incorporates an interface with the arules package, probably the most used package regarding association rules, closely related to FCA. Finally, we show an application of the use of the package to design a recommender system based on logic for diagnosis in neurological pathologies.
Library: https://malaga-fca-group.github.io/fcaR/
ABSTRACT Formal concept analysis (FCA) is a solid mathematical framework to manage information based on logic and lattice theory. It defines two explicit representations of the knowledge present in a dataset as concepts and implications. This paper describes an R package called fcaR that implements FCA's core notions and techniques. Additionally, it implements the extension of FCA to fuzzy datasets and a simplification logic to develop automated reasoning tools. This package is the first to implement FCA techniques in R. Therefore, emphasis has been put on defining classes and methods that could be reusable and extensible by the community. Furthermore, the package incorporates an interface with the arules package, probably the most used package regarding association rules, closely related to FCA. Finally, we show an application of the use of the package to design a recommender system based on logic for diagnosis in neurological pathologies.
Library: https://malaga-fca-group.github.io/fcaR/
Journal paper accepted
24/03/21/08:55
N. Madrid, C. López-Molina and P. Hurtik. Non-linear scale-space based on fuzzy contrast enhancement: Theoretical results. Fuzzy Sets and Systems 421:133-157, 2021.
ABSTRACT This work presents a contrast enhancement operator based on a fuzzy-numerical description of images at the pixel level; this operator is further used to construct a scale-space, whose theoretical and practical properties are reviewed. A very remarkable feature of our scale-space is that, in contrast to many other scale-spaces, it converges to non-trivial stages. Within the study of our scale-space, we present a series of theoretical results that show that the convergence of the scale-space is closely related to the signal's convexity. Specifically, we prove formally that the intensities in convex signals tend to converge to the minimum intensity. As a result, our scale-space increases the contrast in the image and homogenizes images. In addition to theoretical results, we illustrate the scale-space's behaviour in ad-hoc 1D signals and in greyscale images. Finally, to validate the potential application of this theoretical approach, we show that the proposal can be used as a preprocessing that performed before a neural network technique, increasing the accuracy in a classification task.
ABSTRACT This work presents a contrast enhancement operator based on a fuzzy-numerical description of images at the pixel level; this operator is further used to construct a scale-space, whose theoretical and practical properties are reviewed. A very remarkable feature of our scale-space is that, in contrast to many other scale-spaces, it converges to non-trivial stages. Within the study of our scale-space, we present a series of theoretical results that show that the convergence of the scale-space is closely related to the signal's convexity. Specifically, we prove formally that the intensities in convex signals tend to converge to the minimum intensity. As a result, our scale-space increases the contrast in the image and homogenizes images. In addition to theoretical results, we illustrate the scale-space's behaviour in ad-hoc 1D signals and in greyscale images. Finally, to validate the potential application of this theoretical approach, we show that the proposal can be used as a preprocessing that performed before a neural network technique, increasing the accuracy in a classification task.
Journal paper accepted
15/03/21/09:42
N. Madrid and M. Ojeda-Aciego. A Measure of Consistency for Fuzzy Logic Theories. Mathematical Methods in the Applied Sciences, 2021. To appear
ABSTRACT Fuzzy logic has shown to be a suitable framework to handle contradictions in which, unsurprisingly, the notion of inconsistency can be defined in different ways. This paper starts with a short survey of different ways to define the notion of inconsistency in fuzzy logic systems. As a result, we provide a first notion of inconsistency by means of the absence of models. Subsequently, we define two measures of consistency that belong purely to the fuzzy paradigm; in the sense that both measures coincide with the crisp notion of consistency when the set of truth values is {0,1}. Accordingly, we can state that the two provided measures of consistence are notions of consistence based on degrees, bringing back the spirit of fuzzy logic into the notion of consistency.
ABSTRACT Fuzzy logic has shown to be a suitable framework to handle contradictions in which, unsurprisingly, the notion of inconsistency can be defined in different ways. This paper starts with a short survey of different ways to define the notion of inconsistency in fuzzy logic systems. As a result, we provide a first notion of inconsistency by means of the absence of models. Subsequently, we define two measures of consistency that belong purely to the fuzzy paradigm; in the sense that both measures coincide with the crisp notion of consistency when the set of truth values is {0,1}. Accordingly, we can state that the two provided measures of consistence are notions of consistence based on degrees, bringing back the spirit of fuzzy logic into the notion of consistency.
