The φ-index of inclusion has proven to be a suitable generalization of the inclusion in the fuzzy setting. In this paper, the properties of the φ-index of inclusion, when its definition is restricted to a subset of indexes, are analyzed. The theoretical results obtained in this work are necessary in order to develop fuzzy inference systems based on the φ-index of inclusion.
Despite several successful applications of the Extreme Learning Machine (ELM) as a new neural network training method that combines random selection with deterministic computation, we show that some fundamental principles of ELM lack a rigorous mathematical basis. In particular, we refute the proofs of two fundamental claims and construct datasets that serve as counterexamples to the ELM algorithm. Finally, we provide alternative claims to the basic principles that justify the effectiveness of ELM in some theoretical cases.
We continue our research line on the analysis of the properties of the f-indexes of inclusion and contradiction; in this paper, specifically, we show that both notions can be related by means of the, conveniently reformulated, Aristotelian square of opposition. We firstly show that the extreme cases of the f-indexes of inclusion and contradiction coincide with the vertexes of the Aristotelian square of opposition in the crisp case; then, we allocate the rest of f-indexes in the diagonals of the extreme cases and we prove that the Contradiction, Contrariety, Subcontrariety, Subalternation and Superalternation relations also hold between the f-indexes of inclusion and contradiction.
We investigate building a connection between RDF and FCA. The proposed approach transforms an RDF graph, where vertices represent objects of different types and edges represent relationships between these objects, into a series of bipartite graphs. It is achieved by separating edges representing specific relationships, resulting in a clear representation of the relationship of interest without clutter. To address this issue, we propose a bond-based construction of rigorous and benevolent compositions of bipartite graphs. These bipartite graphs are extracted from RDF graphs and combined?using the proposed construction?with external information related to the graphs' entities.
In Formal Concept Analysis (FCA), L-bonds represent relationships between L-formal contexts. Choosing the appropriate bond between L-fuzzy formal contexts is an important challenge for its application in recommendation tasks. Recent work introduced two constructions of bonds, given by direct products of two L-fuzzy formal contexts, and showed their usefulness in a particular application. In this paper, we present further theoretical and experimental results on these constructions; in particular, we provide extended interpretations of both rigorous and benevolent concept-forming operators, introduce new theoretical properties of the proposed bonds to connect two concept lattices given external information, and finally present the experimental study of the upper bounds.
In this paper we provide a sound and complete logic to formalise and reason about f-indices of inclusion. The logic is based on finite-valued Lukasiewicz logic and its S5-like modal extension S5(L) with additional unary operators.
We continue studying the properties of the f-index of inclusion and show that, given a fixed pair of fuzzy sets, their f-index of inclusion can be linked to a fuzzy conjunction which is part of an adjoint pair. We also show that, when this pair is used as the underlying structure to provide a fuzzy interpretation of the modus ponens inference rule, it provides the maximum possible truth-value in the conclusion among all those values obtained by fuzzy modus ponens using any other possible adjoint pair.
We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.
We work on the recently proposed relationship between L-fuzzy measures and integrals and isotone concept-forming operators on a Girard monoid in order to define a generalized notion of Sugeno integral for Atanassov Intuitionistic L-fuzzy sets. We also provide some examples on the practical interpretation of the new proposed notion.
We develop a methodology which allows to detect forgeries in driving licences based on an analysis of two serial codes usually present in licences issued in European countries. Results from the initial analysis of licences issued in France and Italy are presented.
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.
Information does not use to be precise and exact and, in many cases, some data are missed or unknown. To manage with this data, fuzzy logic introduces a set of (in nitely many) values between the two Boolean truth values. Other authors use a three-valued approach, by adding an intermediate value to the set of Boolean truth-values. We propose a formal framework strongly based on Atanassov Fuzzy logic, associating each proposition with a pair of degrees characterizing our knowledge about the two truthfulness values, since this logic does not include the law of the excluded middle. Our starting point is the crisp Formal Concept Analysis, which provides a formal framework for knowledge representation and reasoning. Information is described by means of a binary relation characterizing the relationship among a set of objects and a set of attributes. We extend the crisp framework considering a pair of degrees for each element in the relation.Formal concept analysis provides a twofold representation of the knowledge: the so-called concept lattice and the implication set. In this paper we choice the second option since it better provides a symbolic manipulation of the information. Here, we introduce the syntax and semantics for a new intuitionistic implication. This notion of implication allows a further definition of an intuitionistic logic to manage this kind of information with a powerful expressive power, but avoiding the problems of the classical propositional Logic regarding its execution because of the SAT problem.
This paper is an attempt at bridging two strains of research being developed by the authors: a theory of information flows to subserve intelligence and a theory of affordances for the modelling of Embodied, Embedded, Extended and Enacted Computational Intelligence as provided by FCA. We list previous successes, present challenges, and future avenues of research that suggest themselves.
En este documento presentamos alguno de los últimos resultados teóricos obtenidos sobre el f-índice de inclusión. Estos resultados motivan el uso de dicho índice como una nueva forma de representar la inclusión entre dos conjuntos difusos y como un operador de inferencia lógica. En este resumen recordamos dos: se satisfacen los axiomas de Sinha-Dougherty (convenientemente adaptados al marco teórico del f-índice de inclusión) y, además, corresponde a una elección optimal de una implicación difusa residuada para llevar a cabo la inferencia Modus Ponens.
Presentamos una versión difusa de la noción de conexión de Galois relacional entre grafos dirigidos transitivos difusos (digrafos T-difusos) en el entorno específico en el que el álgebra subyacente de valores de verdad es un álgebra de Heyting completa. Los componentes de dicha conexión de Galois difusa son relaciones difusas que satisfacen ciertas propiedades razonables expresadas en términos de lo que denominamos "full powering". Además, proporcionamos una condición necesaria y suficiente bajo la cual es posible construir un adjunto a la derecha para una relación difusa dada entre un digrafo T-difuso y un conjunto no estructurado.
Contextos formales parciales son contextos con tres valores que nos permite establecer cuando una propiedad se satisface o no. Además, permite representar situaciones en las que existe ignorancia sobre si una propiedad se satisface o no. Esto puede ser bastante útil en diferentes aspectos como cuando hay información desconocida o, también, cuando aparece la información desconocida debido a intentamos reducir el tamaño de un contexto formal agrupando filas. En este artículo extendemos estas nociones e ideas para añadir grados de conocimiento.
In this paper we set out to understand the cognitive basis of Formal Concept Analysis used as an Exploratory Data Analysis frame- work under the guise of the Landscapes of Knowledge metaphor introduced by Wille. We show that it can be re-interpreted and extended in the framework of the Theory of Affordances from Ecological Psychology to provide not only different affordances for different flavours of formal analysis of the information captured by a formal context, but also a theory that sheds light on how we learn to do it, Perceptual Learning. This raises the issue of what it is that a formal analysis of a formal context provides. We introduce the concept of formal qualia as basic, incomparable, privative items of information afforded by each possible analysis and illustrate these concepts by the formal qualia provided by Formal Concept, Independence and Equivalence Analysis.
Partial formal contexts are trivalued contexts that, besides allowing to establish whether a property is satisfied or not, allow to represent situations in which there is ignorance about whether a property is satisfied. This can be useful, not only for when the modeled phenomenon has intrinsically unknown information, but also when summarizing information from a formal context by grouping similar rows. In this paper we prospect for its extension including degrees of knowledge.
In this paper, we continue the study of different generalizations on the notion of Galois connection. In previous works, we focused on cases where the (co)domain has the structure of a transitive digraph or a fuzzy transitive digraph. Now, we extend it to the fuzzy relational framework. Specifically, we present a suitable notion of fuzzy relational Galois connection between fuzzy transitive digraphs where both components are now fuzzy relations and the underlying truth value algebra is a complete Heyting algebra. This notion of fuzzy relational Galois connection inherits the most interesting characterisation of the notion of (crisp) relational Galois connection.
In this paper we provide an embedding of finite lattices into (N,|), the lattice of divisibility of natural numbers. For that purpose, we explore two representations: vector clocks, a device to provide a virtual time used in distributed systems that has gained traction as a finite lattice representation, and the log-prime function that transforms natural numbers into sequences of prime exponents. Using a generalized log-prime function and its inverse we describe how to embed any finite (width, height) lattice into (N, |) and provide examples for such process, prior to analysing the affordances of such encoding vis-a-vis the representation of non-global, distributed time. We also discuss how this representation may help improve the affordances of using complete lattices in data analysis both for time- and non-time related data.
