extending fuzzy logics with many hedges abstract fuzzy logic aims at modeling logical reasoning with vague or imprecise statements which may contain linguistic hedges in fact many hedges e g very highly rather and slightly can be used simultaneously to express different levels of emphasis moreover each hedge might have a dual one e g slightly can be seen as a dual hedge of very thus it is necessary to extend systems of fuzzy logic with multiple hedges this work proposes two axiomatizations for multiple hedges as an expansion of a core fuzzy logic in one axiomatization hedges do not have any dual one while in the other each hedge can have its own dual one it is shown that the proposed logics not only cover a large class of hedge functions but also have all completeness properties as the underlying logic w r t the class of their chains as well as distinguished subclasses of their chains including standard completeness the axiomatizations are also extended to the first order level furthermore we present a method to build linguistic fuzzy logics based on the axiomatizations and a hedge algebra whose corresponding algebras use a linguistic truth domain taken from the hedge algebra for representing and reasoning with linguistically expressed human knowledge where truth values of vague sentences are given in linguistic terms

multi adjoint t concept lattices 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

derivation digraphs for dependencies in ordinal and similarity based data we present graph based method of reasoning with if then rules describing dependencies between attributes in ordinal and similarity based data the rules we consider have two basic interpretations as attribute implications in object attribute incidence data where objects are allowed to have attributes features to degrees and as similarity based functional dependencies in an extension of the codd model of data main results in this paper show that degrees to which if then rules are semantically entailed from sets or graded sets of other if then rules can be characterized by existence of particular directed acyclic graphs with vertices labeled by attributes and degrees coming from complete residuated lattices in addition we show that the construction of directed acyclic graphs can be used to compute closures of sets of attributes and normalized proofs

on g inverses and the nonsingularity of a bordered matrixabc0 abstract we consider the bordered matrix m a b c 0 with the rank additivity condition and g c 1 c 2 c 3 c 4 which is a g inverse of m we give the necessary and sufficient conditions for c 1 to be a g inverse outer inverse reflexive g inverse and moore penrose inverse of a as a special case we give necessary and sufficient conditions for m to be nonsingular and explicit expressions of c i in m 1 moreover we give a method for computing any outer inverse of a matrix

solving constrained matrix equations and cramer rule this paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using cramer rule

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

uninorm logic with the n potency axiom this paper investigates a class of extensions c nul of uninorm logic ul which is obtained by adding the n potency axiom to ul the main result of this paper is that by generalizing jenei and montagna style approach for proving standard completeness for monoidal t norm based logic mtl we obtain the standard completeness for c nul in addition we present the suitable analytic hypersequent calculi for c nul

on g inverses of a bordered matrix revisited abstract in this article we study the structure and properties of g inverse m of the bordered matrix m a b c 0 which is partitioned conforming with m m d 1 d 2 d 3 d 4 by applying the multiple quotient singular value decomposition qq svd from the structure of m we discuss the relations among d1 d2 d3 and d4 and derive equivalent conditions for m to be a bordered matrix by studying d1 in m we obtain some equivalent conditions for d1 to be a g inverse of a

on ordinal sums of triangular norms on bounded lattices ordinal sums have been introduced in many different contexts e g for posets semigroups t norms copulas aggregation operators or quite recently for hoops in this contribution we focus on ordinal sums of t norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t norms by such also the structure of the underlying bounded lattice is investigated further it is shown that up to trivial cases there are no ordinal sum t norms on product lattices in general

dual multi adjoint concept lattices 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

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

isotone fuzzy galois connections with hedges we study isotone fuzzy galois connections and concept lattices parameterized by particular unary operators the operators represent linguistic hedges such as very rather more or less etc isotone fuzzy galois connections and concept lattices provide an alternative to their antitone counterparts which are the fundamental structures behind formal concept analysis of data with fuzzy attributes we show that hedges enable us to control the number of formal concepts in the associated concept lattice we also describe the structure of the concept lattice and provide a counterpoint to the main theorem of concept lattices

construction of rough approximations in fuzzy setting in this paper two l rough approximation operators are defined by an arbitrary mapping from the set of all singletons to the set of all l sets which is a graded extension of jarvinen s method it also provides a convenient method to compute rough sets in fuzzy setting then the algebraic structure of rough sets and the relation to l topology are investigated in the end three special cases are discussed f is extensive symmetric and intensive

