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 paper focuses on logics for order of magnitude reasoning. Our main goal is to formalize the notion of qualitative sum by devising a logic to deal with this arithmetic operation, without losing another important relations such as negligibility and closeness. We discuss the soundness and completeness of this logic and show some its applications on the basis of examples.
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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