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.
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.
The proposed relationship between L-fuzzy measures, integrals and isotone concept-forming operators on a Girard monoid has recently been used to define a generalized notion of Sugeno integral for Atanassov Intuitionistic L-fuzzy sets. We further continue generalizing the construction aiming at the framework of multi-adjoint formal concept analysis. At this level of generality it is still possible to define a multi-adjoint analogy for lower Sugeno integral at the price of slight modifications in the definition of the underlying adjoint triple. The notion of upper Sugeno integral requires a much more complete treatment including negations, implications with many properties, etc. and is an interesting piece of future work.
We work on the recently proposed relationship between L-fuzzy measures and integrals and isotone concept-forming operators on a Girard monoid in order to define a generalized notion of Sugeno integral for Atanassov Intuitionistic L-fuzzy sets. We also provide some examples on the practical interpretation of the new proposed notion.
We develop a methodology which allows to detect forgeries in driving licences based on an analysis of two serial codes usually present in licences issued in European countries. Results from the initial analysis of licences issued in France and Italy are presented.
The origin of the f-index of inclusion can be dated back to the incorporation of negation connectives in multi-adjoint logic programs and, hence, to study inconsistency and, its sibling, coherence. As a result, it was clear that (in-)consistency should not be considered as a crisp notion when applied in (general) fuzzy logic theories, and different approaches were proposed for this goal. We introduced the notion weak-contradiction as a generalization of the notion of coherence in the general framework of fuzzy set theory. Soon after introducing measures for weak-contradiction, we started to imagine some kind of function-based approach to measuring the inclusion between fuzzy sets, and presented the first ideas about the f-index of inclusion. We have recently recovered the idea of relating the two research lines emerged in parallel, namely the weak-contradiction and the f-index of inclusion, with satisfactory results. In this work we show that the f-index of inclusion is very related to fuzzy implications and, in fact, three different residuated structures can be obtained.
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.
We continue analyzing the properties of f-inclusion and f-contradiction, and provide an alternative interpretation of the square of opposition.
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.
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.
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.
En este trabajo nos centramos en un enfoque lógico de la importante noción de cercanía, que no ha recibido mucha atención en la bibliografía. Introduciremos una noción de cercanía basada en intervalos llamados “intervalos de proximidad”, que se utilizarán para decidir los elementos que están cerca unos de otros. Algunas de las intuiciones de esta definición se explican sobre la base de ejemplos. Además, presentamos una noción de despreciabilidad que combinamos con la noción de cercanía. Probaremos la capacidad de decisión de la lógica multimodal introducida y, a continuación, mostraremos algunos aspectos de la potencia expresiva de nuestra lógica: su capacidad para denotar posiciones particulares de los intervalos de proximidad, la cantidad de intervalos que posee una clase cualitativa dada y su capacidad para definir diferentes tipos de conectivas modales sobre intervalos, sean o no de proximidad.
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 -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 -weak-contradiction was introduced to represent the contradiction between two fuzzy sets via a mapping and its definition has many similarities with the -degree of inclusion. This suggests the existence of relations between both -degrees. Specifically, following this line, we analyze the relationship between the -degree of inclusion and the -degree of contradiction via the complement of fuzzy sets and Galois connections.
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 -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 -weak-contradiction was introduced to represent the contradiction between two fuzzy sets via a mapping and its definition has many similarities with the -degree of inclusion. This suggests the existence of relations between both -degrees. Specifically, following this line, we analyze the relationship between the -degree of inclusion and the -degree of contradiction via the complement of fuzzy sets and Galois connections.
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.
The notion of inclusion is one of the most basic relations between sets, however, there is not a consensus about how to extend such a notion in fuzzy set theory. We introduce an alternative approach to previous methods in the literature in which we make use of the so-called φ-index of inclusion. This approach has a main difference with respect to previous ones: instead of a value in [0,1], the measure of inclusion is identified with a function. In this paper, using the φ-index of inclusion we define two measures of inclusion in the standard sense, i.e., taking a value in [0,1] and then, we show that both measures are in accordance with the standard axiomatic approaches about measures of inclusion in the literature.
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.
The construction of Galois connections between unbalanced structures has received considerable attention in the recent years. In a nutshell, the problem is to find a right adjoint of a mapping defined between sets with unbalanced structure; in this paper we survey recent results obtained in this framework, focusing specially on the fuzzy structures that have been considered so far in this context: fuzzy preposets, fuzzy preordered structures, and fuzzy T-digraphs.
We introduce a new framework, Formal Context Analysis (FxA), for the exploratory analysis of data tasks cast in the guise of formal contexts. FxA gathers a number of results from Formal Concept Analysis, Formal Independence Analysis and Formal Equivalence Analysis to enhance the establishment and processing of hypothesis about data. We apply this framework to the study of the Multi-label Classification (MLC) task and obtain a number of results of technical nature about how the induction mechanism for MLC classifiers should proceed. The application is based on an analysis of multilabel classification from the standpoint of FxA.
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.
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 a 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 criterium. 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
In a previous work, we 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); in this paper, we extend the previous relation to the Big Toy Models introduced as a tool to represent quantum systems in terms of Chu spaces
In this paper, we focus 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 as well. Specifically, we introduce the notion of relational Galois connection between two transitive digraphs, study some of its properties and its relationship with other existing approaches in the literature.
Following R. Wille?s lead and suggestion we set out to design a new kind of view onto a formal context analogous but different to Formal Concept Analysis (FCA) and Formal Independent Analysis (FIA). In this instance, we choose to analyse the information in the incidence table in terms of the partitions induced on the sets of objects and attributes by the some functions of single attributes and objects of the context. These functions constitute a left adjunction between sets of objects and attributes and we later lift this left adjunction to partitions of the objects and attributes. Therefore we refer to this new view onto the formal context as Formal Equivalence Analysis (FEA). Rather than looking on the effect of these partitions on set representation, as in Rough Sets, we try to make explicit the information in the context.
In this paper, we give the first steps towards a formal definition of fuzzy relational Galois connection between fuzzy sets with arbitrary fuzzy transitive relations (fuzzy T-digraphs), where the two components of the connection are fuzzy relations. To this end we consider, on the one hand, our definition of relational Galois connection between T-digraphs in the crisp case; and, on the other hand, our definition of fuzzy relational Galois connection between fuzzy preorders. We compare both definitions and conclude that some (fuzzy) generalization of the notion of clique is needed.
We propose an alternative to the standard structure of L-fuzzy Mathematical Morphology (MM) by, on the one hand, considering L-fuzzy relations as structuring elements and, on the other hand, by using adjoint triples to handle membership values. Those modifications lead to a framework based on set-theoretical operations where we can prove a representation theorem for algebraic morphological erosions and dilations. In addition, we also present some new results concerning duality and transformation invariance. Concerning duality, we show that duality and adjointness can coexist in this L-fuzzy relational MM. Concerning transformation invariance, we show sufficient conditions to guarantee the invariance of morphological operators under arbitrary transformations.
We continue our prospective study of the generalization of formal concept analysis in terms of intuitionistic L-fuzzy sets. The main contribution here is an adjoint pair in the set Li of intuitionistic L-fuzzy values associated to a complete residuated lattice, which allows the definition of a pair of derivation operators which form an antitone Galois connection.
We propose the generalization of the notion of bond between two formal contexts to the case of n formal contexts. The first properties of the n-ary bonds are given, together with a method for building n-ary bonds. This construction enables to formalize some inference rules within the research topic of building a sequent calculus for formal contexts.
In this paper we present a new technique for the analysis of data tables by means of Formal Independence Analysis (FIA). This is an analogue of Formal Concept Analysis for the study of independence relations in data, instead of hierarchical relations. A FIA of a context produces, when possible, its block diagonalization by detecting pairs of sets of objects and attributes that are not mutually incident, or tomoi, that partition the context. In this paper we combine this technique with the exploration of contexts with entries in a semifield to find independent sets in contingency matrices. Specifically, we apply it to a number of confusion matrices issued from cognitive experiments to find evidences for the hypothesis of perceptual channels.