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

formal concept analysis via multi adjoint concept lattices several fuzzifications of formal concept analysis have been proposed to deal with uncertain information in this paper we focus on concept lattices under a multi adjoint paradigm which enriches the language providing greater flexibility to the user in that he she can choose from a number of different connectives multi adjoint concept lattices are shown to embed different fuzzy extensions of concept lattices found in the literature the main results of the paper being the representation theorem of this paradigm and the embedding of other well known approaches

on reductants in the framework of multi adjoint logic programming 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 multi adjoint framework we study some of its properties and its relationships with other notions of reductants and provide an algorithm for computing all the reductants associated with a multi adjoint logic program

solution of a system of linear equations with fuzzy numbers in this paper the interval nature of fuzzy numbers is revealed by showing that many interesting results from classical interval analysis transfer also into the fuzzy case the paper deals with a solution of a fuzzy interval system of linear equations i e a system in which fuzzy intervals numbers appear instead of crisp numbers

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

description of sup and inf preserving aggregation functions via families of clusters in data tables abstract connection between the theory of aggregation functions and formal concept analysis is discussed and studied thus filling a gap in the literature by building a bridge between these two theories one of them living in the world of data fusion the second one in the area of data mining we show how galois connections can be used to describe an important class of aggregation functions preserving suprema and by duality to describe aggregation functions preserving infima our discovered method gives an elegant and complete description of these classes also possible applications of our results within certain biclustering fuzzy fca based methods are discussed

fundamental study the multiple facets of the canonical direct unit implicational basis the notion of dependencies between attributes arises in many areas such as relational databases data analysis data mining formal concept analysis knowledge structures formalization of dependencies leads to the notion of so called full implicational systems or full family of functional dependencies which is in one to one correspondence with the other significant notions of closure operator and of closure system an efficient generation of a full implicational system or a closure system can be performed from equivalent implicational systems and in particular from the bases for such systems for example the so called canonical basis this paper shows the equality between five other bases originating from different works and satisfying various properties in particular they are unit implicational systems the three main properties of this unique basis are the directness canonical and minimal properties whence the name canonical direct unit implicational basis given to this unit implicational system the paper also gives a nice characterization of this canonical basis and makes precise its link with the prime implicants of the horn function associated to a closure operator it concludes that it is necessary to compare more closely related works made independently and with a different terminology in order to take advantage of the really new results in these works

on multi adjoint concept lattices based on heterogeneous conjunctors 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

on minimal sets of graded attribute implications we explore the structure of non redundant and minimal sets consisting of graded if then rules the rules serve as graded attribute implications in object attribute incidence data and as similarity based functional dependencies in a similarity based generalization of the relational model of data based on our observations we derive a polynomial time algorithm which transforms a given finite set of rules into an equivalent one which has the least size in terms of the number of rules

logical structure of fuzzy if then rules abstract this paper provides a logical basis for manipulation with fuzzy if then rules our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to lotfi a zadeh but also other kinds of approximate reasoning methods e g perception based deduction provided that there exists a possibility to characterize them within a formal logical system in contrast with other approaches employing variants of multiple valued first order logic the approach presented here employs fuzzy type theory of v novak which has sufficient expressive power to present the essential concepts and results in a compact elegant and justifiable form within the effectively formalized representation developed here based on a complete logical system it is possible to reconstruct numerous well known properties of cri related fuzzy inference methods albeit not from the analytic point of view as usually presented but as formal derivations of the logical system employed the authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy if then rule bases will be represented formalized and backed up by proof within the well founded logical representation presented here an immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy if then rules are first class citizens of the representation there are clear logically founded definitions for fuzzy if then rule bases to be consistent complete or independent

on some questions concerning the axiomatisation of wnm algebras and their subvarieties in a seminal paper esteva and godo introduced monoidal t norm based logic mtl and some of its prominent extensions such as nm and wnm we notice that nm is axiomatisable from imtl and hence mtl with one variable axioms by instantiating the wnm axiom over one variable this observation leads us here to study the logic axiomatised by extending mtl by this one variable axiom we shall refer to its equivalent algebraic semantics as the variety of ghp algebras for those algebras will be shown to form the largest variety of mtl algebras such that the falsum free reducts of the positive cones of their chains are the most general totally ordered godel hoops among other results we obtain a general description of ghp standard algebras and use the latter to characterise those extensions of wnm that can be obtained from ghp via the same set of extending axioms