Journal paper accepted
25/01/21/08:47
N. Madrid and M. Ojeda-Aciego. Measures of inclusion and entropy based on the φ-index of inclusion. Fuzzy Sets and Systems, 2021. To appear
ABSTRACT Surprisingly, despite that fuzzy sets were introduced more than fifty years ago, there is not consensus yet about how to extend the notion of inclusion in such a framework. Recently, alternatively to previous methods in the literature, we introduced an approach in which we make use of the so-called φ-index of inclusion. This approach has a main difference with respect to previous ones: the degree of inclusion is identified with a function instead of with a value in [0,1], although such a feature makes it difficult to compare the φ-index of inclusion with existing axiomatic approaches concerning measures of inclusion. This is the reason why in this paper we define two different and natural measures of inclusion by means of the φ-index of inclusion and, then, show that both measures satisfy some standard axiomatic approaches about measures of inclusion in the literature. In addition, taking into account the relationship of fuzzy entropy with Young axioms for measures of inclusion, we present also a measure of entropy based on the φ-index of inclusion that is in accordance with the axioms of De Luca and Termini.
ABSTRACT Surprisingly, despite that fuzzy sets were introduced more than fifty years ago, there is not consensus yet about how to extend the notion of inclusion in such a framework. Recently, alternatively to previous methods in the literature, we introduced an approach in which we make use of the so-called φ-index of inclusion. This approach has a main difference with respect to previous ones: the degree of inclusion is identified with a function instead of with a value in [0,1], although such a feature makes it difficult to compare the φ-index of inclusion with existing axiomatic approaches concerning measures of inclusion. This is the reason why in this paper we define two different and natural measures of inclusion by means of the φ-index of inclusion and, then, show that both measures satisfy some standard axiomatic approaches about measures of inclusion in the literature. In addition, taking into account the relationship of fuzzy entropy with Young axioms for measures of inclusion, we present also a measure of entropy based on the φ-index of inclusion that is in accordance with the axioms of De Luca and Termini.
Journal paper accepted
09/11/20/10:47
O. Krídlo, M. Ojeda-Aciego. Classifying adjoint pairs and adjoint triples in an Atanassov L-fuzzy framework. IEEE Transactions on Fuzzy Systems, 2021. To appear
ABSTRACT We study and classify a family of adjoint pairs and adjoint triples for Atanassov $L$-fuzzy framework based on a complete residuated lattice satisfying the double negation law.
ABSTRACT We study and classify a family of adjoint pairs and adjoint triples for Atanassov $L$-fuzzy framework based on a complete residuated lattice satisfying the double negation law.
Journal paper accepted
09/09/20/18:39
P. Cordero, M. Enciso, A. Mora, M. Ojeda-Aciego, C. Rossi. A Formal Concept Analysis approach to cooperative conversational recommendation. International Journal of Computational Intelligence Systems, 13(1):1243-1252, 2020.
ABSTRACT We focus on the development of a method to guide the choice of a set of users in an environment where the number of features describing the items is high and user interaction becomes laborious. Using the framework of formal concept analysis, particularly the notion of implication between attributes, we propose a method strongly based on logic which allows to manage the users' preferences by following a conversational paradigm. Concerning complexity, to build the conversation and provide updated information based on the users' previous actions (choices) the method has polynomial delay.
ABSTRACT We focus on the development of a method to guide the choice of a set of users in an environment where the number of features describing the items is high and user interaction becomes laborious. Using the framework of formal concept analysis, particularly the notion of implication between attributes, we propose a method strongly based on logic which allows to manage the users' preferences by following a conversational paradigm. Concerning complexity, to build the conversation and provide updated information based on the users' previous actions (choices) the method has polynomial delay.
Journal paper accepted
03/07/20/10:58
J.M. Rodríguez, M. Ojeda-Aciego. Formal concept analysis with negative attributes for forgery detection. Computational and Mathematical Methods, 2020.