Qualitative reasoning deals with information expressed in terms of qualitative classes and relations among them, such as comparability, negligibility or closeness. In this work, we focus on the different logic-based approaches to the notions of negligibility developed by our group.
A Hilbert space H induces a formal context, the Hilbert formal context, whose associated concept lattice is isomorphic to the lattice of closed subspaces of H. This set of closed subspaces, denoted C(H), is important in the development of quantum logic and, as an algebraic structure, corresponds to a so-called “propositional system”, that is, a complete, atomistic, orthomodular lattice which satisfies the covering law. In this paper, we continue with our study of the Chu construction by introducing the Chu correspondences between Hilbert contexts, and showing that the category of Propositional Systems, PropSys, is equivalent to the category of Chu correspondences between Hilbert contexts.
In this paper we propose a new lens through which to observe the information contained in a formal context. Instead of focusing on the hierarchical relation between objects or attributes induced by their incidence, we focus on the “unrelatedness” of the objects with respect to those attributes with which they are not incident. The crucial order concept for this is that of maximal anti-chain and the corresponding representation capabilities are provided by Behrendt's theorem. With these tools we introduce the fundamental theorem of Formal Independence Analysis and use it to provide an example of what its affordances are for the analysis of data tables. We also discuss its relation to Formal Concept Analysis.
We continue the study of (isotone) Galois connections, also called adjunctions, in the framework of fuzzy preordered structures, which generalize fuzzy preposets by considering underlying fuzzy equivalence relations. Specifically, we present necessary and sufficient conditions so that, given a mapping f: A -> B from a fuzzy preordered structure A into a fuzzy structure B, it is possible to construct a fuzzy relation rho_B that induces a suitable fuzzy preorder structure on B and such that there exists a mapping g: B ->A such that the pair (f,g) constitutes an Galois connection..
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 chapter analyses the notion of inconsistency in general residuated logic programming under the answer-set semantics, shows that inconsistency can be somehow decomposed into instability and incoherence and, finally, shows that each of these notions can be associated with some natural measures of inconsistency. Finally, we focus on measures of inconsistency in the particular framework of fuzzy logic programming.
We introduce a multimodal logic for order of magnitude reasoning which considers a new logic-based alternative to the notion of closeness, we provide an axiom system and prove its soundness and completeness.
We continue our prospective study of the generalization of formal concept analysis in terms of intuitionistic L-fuzzy sets. The main contribution here is an adjoint pair in the set Li of intuitionistic L-fuzzy values associated to a complete residuated lattice, which allows the definition of a pair of derivation operators which form an antitone Galois connection.
We explore a suitable generalization of the notion of Galois connection in which their components are binary relations. Many different approaches are possible depending both on the (pre-)order relation between subsets in the underlying powerdomain and the chosen type of relational composition.
The construction of Galois connections between different structures provides a number of advantages, both from the theoretical and the applied standpoints. In this paper, we survey some works on Galois connections focused essentially on certain aspects of Computational Intelligence.
This paper introduces a representation of fuzzy partitions in term of fuzzy logic programming. This representation models relationships among the different classes that define the fuzzy partition. There are essentially two such relationships. The first one is that classes are disjoint and, therefore, contradictory each other; on the other hand, the second one is that classes cover all the universe. These two relationships are modeled via two different negations, namely, the explicit and default negation. Last but not least, the semantics used to model both negations is the fuzzy answer set semantics.
In this paper we analyze the novel constructive definition of f-index of inclusion with respect to four of the most common axiomatic definitions of inclusion measure, namely Sinha-Dougherty, Kitainik, Young and Fan-Xie-Pei. There exist an important difference between the f-index and these axiomatic definitions of inclusion measure: the f-index represents the inclusion in terms of a mapping in unit interval, whereas the inclusion measure represents such an inclusion as a value in the unit interval.
We propose a suitable generalization of the notion of Galois connection whose components are fuzzy relations. We prove that the construction embeds Yao's notion of fuzzy Galois connection as a particular case. Although the natural framework for the proposed notion is that of fuzzy preposets, we also prove that it behaves properly with respect to the formation of quotient with respect to the fuzzy symmetric kernel relation.
We continue our study of intuitionistic L-fuzzy formal concept analysis by presenting a construction of an adjoint triple based on a non-commutative conjunctor, so that it enables the construction of intuitionistic L-fuzzy t-formal concepts.
The problem of studying the existence of a right adjoint for a mapping defined between sets with different fuzzy structure naturally leads to the search of new notions of adjunction which fit better with the underlying structure of domain and codomain. In this work, we introduce a version of relational fuzzy adjunction between fuzzy preposets which generalizes previous approaches in that its components are fuzzy relations. We also prove that the construction behaves properly with respect to the formation of quotient with respect to the symmetric kernel relation and, thus, giving rise to a relational fuzzy adjunction between fuzzy posets.
A two-fold general approach to the theory of formal concept analysis is introduced by considering intuitionistic fuzzy sets valued on a residuated lattice as underlying structure for the construction.
We continue studying the connections between the Chu construction on the category ChuCors of formal contexts and Chu correspondences, and generalizations of Formal Concept Analysis (FCA). All the required constructions like categorical product, tensor product, together with its bifunctor properties are introduced and proved. The final section focuses on how the second-order generalization of FCA can be built up in terms of the Chu construction.
In this work, we focus on adjunctions, also called isotone Galois connections, in the framework of fuzzy preordered sets (hereafter, fuzzy preposets). Specifically, we present necessary and sufficient conditions so that, given a mapping f : A -> B from a fuzzy preposet A into an unstructured set B, it is possible to construct a suitable fuzzy preorder relation on B for which there exists a mapping g: B -> A such that the pair (f,g) constitutes an adjunction.
Reductants are a special kind of fuzzy rules which constitute an essential theoretical tool for proving correctness properties. As it has been reported, when interpreted on a partially ordered structure, a multi-adjoint logic program has to include all its reductants in order to preserve the (approximate) completeness property. After a short survey of the different notions of reductant that have been developed for multi-adjoint logic programs, we introduce a new and more adequate, notion of reductant in the mult-adjoint framework. We study some of its properties and its relationships with other notions of reductants. In addition, we give an efficient algorithm for computing all the reductants associated with a multi-adjoint logic program.
The Dedekind-MacNeille completion of a poset P can be seen as the least complete lattice containing P. In this work, we analyze some results concerning the use of this completion within the framework of Formal Concept Analysis in terms of the poset of concepts associated with a Galois connection between posets. Specifically, we show an interesting property of the Dedekind-MacNeille completion, in that the completion of the concept poset of a Galois connection between posets coincides with the concept lattice of the Galois connection extended to the corresponding completions. Moreover, we study the specific case when P has multilattice structure and state and prove the corresponding representation theorem.
In this paper we deal with suitable generalizations of the notion of bond between contexts, as part of the research area of Formal Concept Analysis. We study different generalizations of the notion of bond within the L-fuzzy setting. Specifically, given a formal context there are three prototypical pairs of concept-forming operators, and this immediately leads to three possible versions of the notion of bond (so-called homogeneous bond wrt certain pair of concept-forming operators). The first results show a close correspondence between a homogeneous bond between two contexts and certain special types of mappings between the sets of extents (or intents) of the corresponding concept lattices. Later, we introduce the so-called heterogeneous bonds (considering simultaneously two types of concept-forming operators) and generalize the previous relationship to mappings between the sets of extents (or intents) of the corresponding concept lattices.
In this paper we present a definition of erosions and dilations in terms of fuzzy relations and adjoint triples. We firstly show that we can represent any algebraic erosion and dilation in such a terms and secondly, we present a set of approaches that can be covered by our definition of relational erosions and dilations.
Qualitative reasoning deals with information expressed in terms of qualitative classes and relations among them, such as comparability, negligibility or closeness. In this paper, we focus on the notion of closeness using a hybrid approach which is based on logic, order-of-magnitude reasoning, and on the so-called proximity structures; these structures will be used to decide the elements that are close to each other. Some of the intuitions of this approach are explained on the basis of examples. Moreover, we show some capabilities of the logic with respect to expressivity in order to denote particular positions of the proximity intervals.