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

rotation invariant t norms where triple rotation and rotation annihilation meet generalizing the notion of a zoom of a t norm not only allows to extend the triple rotation method and its corresponding decomposition but also allows to describe the interface between the triple rotation method and the rotation annihilation method an alternative view on the partition behind the rotation annihilation method is required to make both methods compatible besides zooms we also use contour lines and the companion of a t norm to concisely formulate the results

on improving density estimators which are not bona fide functions in order to improve the rate of decrease of the imse for nonparametric kernel density estimators with nonrandom bandwidth beyond o n 4 5 all current methods must relax the constraint that the density estimate be a bona fide function that is be nonnegative and integrate to one in this paper we show how to achieve similar improvement without relaxing any of these constraints the method can also be applied for orthogonal series adaptive orthogonal series spline jackknife and other density estimators and assures an improvement of the imse for each sample size

relations of reduction between covering generalized rough sets and concept lattices the reduction theory plays an important role in data analysis this paper studies the relation between the reduction of a covering and the attribute reduction of a concept lattice the reduction of a covering from the perspective of concept lattices is investigated conversely the attribute reduction of a formal context is studied in the framework of covering generalized rough sets the results in this paper show that the reduction of a covering can be viewed as the attribute reduction of a derivative formal context moreover every reduct of a given formal context can be seen as the reduct of an induced covering as an application of the theoretical results an approach to the attribute reduction of concept lattices based on covering generalized rough sets is proposed furthermore experiments are given to show the effectiveness of the proposed method

topology on bl algebras in this paper by using a special family of filters on a bl algebra l we construct a topology on l show that l is a topological bl algebra find the closure of any subsets of l and obtain some conditions under which that l is a hausdorff space also we prove some theorems about connected components of this topological space and use them to construct a quotient semitopological bl algebras finally we introduce the concept of complete bl algebra and attempt to construct a complete bl algebra from l

discrete dualities for n potent mtl algebras and 2 potent bl algebras abstract discrete dualities are developed for n potent mtl algebras and for 2 potent bl algebras that is classes of frames or relational systems are defined that serve as dual counterparts to these classes of algebras the frames defined here are extensions of the frames that were developed for mtl algebras in 24 and 25 the additional frame conditions required are given here and also the proofs that discrete dualities hold with respect to such frames the duality also provides an embedding from an n potent mtl algebra or 2 potent bl algebra into the complex algebra of its canonical frame which is a complete algebra in the lattice sense

constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges the paper deals with the isomorphism between the constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges the essential point of the former approach encompasses the full diversification of data structures within a formal context in particular we use a different complete lattice for diverse objects a different complete lattice for diverse attributes and a different poset for diverse matrix fields the latter framework with heterogeneous hedges results in a reduction in the size of the corresponding concept lattice we present the properties of constraint heterogeneous approach that is associated with the fixpoints of hedges and we add remarks to the related studies

triangular norm based predicate fuzzy logics the paper surveys the present state of knowledge on t norm based predicate fuzzy logics with their double semantics standard the set of truth values being the real interval 0 1 and general with abstract algebras of truth functions

density elimination density elimination a close relative of cut elimination consists of removing applications of the takeuti titani density rule from derivations in gentzen style hypersequent calculi its most important use is as a crucial step in establishing standard completeness for syntactic presentations of fuzzy logics that is completeness with respect to algebras based on the real unit interval 0 1 this paper introduces the method of density elimination by substitutions for general classes of first order hypersequent calculi it is shown that density elimination by substitutions is guaranteed by known sufficient conditions for cut elimination these results provide the basis for uniform characterizations of calculi complete with respect to densely and linearly ordered algebras standard completeness follows for many first order fuzzy logics using a dedekind macneille style completion and embedding

a characterization of truth functions in the nilpotent minimum logic abstract by introducing a new family of partitions into the n cube 0 1 n the problem of characterizing truth tables of formulas in the nilpotent minimum logic is solved and their normal forms are presented so far only this kind of fuzzy truth functions have normal forms among all fuzzy propositional calculi which are based on left continuous but discontinuous t norm

the use of linguistic variables and fuzzy propositions in the l fuzzy concept theory the use of linguistic variables and fuzzy propositions in the interval valued l fuzzy contexts can be an interesting tool to extract a more complete information from them in this paper we analyze three different situations first we obtain significant relations in order to study all the objects and attributes of the interval valued l fuzzy context by means of the interval valued l fuzzy concepts after that we show how to replace the erroneous values to be able to study in a suitable way the context finally we use the linguistic labels to obtain a subcontext that represents our interest of study we also show an experimental evaluation in the paper

prenex normal form in linguistic quantifiers modeled by sugeno integrals in this paper ying s prenex normal form theorem for linguistic quantifiers is strengthened so that it behaves like the one in the classical logic this result further shows elegant logical properties of ying s framework