ABSTRACT Europe’s system of open frontiers, commonly known as "Schengen", let people from different countries travel and cross the inner frontiers without problems. Different documents from these countries, not only European, can be found in road checkpoints and there is no international database to help Police forces to detect whether they are false or not. People who need a driver license to access to specific jobs, or a new identity because of legal problems, often contact forgers who provide false documents with different levels of authenticity. Governments and Police Forces should improve their methodologies, by ensuring that staff is increasingly better able to detect false or falsified documents through their examination, and follow patterns to detect and situate these forgers. In this work we propose a method, based in Formal Concept Analysis using negative attributes, that allows Police forces analysing false documents and provides a guide to enforce the detection of forgers.
ABSTRACT Europe’s system of open frontiers, commonly known as "Schengen", let people from different countries travel and cross the inner frontiers without problems. Different documents from these countries, not only European, can be found in road checkpoints and there is no international database to help Police forces to detect whether they are false or not. People who need a driver license to access to specific jobs, or a new identity because of legal problems, often contact forgers who provide false documents with different levels of authenticity. Governments and Police Forces should improve their methodologies, by ensuring that staff is increasingly better able to detect false or falsified documents through their examination, and follow patterns to detect and situate these forgers. In this work we propose a method, based in Formal Concept Analysis using negative attributes, that allows Police forces analysing false documents and provides a guide to enforce the detection of forgers.
Journal paper accepted
20/04/20/10:04
P. Cordero, M. Enciso, D. López, A. Mora. A conversational recommender system for diagnosis using fuzzy rules. Expert Systems with Applications, 154:113449, 2020.
ABSTRACT Graded implications in the framework of Fuzzy Formal Concept Analysis are used as the knowledge guiding the recommendations. An automated engine based on fuzzy Simplification Logic is proposed to make the suggestions to the users. Conversational recommender systems have proven to be a good approach in telemedicine, building a dialogue between the user and the recommender based on user preferences provided at each step of the conversation. Here, we propose a conversational recommender system for medical diagnosis using fuzzy logic. Specifically, fuzzy implications in the framework of Formal Concept Analysis are used to store the knowledge about symptoms and diseases and Fuzzy Simplification Logic is selected as an appropriate engine to guide the conversation to a final diagnosis. The recommender system has been used to provide differential diagnosis between schizophrenia and schizoaffective and bipolar disorders. In addition, we have enriched the conversational strategy with two strategies (namely critiquing and elicitation mechanism) for a better understanding of the knowledge-driven conversation, allowing user’s feedback in each step of the conversation and improving the performance of the method.
ABSTRACT Graded implications in the framework of Fuzzy Formal Concept Analysis are used as the knowledge guiding the recommendations. An automated engine based on fuzzy Simplification Logic is proposed to make the suggestions to the users. Conversational recommender systems have proven to be a good approach in telemedicine, building a dialogue between the user and the recommender based on user preferences provided at each step of the conversation. Here, we propose a conversational recommender system for medical diagnosis using fuzzy logic. Specifically, fuzzy implications in the framework of Formal Concept Analysis are used to store the knowledge about symptoms and diseases and Fuzzy Simplification Logic is selected as an appropriate engine to guide the conversation to a final diagnosis. The recommender system has been used to provide differential diagnosis between schizophrenia and schizoaffective and bipolar disorders. In addition, we have enriched the conversational strategy with two strategies (namely critiquing and elicitation mechanism) for a better understanding of the knowledge-driven conversation, allowing user’s feedback in each step of the conversation and improving the performance of the method.
Journal paper accepted
13/03/20/10:32
N. Madrid, M. Ojeda. On contradiction and inclusion using functional degrees. International Journal of Computational Intelligence Systems 13(1):464-471, 2020.
ABSTRACT The notion of inclusion is a cornerstone in set theory and therefore, its generalization in fuzzy set theory is of great interest. The degree of f-inclusion is one generalization of such a notion that differs from others existing in the literature because the degree of inclusion is considered as a mapping instead of a value in the unit interval. On the other hand, the degree of f-weak-contradiction was introduced to represent the contradiction between two fuzzy sets via a mapping and its definition has many similarities with the f-degree of inclusion. This suggests the existence of relations between both f-degrees. Specifically, following this line, we analyse the relationship between the f-degree of inclusion and the f-degree of contradiction via the complement of fuzzy sets and Galois connections.