We propose the generalization of the notion of bond between two formal contexts to the case of n formal contexts. The first properties of the n-ary bonds are given, together with a method for building n-ary bonds. This construction enables to formalize some inference rules within the research topic of building a sequent calculus for formal contexts.
Formal Concept Analysis (FCA) has become a very active research topic, both theoretical and practical; its wide applicability justifies the need of a deeper knowledge of its underlying mechanisms, and one important way to obtain this extra knowledge turns out to be via generalization. Several fuzzy variants of generalized FCA have been introduced and developed both from the theoretical and the practical side. Most of the generalizations focus on including extra features (fuzzy, possibilistic, rough, etc.); however, not much have been published on the suitable general version of certain specific notions, such as the bonds between formal contexts. One of the motivations for introducing the notion of bond was to provide a tool for studying mappings between formal contexts, somehow mimicking the behavior of Galois connections between their corresponding concept lattices. In this talk we will deal with generalizations of the notion of bond in an L-fuzzy setting.
This work focuses on the definition of a consequence relation between contexts with which we can decide whether certain contextual information is a logical consequence from a set of contexts considered as underlying hypotheses.
We continue our study of the characterization of existence of adjunctions (isotone Galois connections) whose codomain is insufficiently structured. This paper focuses on the fuzzy case in which we have a fuzzy ordering rho_A on A and a surjective mapping f: ( A, approx_A) -> ( B, approx_B) compatible with respect to the fuzzy equivalences approx_A and approx_B. Specifically, the problem is to find a fuzzy ordering rho_B and a compatible mapping g: (B, approx_B) -> (A, approx_A) such that the pair (f,g) is a fuzzy adjunction.
In this work, we focus on the study of necessary and sufficient conditions in order to ensure the existence (under some constraints) of monotone Galois connections between fuzzy preordered sets.
Qualitative reasoning is an area of AI which provides solutions to problems where the quantitative information either is not available or can not be used; in particular, order of magnitude qualitative reasoning assumes different qualitative classes and relations such as negligibility and closeness. In this paper, we focus mainly on the very important notion of closeness from the logical point of view, 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 introduce a multimodal logic for order of magnitude reasoning which includes the notions of closeness and negligibility, we provide an axiom system, which is sound and complete.
The Dedekind-MacNeille completion of a poset P can be seen as the least complete lattice containing P . In this work, we analyze some results concerning the use of this completion within the framework of Formal Concept Analysis, notably the distributivity of the Dedekind- MacNeille completion and the construction of the poset of concepts as- sociated with a Galois connection between posets.
In this work, we focus on the study of necessary and sufficient conditions in order to ensure the existence (under some constraints) of monotone Galois connections between fuzzy preordered sets.
The goal of this paper is to show a connection between FCA generalisations and the Chu construction on the category ChuCors, the category of formal contexts and Chu correspondences. All needed categorical properties like categorical product, tensor product and its bifunctor properties are presented and proved. Finally, the second order generalisation of FCA is represented by a category built up in terms of the Chu construction.
This work can be seen as a contribution to the area of social network analysis. By considering Formal Concept Analysis (FCA) as the underlying formalizing tool, we use logic-based techniques in order to offer novel solutions to identify user's influence in a social network. We propose the use of the Simplification Logic SLFD for attribute implications as the core of an automated method to build a structure containing the complete set of influences among users.
In this work we present a way to represent contradiction between fuzzy sets. This representation is given in terms of the notion of f-weak contradiction. Unlike other approaches, we do not define contradiction just by using one of the relations of f- weak-contradiction, but by considering the whole set of relations. This consideration avoids the need to fix an operator beforehand in order to take into account all the information between two fuzzy sets. As a result, we characterize the contradiction between fuzzy sets and define a family of measures of contradiction satisfying four interesting properties: symmetry, antitonicity, if the intersection is empty then the measure is one; and if there is an element in the intersection with degree of membership 1 then the measure is zero.
The aim of this work is providing a characterization in terms of closure systems, for the construction, given a mapping f: A -> B from a fuzzy preordered set A into an unstructured set B, of a suitable fuzzy preordering on B for which there exists a mapping g: B -> A such that the pair (f,g) constitutes an adjunction (isotone Galois connection). This contribution continues our research line on the construction of adjunctions in which the theory of fuzzy closure systems is used in order to provide a more meaningful framework for the extension to the fuzzy case of previous results.
We introduce the notion of f-inclusion, which is used to describe different kinds of subsethood relations between fuzzy sets by means of monotonic functions f: [0,1] -> [0,1]. We show that these monotonic functions can be considered indexes of inclusion, since the greater the function considered, the more restrictive is the relationship. Finally, we propose a general index of inclusion by proving the existence of a representative f-inclusion for any two ordered pairs of fuzzy sets. In such a way, our approach is different to others in the literature in no taking a priori assumptions like residuated implications or t-norms.
Due to its solid mathematical foundations, Formal Concept Analysis (FCA) has become an emergent topic in the area of data analysis and knowledge discovering. Information is represented in a binary table defining a relation between a set of objects and a set of attributes?the formal context. The knowledge extracted from the formal context allows to identify useful patterns in data in different forms. One very useful knowledge representation in FCA are implications among attributes which are validated over the objects. The most outstanding feature of implications is that they can be managed by means of inference systems. Equivalent sets of implications can be obtained using different logic-based transformations. The aim of these transformations is to turn the original set of implications into an equivalent one fulfilling some desired properties. Among them, the directness and optimality are very popular targets because getting a direct-optimal basis ensures that the closure of a set of attributes may be computed with lower cost (time and resources). In this work, we introduce a new method to compute the direct-optimal basis which improves the existing ones. The new method reduces the input in a first stage and is guided by the idea of limiting the growth of the intermediate sets of implications as a way to improve the performance. We illustrate the good features of the new method with both a detailed example and by experimental evaluation.
Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.
This paper is related, on the one hand, to the framework of multi-adjoint concept lattices with heterogeneous conjunctors and, on the other hand, to the use of intensifying hedges as truth-stressers. Specifically, we continue on the line of recent works by Belohlavek and Vychodil, which use intensifying hedges as a tool to reduce the size of a concept lattice. In this paper we use hedges as a reduction tool in the general framework of multi-adjoint concept lattices with heterogeneous conjunctors.
We apply recent results on the construction of suitable orderings for the existence of right adjoint to the analysis of the following problem: given a preference ordering on the set of attributes of a given context, we seek an induced preference among the objects which is compatible with the information provided by the context.
The theory of fuzzy property-oriented concept lattices is a formal tool for modeling and processing incomplete knowledge in information systems. This paper relates this research topic to that of mathematical morphology, a theory whose scope is to process and analyze images and signals. Consequently, the theory developed in the concept lattice framework can be used in these particular settings.
Formal Concept Analysis (FCA) as inherently relational can be formalized and generalized by using categorical constructions. This provides a categorical view of the relation between 'object' and 'attributes', which can be further extended to a more generalized view on relations as morphisms in Kleisli categories of suitable monads. Structure of sets of 'objects' and 'attributes' can be provided e.g. by term monads over particular signatures, and specific signatures drawn from and developed within social and health care can be used to illuminate the use of the categorical approach.
There exists a direct relation between fuzzy rough sets and fuzzy preorders. On the other hand, it is well known the existing parallelism between Formal Concept Analysis and Rough Set Theory. In both cases, Galois connections play a central role. In this work, we focus on adjunctions (also named isotone Galois connections) between fuzzy preordered sets; specifically, we study necessary conditions that have to be fulfilled in order such an adjunction to exist.
Given a mapping from a preordered set A into an unstructured set B, we study the problem of defining a suitable preordering relation on B such that there exists a mapping g from B to A such that the pair (f, g) forms an adjunction between preordered sets.
Given a mapping f from a partially ordered set A into an unstructured set B, we study the problem of defining a suitable partial ordering relation on B such that there exists a mapping g from B to A such that the pair of mappings (f, g) forms an isotone Galois connection between partially ordered sets.
We present a logic approach to reason with moving objects under fuzzy qualitative representation. This way, we can deal both with qualitative and quantitative information, and consequently, to obtain more accurate results. The proposed logic system is introduced as an extension of Propositional Dynamic Logic: this choice, on the one hand, simplifies the theoretical study concerning soundness, completeness and decidability; on the other hand, provides the possibility of constructing complex relations from simpler ones and the use of a language very close to programming languages.