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

orthogonal polynomials random matrices and the numerical inversion of the laplace transform of positive functions a method for the numerical inversion of the laplace transform of a continuous positive function f t is proposed random matrices distributed according to a gibbs law whose energy v x is a function of f t are considered as well as random polynomials orthogonal with respect to w x e v x the equation relating w x to the reproducing kernel and to the condensed density of the roots of the random orthogonal polynomials is exploited basic results from the theories of orthogonal polynomials random matrices and random polynomials are revisited in order to provide a unified and almost self contained context the qualitative behavior of the solutions provided by the proposed method is illustrated by numerical examples and discussed by using logarithmic potentials with external fields that give insight into the asymptotic behavior of the condensed density when the number of data points goes to infinity

on least squares g inverses and minimum norm g inverses of a bordered matrix abstract by applying the restricted singular value decomposition r svd qq svd in this article we derive structures of least squares g inverses and minimum norm g inverses of a bordered matrix m a b c 0 we also obtain the specific formulation of the pesudo inverse m

on the characteristic polynomials and eigenvalues of some specially bordered matrices abstract using a determinant identity and leverrier s algorithm for m d system we obtain the explicit expressions of the characteristic polynomial and the eigenvalues of some specially structured bordered matrices m a u v d

an equivalence between relational database dependencies and a fragment of propositional logic it is known that there is an eqmvalence between functional dependencies m a relatmonal database and a certain fragment of proposmonal logic thins eqmvalence is extended to include both functional and multivalued dependencmes thus for each dependency there is a corresponding statement m proposmonal logic it ms then shown that a dependency funcuonal or multivalued is a consequence of a set of dependencies ff and only ff the corresponding proposiuonal statement s a consequence of the corresponding set of proposmonal statements examples are given to show that these techniques are valuable mn provmdmg much shorter proofs of theorems about dependencies than have been obtained by more tradmonal means it is shown that this eqmvalence cannot be extended to include either join dependencies or embedded multmvalued dependencies

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

fuzzy logic and arithmetical hierarchy fuzzy logic is understood as a logic with a comparative and truth functional notion of truth arithmetical complexity of sets of tautologies identically true sentences and satisfiable sentences sentences true in at least one interpretation as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated

on stability of fuzzy formal concepts over randomized one sided formal context abstract we propose a probabilistic approach to the issue of one sided fuzzy formal concepts stability the modified rice siff algorithm represents a crisp index how to select the relevant concepts from the set of all one sided fuzzy formal concepts we suggest to explore the formal concepts stability affected by the random fluctuation of values in a formal context we describe the algorithm and study the properties of the concept stability using random variables with the gaussian normal distribution in combination with the modified rice siff algorithm the gaussian probabilistic index improves the analysis of the most relevant one sided formal concepts from the original one sided formal context the connections to recent works in the related directions are presented

issues on adjointness in multiple valued logics we contribute to the theory of implications and conjunctions related by adjointness in multiple valued logics we suggest their use in zadeh s compositional rule of inference to interpret generalized modus ponens inference schemata we provide new complete characterizations of implications that distinguish left arguments implications that satisfy the exchange principle divisible conjunctions commutative conjunctions associative conjunctions and triangular norms we also introduce and characterize pseudo strict and pseudo continuous implications and conjunctions and we explore the close relationship between these two notions

some kinds of c ccq interval valued fuzzy ideals of bci algebras in this paper we first introduce the notions of positive implicative implicative and commutative interval valued fuzzy ideals of bci algebras which are generalizations of positive implicative implicative and commutative fuzzy ideals respectively and investigate some of their related properties the concept of quasi coincidence of an interval valued fuzzy point within an interval valued fuzzy set is introduced in fact the concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we further introduce the notions of positive implicative implicative and commutative q interval valued fuzzy ideals of bci algebras and investigate some of their related properties some characterization theorems of these generalized interval valued fuzzy ideals are derived the relationship among these generalized interval valued fuzzy ideals of bci algebras is also considered

on some matrix equalities for generalized inverses with applications abstract necessary and sufficient conditions are derived for the matrix equality a pn q to hold where a and n are generalized inverses of matrices some consequences and applications are also given in particular necessary and sufficient conditions are derived for the additive decompositions c a b and a a 1 a k to hold

discretized and interpolated kernel density estimates abstract the kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator nevertheless for computational and especially graphical reasons exact kernel density estimates are often presented in piecewise constant discretized or piecewise linear interpolated form using the exact estimate only at some grid of points the asymptotic integrated mean squared error properties of these modifications are studied it is seen that discretization may adversely affect the order of magnitude of this risk criterion so care needs to be taken in practice to limit the degree of discretization employed this roughness effect essentially disappears when interpolation is used instead then it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate s performance data discretization prior to applying the kernel density estimation prescription is also addressed such prebinning does not affect the smoot