ABSTRACT The notion of inclusion is a cornerstone in set theory and therefore, its generalization in fuzzy set theory is of great interest. The degree of f-inclusion is one generalization of such a notion that differs from others existing in the literature because the degree of inclusion is considered as a mapping instead of a value in the unit interval. On the other hand, the degree of f-weak-contradiction was introduced to represent the contradiction between two fuzzy sets via a mapping and its definition has many similarities with the f-degree of inclusion. This suggests the existence of relations between both f-degrees. Specifically, following this line, we analyse the relationship between the f-degree of inclusion and the f-degree of contradiction via the complement of fuzzy sets and Galois connections.
Journal paper accepted
26/02/20/08:36
E. Ramírez-Poussa, N. Madrid, J. Medina. Rough Sets based on Galois connections. Applied Mathematics and Computer Science 30(2):299-313, 2020.
ABSTRACT Rough set theory is an important tool to extract knowledge from relational databases. The original definitions of the approximation operators are based on an indiscernibility relation, which is an equivalence relation. Later, different papers have motivated the possibility of considering arbitrary relations nevertheless, when arbitrary relations are considered, the original definitions given by Pawlak may lose fundamental properties. This paper proposes a possible solution of the arisen problems by presenting an alternative definition of the approximation operators based on the closure and the interior operators obtained from an isotone Galois connection. We prove that the proposed definition satisfies interesting properties and that it also improves object classification tasks.
ABSTRACT Rough set theory is an important tool to extract knowledge from relational databases. The original definitions of the approximation operators are based on an indiscernibility relation, which is an equivalence relation. Later, different papers have motivated the possibility of considering arbitrary relations nevertheless, when arbitrary relations are considered, the original definitions given by Pawlak may lose fundamental properties. This paper proposes a possible solution of the arisen problems by presenting an alternative definition of the approximation operators based on the closure and the interior operators obtained from an isotone Galois connection. We prove that the proposed definition satisfies interesting properties and that it also improves object classification tasks.
Journal paper accepted
10/02/20/08:47
I.P. Cabrera, P. Cordero, E. Muñoz-Velasco, M. Ojeda-Aciego, B. De Baets. Relational Galois connections between transitive fuzzy digraphs. Mathematical Methods in the Applied Sciences 43(9):5673-5680, 2020.
ABSTRACT Fuzzy directed graphs are often chosen as the datatype to model and implement solutions of several problems in the applied sciences. Galois connections have also shown to be useful both in theoretical and in practical problems. In this paper, the notion of relational Galois connection is extended to be applied between transitive fuzzy directed graphs. In this framework, the components of the connection are crisp relations satisfying certain reasonable properties given in terms of the so-called full powering.
ABSTRACT Fuzzy directed graphs are often chosen as the datatype to model and implement solutions of several problems in the applied sciences. Galois connections have also shown to be useful both in theoretical and in practical problems. In this paper, the notion of relational Galois connection is extended to be applied between transitive fuzzy directed graphs. In this framework, the components of the connection are crisp relations satisfying certain reasonable properties given in terms of the so-called full powering.
Journal paper accepted
20/01/20/11:07
I.P. Cabrera, P. Cordero, E. Muñoz-Velasco, M. Ojeda-Aciego, B. De Baets. Relational Galois connections between transitive digraphs: characterization and construction. Information Sciences 519:439-450, 2020.
ABSTRACT This paper focuses on a twofold relational generalization of the notion of Galois connection. It is twofold because it is defined between sets endowed with arbitrary transitive relations and, moreover, both components of the connection are relations, not necessarily functions. A characterization theorem of the notion of relational Galois connection is provided and, then, it is proved that a suitable notion of closure can be obtained within this framework. Finally, we state a necessary and sufficient condition that allows to build a relational Galois connection starting from a single transitive digraph and a single binary relation.
ABSTRACT This paper focuses on a twofold relational generalization of the notion of Galois connection. It is twofold because it is defined between sets endowed with arbitrary transitive relations and, moreover, both components of the connection are relations, not necessarily functions. A characterization theorem of the notion of relational Galois connection is provided and, then, it is proved that a suitable notion of closure can be obtained within this framework. Finally, we state a necessary and sufficient condition that allows to build a relational Galois connection starting from a single transitive digraph and a single binary relation.