This paper introduces sufficient and necessary conditions with respect to the fuzzy operators considered in a multi-adjoint frame under which the standard combinations of multi-adjoint sufficiency, possibility and necessity operators form (antitone or isotone) Galois connections. The underlying idea is to study the minimal algebraic requirements so that the concept-forming operators (defined using the same syntactical form than the extension and intension operators of multi-adjoint concept lattices) form a Galois connection. As a consequence, given a relational database, we have much more possibilities to construct concept lattices associated with it, so that we can choose the specific version which better suits the situation.
Given a mapping from a fuzzy poset (A, ?) to any set B, we introduce conditions which allow for defining a fuzzy ordering on B and a mapping from B to A such that the pair (f ,g) forms a fuzzy Galois connection
Continuing with our general study of algebraic hyperstructures, we focus on the residuated operation in the framework of multilattices. Firstly, we recall the existing relation between filters, homomorphisms and congruences in the framework of multilattices; then, introduce the notion of residuated multilattice and further study the notion of filter, which has to be suitably modified so that the results in the first section are conveniently preserved also in the residuated case.
The most popular basis in Formal Concept Analysis is the Duquenne-Guigues basis, which ensure minimality in the number of dependencies and it is built with pseudo-intents, and some method to calculate these basis from an arbitrary set of implications have been introduced. 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. The new basis also has the minimal property demanded in the Duquenne-Guigues basis. It is minimal in the cardinal of the set of implications, and minimal in the size of the left-hand side of the implications.
We continue our study of the general notion of L-Chu correspondence by introducing the category CRL-ChuCors incorporating residuation to the underlying complete lattice L, specifically, on the basis of a residuation-preserving isotone Galois connection ?. Then, the L-bonds are generalized within this same framework, and its structure is related to that of the extent of a suitably defined ?-direct product.
L-bonds represent relationships between formal contexts. We study properties of these intercontextual structures w.r.t. isotone concept- forming operators in fuzzy setting. We also focus on the direct product of two formal fuzzy contexts and show conditions under which a bond can be obtained as an intent of the product. In addition, we show that the previously studied properties of their antitone counterparts can be easily derived from the present results.
In this work we elaborate on the notion of contradiction between fuzzy sets introduced by Trillas et al in a fuzzy logic context. Our approach is parametric in that the operator used to define contradiction is rather a variable than a constant introduced prior to the analysis of contradiction. We give several motivations to consider weaker operators than the usual involutive negations, and obtain some preliminary results which validate this proposal.
In this work, we introduce sufficient and necessary conditions for a pair of fuzzy conjunctor and implication considered in a multi-adjoint frame under which the usual combinations of multi-adjoint sufficiency, possibility and necessity operators form either antitone or isotone Galois connections.
The concept lattice corresponding to a context may be alternatively specified by means of attribute implications. 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 of approaches to basis in the literature). 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
After recalling the different interpretations usually assigned to the term Galois connection, both in the crisp and in the fuzzy case, we survey on several of their applications in Computer Science and, specifically, in Soft Computing.
We have recently proposed a technique for generating thresholds (filters) useful for avoiding useless computations when executing fuzzy logic programs in a tabulated way. The method was conceived as a static preprocess practicable on program rules before being executed with our fuzzy thresholded tabulation principle, thus increasing the opportunities of prematurely disregarding those computation steps which are redundant (tabulation) or directly lead to non-significant solutions (thresholding). In this paper we reinforce the power of such static preprocess-which obviously does not require the consumption of extra computational resources at execution time-by re-formulating it in terms of the fuzzy unfolding technique initially designed in our group for transforming and optimizing fuzzy logic programs.
Adjoint pairs or adjoint triples defined on lattices have proven to be a useful tool when working in fuzzy formal concept analysis. This paper shows that adjoint pairs and triples can play as well an important role within the framework of multilattices, especially in order to form the Galois connections needed to build concept multilattices.
In this paper we continue analyzing the introduction of negation into the framework of residuated logic programming; specifically, we focus on extended programs, that is we consider programs with strong negation. The classical approach to extended logic programs consists in considering negated literals as new, independent, ones and, then apply the usual monotonic approach (based on the fix-point semantics and the TP operator); if the least fix-point so obtained is inconsistent, then the approach fails and no meaning is attached to the program. This paper introduces several approaches to measure consistency (under the term coherence) into a multi-adjoint setting.
Several papers relate different alternative approaches to classical concept lattices: such as property-oriented and object-oriented concept lattices and the dual concept lattices. Whereas the usual approach to the latter is via a negation operator, this paper presents a fuzzy generalization of the dual concept lattice, the dual multi-adjoint concept lattice, in which the philosophy of the multi-adjoint paradigm is applied and no negation on the lattices is needed.
Sets of attribute implications may have a certain degree of redundancy and the notion of basis appears as a way to characterize the implication set with less redundancy. 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.The main problem is to enumerate all the minimal generators from a set of implications. We introduce a method to compute all the minimal generators which is based on the Simplification Rule for implications. The simplification paradigm allows us to remove redundancy in the implications by deleting attributes inside the implication without removing the whole implication itself. In this work, the application of the Simplification Rule to the set of implications guides the search of the minimal generators in a logic-based style, providing a deterministic approach.
Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.
In this work we focus on the use of intensifying hedges as a tool to reduce the size of the recently introduced multi-adjoint concept lattices with heterogeneous conjunctors.
We focus on the notion of coherent L-interpretations with respect to a negation operator, as a convenient generalization to a fuzzy or multiple-valued environment of the classical notion of consistent interpretation. We show that, given an L-interpretation I, the set of negation operators n satisfying that I is coherent wrt n has a structure of complete lattice; so there exists the greatest and the least negation operators satisfying such property; moreover, the expression of the least negation operator n satisfying that I is coherent wrt n is presented. Finally, for the case in which the underlying set of truth-values is the real unit interval [0,1], we describe a method to achieve a practical expression for the least coherence-preserving negation.
We continue the study of the residuated operations in the framework of hyperstructures. We focus on the case of a multilattice as underlying algebraic structure and introduce the notions of filter and deductive system. They differ from the analogous concepts in a pocrim due to the connection to congruence relations. Finally, we prove that the set of filters of a residuated multilattice is a complete lattice.
Sets of attributes and objects in fuzzy formal concept analysis are usually different and, hence, it might not make sense to evaluate them on the same carrier. In this context, the operators used to obtain the concept lattice could be defined by associating different lattices to attributes and objects; several reasons exist for which we need to evaluate the sets of attributes and objects in the same carrier. Following this direction, we introduce a new definition of a concept lattice, where objects and attributes are evaluated on the same lattice L, although operators evaluating objects and attributes in different carriers are used. Moreover, we study the relationship between this new concept lattice and the alternative one which can be obtained directly by using different carriers for the sets of attributes and objects.
We introduce a sufficient condition which guarantees the existence of stable models for a normal residuated logic program interpreted on the truth-space [0, 1]n. Specifically, the continuity of the connectives involved in the program ensures the existence of stable models. Then, we study conditions which guarantee the uniqueness of stable models in the particular case of the product t-norm, its residuated implication, and the standard negation.
In this paper we continue the coalgebraization of the structure of multilattice. Specifically, we introduce a coalgebraic characterization of the notion of finitary multi(semi)lattice, a generalization of that of semilattice which arises naturally in several areas of computer science and provides the possibility of handling non-determinism.
An L-fuzzy generalization of the so-called Chu correspondences between formal contexts forms a category called L-ChuCors. In this work we show that this category naturally embeds ChuCors, and prove that it is *-autonomous. We also focus on the direct product of two L-fuzzy contexts, which is defined with the help of a binary operation, essentially a disjunction, on a lattice of truth-values L.
We initiate the exploration of the residuated operations in the framework of hyperstructures. We focus on the case of a multilattice as underlying algebraic structure, introduce the notion of residuated multilattice and study some of its properties, among which we have shown that the idempotency of the monoidal operation characterises the subclass of Heyting algebras.
Different notions of coherence and consistence have been proposed in the literature on fuzzy systems. In this work we focus on the relationship between some of the approaches developed, on the one hand, based of residuated lattices and, on the other hand, based on the theory of bilattices.
The need of considering non-determinism in theoretical computer science has been claimed by several authors in the literature. The notion of non-deterministic automata as a formal model of computation is widely used, but the specific study of non-determinism is useful, for instance, for natural language processing, in describing interactive systems, for characterizing the flexibility allowed in the design of a circuit or a network, etc. The most suitable structures for constituting the foundation of this theoretical model of computation are non-deterministic algebras. The interest on these generalized algebras has been growing in recent years, both from a crisp and a fuzzy standpoint. This paper presents a survey of these structures in order to foster its applicability for the development of new soft computing techniques.
The category of L-Chu correspondences between formal L-fuzzy contexts provides a categorical view on Formal Context Analysis. In this paper some interesting and useful properties are shown. The main aim of this paper is to introduce a functor between L-ChuCors and a category of supremum preserving mappings between completely L-ordered sets.
Recent approaches have shown that the measurement of the amount of inconsistent information contained in a logic theory can be useful to infer positive information.This paper deals with the definition of measures of inconsistency in the residuated logic programming paradigm under the fuzzy answer set semantics. This fuzzy framework provides a soft mechanism of controlling the amount of information inferred and thus, controlling the inconsistencies by modifying slightly the truth-values of some rules.
We introduce a sufficient condition which guarantees the existence of stable models for a normal residuated logic program interpreted on the truth-space [0,1]n. Specifically, the continuity of the connectives involved in the program ensures the existence of stable models. Then, we focus on the assignment of a fuzzy stable model semantics to inconsistent classical logic programs on the basis of the separation of the notion of inconsistence and uncertainty.
In this paper, we focus on the framework of Chu correspondences introduced by Mori for classical formal concept analysis, and we propose a suitable extension of the framework in a more general and flexible environment based on L-fuzzy sets, and define the notions of L-Chu correspondence and of L-bond. After introducing the generalized framework, the sets of L-Chu correspondences and of L-bonds are proved to have the structure of complete lattice and, furthermore, there exists a natural anti-isomorphism between them.
We introduce the syntax, semantics, and an axiom system for a PDL-based extension of the logic for order of magnitude qualitative reasoning, developed in order to deal with the concept of qualitative velocity, which together with qualitative distance and orientation, are important notions in order to represent spatial reasoning for moving objects, such as robots. The main advantages of using a PDL-based approach are, on the one hand, all the well-known advantages of using logic in AI, and, on the other hand, the possibility of constructing complex relations from simpler ones, the flexibility for using different levels of granularity, its possible extension by adding other spatial components, and the use of a language close to programming languages.
Multilattices are a suitable generalization of lattices which enables to accommodate the formalization of non-deterministic computation; specifically, the algebraic characterization for multilattices provides a formal framework to develop tools in several fields of computer science. On the other hand, the usefulness of coalgebra theory has been increasing in the recent years, and its importance is undeniable. In this paper, somehow mimicking the use of universal algebra, we define a new kind of coalgebras (the ND-coalgebras) that allows to formalize non-determinism, and show that several concepts, widely used in computer science are, indeed, ND-coalgebras. Within this formal context, we study a minimal set of properties which provides a coalgebraic definition of multilattices.
We focus on the direct product of two L-fuzzy contexts, which are defined with the help of a binary operation on a lattice of truth-values L. This operation, essentially a disjunction, is defined as kl=¬k->l, for k,linL where negation is interpreted as ¬l=l->0. We provide some results which extend previous work by Krötzsch, Hitzler and Zhang.
In Formal Concept Analysis, attribute reduction is a important step in order to reduce the complexity of the computation of the concept lattice. This reduction is more complex in fuzzy environments. In this paper, we will present a first approximation to reduce the set of attributes in the multi-adjoint concept lattice. The solution found is based on the development of specific results which allow us to reduce the number of attributes in the classical case, by detecting some relatively necessary and absolutely unnecessary attributes and, then, use linguistic labels in order to obtain a method to reduce the number of attributes in a multi-adjoint context, working up to some level of tolerance, and preserving the original lattice structure of the set of concepts.
An L-fuzzy generalization of the so-called Chu correspondences between formal contexts forms a category called L-ChuCors. In this work we show that this category naturally embeds ChuCors.
We introduce a sufficient condition which guarantees the existence of stable models for a normal residuated logic program interpreted on the truth-space [0,1]n. Specifically, the continuity of the connectives involved in the program ensures the existence of stable models.
In formal concept analysis, the sets of attributes and objects are usually different, with different meaning and, hence, it might not make sense to evaluate them on the same carrier. In this context, the operators used to obtain the concept lattice could be defined by considering different lattices associated to attributes and objects. Anyway there exist several reasons for which we need to evaluate the set of attributes and objects in the same carrier. In this direction, we present in this paper a new concept lattice, where the objects and attributes are evaluated on the same lattice L, although operators which evaluate objects and attributes in different carriers are used. Moreover, we have studied the relationship between the new concept lattice and the other one obtained directly considered different carriers to both set of attributes and objects.
Multilattices are a suitable generalization of lattices which enables to accommodate the formalization of non-deterministic computation; specifically, the algebraic characterization for multilattices provides a formal framework to develop tools in several fields of computer science. On the other hand, the usefulness of coalgebra theory has been increasing in the recent years, and its importance is undeniable. In this work, we define a new kind of coalgebras (the ND-coalgebras) that allows to formalize non-determinism, and show that several concepts, widely used in computer science are, indeed, ND-coalgebras. Within this formal context, we study a minimal set of properties which provides a coalgebraic definition of multilattices.
Fuzzy logic programming represents a flexible and powerful declarative paradigm amalgamating fuzzy logic and logic programming, for which there exists different promising approaches described in the literature. In this work we propose an improved fuzzy query answering procedure for the so-called multi-adjoint logic programming approach, which avoids the re-evaluation of goals and the generation of useless computations thanks to the combined use of tabulation with thresholding techniques. The general idea is that, when trying to perform a computation step by using a given program rule R, we firstly analyze if such step might contribute to reach further significant solutions (non-tabulated yet). When it is the case, it is possible to avoid useless computation steps via rule R by using thresholds and filters based on the truth degree of R, as well as a safe, accurate and dynamic estimation of the maximum truth degree associated to its body.
Inconsistency in the framework of general residuated logic programs can be, somehow, decomposed in two notions: incoherence and instability. In this work, we focus on the measure of instability of normal residuated programs. Some measures were already provided and initial results obtained in terms of the amount of information that have to be discarded in order to recover stability; in this paper, our interest is focused precisely on the case in which stability can be recovered by adding information to our program.
Inconsistency in the framework of general residuated logic programs can be, somehow, decomposed in two notions: incoherence and instability. In this work, we focus on the measure of instability of normal residuated programs. Some measures are provided and initial results are obtained in terms of the amount of information that have to be discarded in order to recover stability.
We focus on the study of the structure of hyperrings; in this paper, we recall the basics of crisp homomorphisms between hyperstructures, particularly, between hyperrings and, then, the notion of fuzzy homomorphism between hyperrings is established and its main properties are analysed.
The notion of coherence, introduced in the context of fuzzy answer set programming (FASP), provides a metalogic condition on the obtained models in FASP. In this work, we relate it with the concept of N-contradiction which is used in the definition of antonyms.
The t-concept lattice is introduced as a set of triples associated to graded tabular information interpreted in a non-commutative fuzzy logic. Following the general techniques of formal concept analysis, and based on the works by Georgescu and Popescu, given a non-commutative conjunctor it is possible to provide generalizations of the mappings for the intension and the extension in two different ways, and this generates a pair of concept lattices. In this paper, we show that the information common to both concept lattices can be seen as a sublattice of the Cartesian product of both concept lattices. The multi-adjoint framework can be applied to this general t-concept lattice, and its usefulness is illustrated by a working example.