Journal paper accepted
28/11/19/13:33
N. Madrid and M. Ojeda-Aciego. Multi-adjoint lattices from adjoint triples with involutive negation. Fuzzy Sets and Systems 405:88-105, 2021.
ABSTRACT We focus primarily on the use of involutive negations in adjoint triples and the satisfiability of the contraposition law. Instead of considering natural negations, such as n(x)=x -> 0, we consider an arbitrary involutive negation and an arbitrary adjoint triple. Then, we construct a multiadjoint lattice (an algebraic structure with several conjunctions and implications) with the help of two new adjoint triples defined from the original one and the involutive negation considered. Finally, we present several results that relate the different implications and conjunctions appearing in the mentioned multi-adjoint lattice in terms of the logical laws of contraposition, interchange and exportation.
ABSTRACT We focus primarily on the use of involutive negations in adjoint triples and the satisfiability of the contraposition law. Instead of considering natural negations, such as n(x)=x -> 0, we consider an arbitrary involutive negation and an arbitrary adjoint triple. Then, we construct a multiadjoint lattice (an algebraic structure with several conjunctions and implications) with the help of two new adjoint triples defined from the original one and the involutive negation considered. Finally, we present several results that relate the different implications and conjunctions appearing in the mentioned multi-adjoint lattice in terms of the logical laws of contraposition, interchange and exportation.
Journal paper accepted
12/11/19/13:33
A. Burrieza, E. Muñoz-Velasco and M. Ojeda-Aciego. A flexible logic-based approach to closeness using order of magnitude qualitative reasoning. Logic Journal of the IGPL 28(1):121-133, 2020.
ABSTRACT In this paper, we focus on a logical approach to the important notion of closeness, which has not received much attention in the literature. Our notion of closeness is based on the so-called proximity intervals, which will be used to decide the elements that are close to each other. Some of the intuitions of this definition are explained on the basis of examples. We prove the decidability of the recently introduced multimodal logic for closeness and, then, we show some capabilities of the logic with respect to expressivity in order to denote particular positions of the proximity intervals.
ABSTRACT In this paper, we focus on a logical approach to the important notion of closeness, which has not received much attention in the literature. Our notion of closeness is based on the so-called proximity intervals, which will be used to decide the elements that are close to each other. Some of the intuitions of this definition are explained on the basis of examples. We prove the decidability of the recently introduced multimodal logic for closeness and, then, we show some capabilities of the logic with respect to expressivity in order to denote particular positions of the proximity intervals.
Journal paper accepted
25/03/19/22:48
N. Madrid and M. Ojeda-Aciego. Functional degrees of inclusion and similarity between L-fuzzy sets. Fuzzy Sets and Systems 390:1-22, 2020
ABSTRACT Inclusion is one of the most basic relations between sets. In this paper, we show how to represent the degree of inclusion between two L-fuzzy sets via a function. Specifically, such a function determines the minimal modifications needed in an L-fuzzy set to be included (in Zadeh's sense) into another. To reach such a goal, firstly we present the notion of f-inclusion, which defines a family of crisp binary relations between L-fuzzy sets that are used as indexes of inclusion and, subsequently, we define the φ-degree of inclusion as the most suitable f-inclusion under certain criterion. In addition, we also present three φ-degrees of similarity definable from the φ-degree of inclusion. We show that the φ-degree of inclusion and the φ-degrees of similarities satisfy versions of many common axioms usually required for measures of inclusion and similarity in the literature.
ABSTRACT Inclusion is one of the most basic relations between sets. In this paper, we show how to represent the degree of inclusion between two L-fuzzy sets via a function. Specifically, such a function determines the minimal modifications needed in an L-fuzzy set to be included (in Zadeh's sense) into another. To reach such a goal, firstly we present the notion of f-inclusion, which defines a family of crisp binary relations between L-fuzzy sets that are used as indexes of inclusion and, subsequently, we define the φ-degree of inclusion as the most suitable f-inclusion under certain criterion. In addition, we also present three φ-degrees of similarity definable from the φ-degree of inclusion. We show that the φ-degree of inclusion and the φ-degrees of similarities satisfy versions of many common axioms usually required for measures of inclusion and similarity in the literature.