Uncertainty exists almost everywhere, except in the most idealized situations; it is not only an inevitable and ubiquitous phenomenon, but is also a fundamental scientific principle. Furthermore, uncertainty is an attribute of information and, usually, decision-relevant information is uncertain and/or imprecise, therefore the abilities to handle uncertain information and to reason from incomplete knowledge are crucial features of intelligent behaviour in complex and dynamic environments. By carefully exploiting our tolerance for imprecision and approximation we can often achieve tractability, robustness, and better descriptions of reality than traditional deductive methods would allow us to obtain. In conclusion, as we move further into the age of machine intelligence, the problem of reasoning under uncertainty, in other words, drawing conclusions from partial knowledge, has become a major research theme.Not surprisingly, the rigorous treatment of uncertainty requires sophisticated machinery, and the present volume is conceived as a contribution to a better understanding of the foundations of information processing and decision-making in an environment of uncertainty, imprecision and partiality of truth.
This volume draws on papers presented at the 2008 Conference on Information Processing and Management of Uncertainty (IPMU) which was held in Málaga, Spain, organized by the University of Málaga, and brought together some of the world's leading experts in uncertainty handling.
Since its first edition, held in 1986, the focus of IPMU conferences has been on the development of foundations and technology needed for the construction of intelligent systems. Over the years, IPMU has grown steadily in visibility and importance, and has evolved into a leading conference in its field, embracing a wide-variety of methodologies for dealing with uncertainty and imprecision, and this explains the unusually wide variety of concepts, methods and techniques which are discussed in the book. The growth in importance of IPMU reflects the fact that as we move further into the age of machine intelligence and mechanized decision-making, the issue of how to deal with uncertain information becomes an issue of paramount concern.
The book starts with a revisited approach for possibilistic fuzzy regression methods proposed by Bisserier et al., in which the identification problem is reformulated according to a new criterion that assesses the model fuzziness independently of the collected data. Later, Bonissone et al. propose the fundamentals to design and construct a “forest” of randomly generated fuzzy decision trees in an approach which combines the robustness of multi-classifiers, the construction efficiency of decision trees, the power of the randomness to increase the diversity of the trees in the forest, and the flexibility of fuzzy logic and the fuzzy sets for data managing. The third contribution, by Delgado et al., is related to the well-known framework of mining association rules for extracting useful knowledge from databases; they introduce so-called double rules as a new type of rules which in conjunction with exception rules will describe in more detail the relationship between two sets of items. Next, Dubois discusses ignorance and contradiction, and argues that they cannot be viewed either as additional truth-values or processed in a truth-functional manner, and that doing it leads to weak or debatable uncertainty handling approaches.
The volume continues with Grzegorzewski's work, which introduces new algorithms for calculating the proper approximations by trapezoidal fuzzy numbers which preserves the expected interval. Next, Jenhani et al. investigate the problem of measuring the similarity degree between two normalized possibility distributions encoding preferences or uncertain knowledge. Later, Julián et al. propose an improved fuzzy query answering procedure for multi-adjoint logic programming which avoids the re-evaluation of goals and the generation of useless computations thanks to the combined use of tabulation with thresholding techniques. Then, Kacprzyk and Wilbik, focus on an extension of linguistic summarization of time series; ; in addition to the basic criterion of a degree of truth (validity), they also use a degree of imprecision, specificity, fuzziness and focus as an additional criteria.
In the final part of the volume, Kalina et al. discuss the possibility of applying the modified level-dependent Choquet integral to a monopersonal multicriterial decision-making problem; they propose an algorithm which produces an outranking of objects taking into account an interaction between criteria. Next, Llamazares and Marques Pereira consider mixture operators to aggregate individual preferences and characterize those that allow to extend some majorities rules, such as simple, Pareto and absolute special majorities, to the field of gradual preferences. Later, Mercier et al. concentrate on the links between the different operations that can be used in the theory of belief functions to correct the information provided by a source, given meta-knowledge about that source. Then, Miranda compares the different notions of conditional coherence within the behavioural theory of imprecise probabilities when all the referential spaces are finite. Finally, Soubaras focuses on evidential Markov chains as a suitable generalization of classical Markov chains to the Dempster-Shafer theory, replacing the involved states by sets of states.
Last, but not least, we would like to thank the following institutions for their help with the organization of the 12th IPMU Conference: Ministerio de Educación y Ciencia, grant TIN2007-30838-E, Junta de Andalucí a, grant RES. 2/07-OC, Universidad de Málaga, Diputación Provincial de Málaga, Patronato de Turismo de la Costa del Sol, Ayuntamiento de Málaga, Ayuntamiento de Torremolinos, European Society for Fuzzy Logic and Technology, EUSFLAT, IEEE Computational Intelligence Society.
Fuzzy logic programming represents a flexible and powerful declarative paradigm amalgamating fuzzy logic and logic programming, for which there exists different promising approaches described in the literature. In this paper we propose an improved fuzzy query answering procedure for the so called multi-adjoint logic programming approach, which avoids the re-evaluation of goals and the generation of useless computations thanks to the combined use of tabulation with thresholding techniques. The general idea is that, when trying to perform a computation step by using a given program rule R, we firstly analyze if such step might contribute to reach further significant solutions (non tabulated yet). When it is the case, it is possible to avoid a useless computation step via a rule R by using thresholds and filters based on the truth degree of R, as well as a safe, accurate and dynamic estimation of the maximum truth degree associated to its body.
The syntax, semantics and an axiom system for an extension of Propositional Dynamic Logic (PDL) for order of magnitude qualitative reasoning which formalizes the concepts of closeness and distance is introduced in this paper. In doing this, we use some of the advantages of PDL: firstly, we exploit the possibility of constructing complex relations from simpler ones for defining the concept of closeness and other programming commands such as while ... do and repeat ... until; secondly, we employ its theoretical support in order to show that the satisfiability problem is decidable. Moreover, the specific axioms of our logic have been obtained from the minimal set of formulas needed in our definition of qualitative sum of small, medium and large numbers. We also present some of the advantages of our approach on the basis of an example.
In this work we study the structure of the set of congruences on several hyperstructures with one and two (hyper-)operations. On the one hand, we show sufficient conditions guaranteeing that the set of congruences of an nd-groupoid forms a complete lattice (which, in turn, is a sublattice of the lattice of equivalence relations on the nd-groupoid). On the other hand, we focus on the study of the congruences on a multilattice; specifically, we prove that the set of congruences on an m-distributive multilattice forms a complete lattice and, moreover, show that the classical relationship between homomorphisms and congruences can be adequately adapted to work with multilattices under suitable restrictions.
Unlike monotone single-valued functions, multivalued mappings may have zero, one, or (possibly infinitely) many minimal fixed-points. The contribution of this work is twofold. First, we overview and investigate the existence and computation of minimal fixed-points of multivalued mappings, whose domain is a complete lattice and whose range is its power set. Second, we show how these results are applied to a general form of logic programs, where the truth space is a complete lattice. We show that a multivalued operator can be defined whose fixed-points are in one-to-one correspondence with the models of the logic program.
In this paper we continue analyzing the introduction of negation into the framework of residuated logic programming; specifically, we focus on extended programs, in which strong negation is introduced. The classical approach to extended logic programs consists in considering negated literals as new, independent, ones and, then apply the usual monotonic approach (based on the fix-point semantics and the TP operator); if the least fix-point so obtained is inconsistent, then the approach fails and no meaning is attached to the program. This paper introduces several approaches to considering consistence (under the term coherence) into a fuzzy setting, and studies some of their properties.
Sometimes, in real applications, we have to consider the use of non-commutative operators. However, it is interesting to be able to “balance" of “soften" the non-commutative character of the involved operators.There exist some approaches to the construction of concept lattices based on non-commutative conjunctors L×L->L , but are based on the fact that the supports (or carriers) of the fuzzy subsets of both objects and attributes have to coincide.
Our contribution in this work is to present sufficient conditions in order to be able to construct concepts in a generalized fuzzy context in which the domain of the underlying conjunctors can be L1×L2 with L1<>L2.
Starting with the underlying motivation of developing a general theory of L-fuzzy sets where L is a multilattice (a particular case of non-deterministic algebra), we study the relationship between the crisp notions of congruence, homomorphism and substructure on some non-deterministic algebras which have been used in the literature, i.e. hypergroups, and join spaces. Moreover, we provide suitable extensions of these notions to the fuzzy case.
We survey on the theoretical and practical developments of the theory of fuzzy logic and soft computing. Specifically, we briefly review the history and main milestones of fuzzy logic (in the wide sense), the more recent development of soft computing, and finalise by presenting a panoramic view of applications: from the most abstract to the most practical ones.
Formal concept analysis has become an important and appealing research topic. There exist a number of different fuzzy extensions of formal concept analysis and of its representation theorem, which gives conditions for a complete lattice in order to be isomorphic to a concept lattice. In this paper we concentrate on the study of operational properties of the mappings α and β required in the representation theorem.
In this position paper, we focus on the framework of Chu correspondences extending Mori's approach to formal concept analysis by proposing suitable definitions of the required concepts in an L-fuzzy environment.
In this paper, we focus on the notions of congruence, ideal and homomorphism on the generalized structure of multilattice. We provide suitable definitions of these notions in order to guarantee the classical relationship between these concepts.
In this work we recall the first steps towards the definition of an answer set semantics for residuated logic programs with negation, and concentrate on the development of relationships between the notions of coherence and consistence of an interpretation.
Logic programming has been used as a natural framework to automate deduction in the logic of order-of-magnitude reasoning. Specifically, we introduce a Prolog implementation of the Rasiowa-Sikorski proof system associated to the relational translation Re(OM) of the multimodal logic of order-of-magnitude qualitative reasoning OM.
Fuzzy logic programming represents a flexible and powerful declarative paradigm amalgamating fuzzy logic and logic programming, for which there exists different promising approaches described in the literature. In this paper we propose an improved fuzzy query answering procedure for the so called multi-adjoint logic programming approach, which avoids the re-evaluation of goals and the generation of useless computations thanks to the combined use of tabulation with thresholding techniques. The general idea is that, when trying to perform a computation step by using a given program rule R, we firstly analyze if such step might contribute to reach further significant solutions (non tabulated yet). When it is the case, it is possible to avoid a useless computation step via a rule R by using thresholds and filters based on the truth degree of R, as well as a safe, accurate and dynamic estimation of the maximum truth degree associated to its body.
In this work we introduce the notion of fuzzy congruence relation on an nd-groupoid and study conditions on the nd-groupoid which guarantee a complete lattice structure on the set of fuzzy congruence relations. The study of these conditions allowed to construct a counterexample to the statement that the set of fuzzy congruences on a hypergroupoid is a complete lattice.
We investigate the use of multilattices as the set of truth-values underlying a general fuzzy logic programming framework. On the one hand, some theoretical results about ideals of a multilattice are presented in order to provide an ideal-based semantics; on the other hand, a restricted semantics, in which interpretations assign elements of a multilattice to each propositional symbol, is presented and analysed.
In this work we introduce the first steps towards the def- inition of an answer set semantics for residuated logic pro- grams with negation.
Generalized concept lattices have been recently proposed to deal with uncertainty or incomplete information as a non-symmetric generalization of the theory of fuzzy formal concept analysis. On the other hand, concept lattices have been defined as well in the framework of fuzzy logics with non-commutative conjunctors. The contribution of this paper is to prove that any concept lattice for non- commutative fuzzy logic can be interpreted inside the framewok of generalized con- cept lattices, specifically, it is isomorphic to a sublattice of the cartesian product of two generalized concepts lattices.
We focus on a possible generalisation of the theory of congruences on a lattice to a more general framework. In this paper, we prove that the set of congruences on an m-distributive multilattice forms a complete lattice and, moreover, show that the classical relationship between homomorphisms and congruences can be adequately adapted to work with multilattices.
In this work we introduce the notion of fuzzy congruence relation on an nd- groupoid and study conditions on the nd-groupoid which guarantee a complete lattice structure on the set of fuzzy congruence relations. The study of these conditions allowed to construct a counterexample to the statement that the set of fuzzy congruences on a hypergroupoid is a complete lattice.
This work concentrates on the automated deduction of logics of order-of-magnitude reasoning. Specifically, a Prolog implementation is presented for the Rasiowa- Sikorski proof system associated to the relational translation Re(OM) of the mul- timodal logic of qualitative order-of-magnitude reasoning OM.
In this work we prove that the set of congruences on an nd-groupoid under suitable conditions is a complete lattice which is a sublattice of the lattice of equivalence relations on the nd-groupoid. The study of these conditions allowed to construct a counterexample to the statement that the set of (fuzzy) congruences on a hypergroupoid is a complete lattice.
In this work we introduce the first steps towards the definition of an answer set semantics for residuated logic programs with negation.
We introduce a Propositional Dynamic Logic for order of magnitude reasoning in order to formalize qualitative operations of sum and product. This new logic has enough expressive power to consider, for example, the concept of closeness, and to study some interesting proper- ties for the qualitative operations, together with the logical definability of these properties. Finally, we show the applicability of our approach on the basis of some examples.
Multi-adjoint logic programming represents an extremely flexible attempt for fuzzifying logic programming, where the classical SLD-resolution principle has been extended to cope with imperfect information. In this paper we propose an enhanced tabulation-based query answering procedure, which avoids the generation of useless computations via thresholding techniques.
Generalisation of the foundational basis for many-valued logic programming builds upon generalised terms in the form of powersets of terms. A categorical approach involving set and term functors as monads allows for a study of monad compositions that provide variable substitutions and compositions thereof. In this paper, substitutions and unifiers appear as constructs in Kleisli categories related to particular composed powerset term monads. Specifically, we show that a frequently used similarity-based approach to fuzzy unification is compatible with the categorical approach, and can be adequately extended in this setting; also some examples are included in order to illuminate the definitions.
In this paper some results are obtained regarding the existence and reachability of minimal fixed points for multiple-valued functions on a multilattice. The concept of inf-preserving multi-valued function is introduced, and shown to be a sufficient condition for the existence of minimal fixed point; then, we identify a sufficient condition granting that the immediate consequence operator for multilattice-based fuzzy logic programs is sup-preserving and, hence, computes minimal models in at most ω iterations.
This paper continues the research line on the multimodal logic of qualitative reasoning; specifically, it deals with the introduction of the notions non-closeness and distance. These concepts allow us to consider qualitative sum of medium and large numbers. We present a sound and complete axiomatization for this logic, together with some of its advantages by means of an example.
Generalized concept lattices have been recently proposed to deal with uncertainty or incomplete information as a non-symmetric generalization of the theory of fuzzy formal concept analysis. On the other hand, concept lattices have been defined as well in the framework of fuzzy logics with non- commutative conjunctors. The contribution of this paper is to prove that any concept lattice for non-commutative fuzzy logic can be interpreted inside the framewok of generalized concept lattices, specifically, it is isomorphic to a sublattice of the cartesian product of two generalized concepts lattices.
A general logic programming framework allowing for the combination of several adjoint lattices of truth-values is presented. The language is sorted, enabling the combination of several reasoning forms in the same knowledge base. The contribution of the paper is two-fold: on the one hand, sufficient conditions guaranteeing termination of all queries for the fix-point semantics for a wide class of sorted multi-adjoint logic programs are presented and related to some well-known probability-based formalisms; in addition, we specify a general non-deterministic tabulation goal-oriented query procedure for sorted multi-adjoint logic programs over complete lattices. We prove its soundness and completeness as well as independence of the selection ordering. We apply the termination results to probabilistic and fuzzy logic programming languages, enabling the use of the tabulation proof procedure for query answering.
This paper presents a computability theorem for fixed points of multi-valued functions defined on multilattices, which is later used in order to obtain conditions which ensure that the immediate consequence operator computes minimal models of multilattice-based logic programs in at most ? iterations.
Several fuzzifications of formal concept analysis have been proposed to deal with uncertainty or incomplete information. In this paper, we focus on the new paradigm of multi-adjoint concept lattices which embeds different fuzzy extensions of concept lattices, our main result being the representation theorem of this paradigm. As a consequence of this theorem, the representation theorems of the other paradigms can be proved more directly. Moreover, the multi-adjoint paradigm enriches the language providing greater flexibility to the user.
We investigate the use of multilattices as the set of truth-values underlying a general fuzzy logic programming framework. On the one hand, some theoretical results about ideals of a multilattice are presented in order to provide an ideal-based semantics; on the other hand, a restricted semantics, in which interpretations assign elements of a multilattice to each propositional symbol, is presented and analysed.
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