evolution in time of l fuzzy context sequences in this work we consider a complete lattice l and we study l fuzzy context sequences which represent the evolution in time of an l fuzzy context to carry out this study in the first part of the paper we consider n ary owa operators in complete lattices which enable us to make a general analysis and a temporal analysis at any moment in time of l fuzzy context sequences after that evolution in time of the relationship between the objects and the attributes is considered in particular we analyze the concepts of trend and persistent formal contexts finally we illustrate our results with an example where we consider the particular lattice l j 0 1

fuzzy logic programming reduced to reasoning with attribute implications abstract we present a link between two types of logic systems for reasoning with graded if then rules the system of fuzzy logic programming flp in sense of vojtas and the system of fuzzy attribute logic fal in sense of belohlavek and vychodil we show that each finite theory consisting of formulas of fal can be represented by a definite program so that the semantic entailment in fal can be characterized by correct answers for the program conversely we show that for each definite program there is a collection of formulas of fal so that the correct answers can be represented by the entailment in fal using the link we can transport results from fal to flp and vice versa which gives us e g a syntactic characterization of correct answers based on pavelka style armstrong like axiomatization of fal we further show that entailment in flp is reducible to reasoning with boolean attribute implications and elaborate on related issues including properties of least models

the influence of rounding errors on some nonparametric estimators of a density and its derivatives all continuous data is rounded to some extent and this rounding introduces errors into the estimation of the parent density and its derivatives in this paper we examine the influence of rounding errors on the two most popular types of nonparametric density estimator those based on kernels and those based on generalized fourier series our results lead directly to several important conclusions which are very relevant to the practical problems of density estimation

frequency polygons theory and application abstract in this article i investigate the theoretical properties and applications of the frequency polygon which is constructed by connecting with straight lines the mid bin values of a histogram for estimating an unknown probability density function using a random sample the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error achieving the same rate of convergence to zero of the integrated mean squared error as non negative kernel estimators data based algorithms for constructing frequency polygons are discussed and illustrated one is based on a histogram with bin width equal to 2 15sn 1 5 where s is an estimate of the standard deviation from a sample of size n another approach is based on the method of generalized crossvalidation the bivariate frequency polygon is also investigated comparisons are made between the frequency polygon and other density estimators examples with data in one and two dimensions are presented

multi adjoint property oriented and object oriented concept lattices this paper presents a generalisation of the classical property and object oriented concept lattices to a fuzzy environment based on the philosophy of the multi adjoint paradigm these concept lattices are generalisations of rough set theory used to consider two different sets the set of objects and the set of attributes to apply the corresponding modal operators as in formal concept analysis first of all the paper presents several specific properties of adjoint triples in which the duals of several supports are considered these properties are then used to introduce the multi adjoint property and object oriented concept lattices in which different adjoint triples can be used in non linear sets as well as the corresponding representation fundamental theorems moreover these lattices are related to the multi adjoint concept lattice in which negation operators are not needed as a result this relation allows the properties to be translated from one to another

the use of two relations in l fuzzy contexts in the analysis of relations among the elements of two sets it is usual to obtain different values depending on the point of view from which these relations are measured the main goal of the paper is the modelization of these situations by means of a generalization of the l fuzzy concept analysis called l fuzzy bicontext we study the l fuzzy concepts of these l fuzzy bicontexts obtaining some interesting results specifically we will be able to classify the biconcepts of the l fuzzy bicontext finally a practical case is developed using this new tool

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

computing the lattice of all fixpoints of a fuzzy closure operator we present a fast bottom up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees along with the partial order on the set of all fixpoints fuzzy closure operators appear in several areas of fuzzy logic and its applications including formal concept analysis fca that we use as a reference area of application in this paper several problems in fca such as computing all formal concepts from data with graded attributes or computing non redundant bases of all attribute dependencies can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data the development of a general algorithm that is applicable in particular to these problems is the ultimate purpose of this paper we present the algorithm its theoretical foundations and experimental evaluation

the genetic development of ordinal sums abstract the aim of this study is to develop an operational model of an ordinal sum of triangular norms the essence of this construct lies in the use of different t norms and or t conorms defined over disjoint subintervals of the unit interval the result of such aggregation is a highly versatile logic operator that can be easily adapted to the existing experimental evidence we propose a genetic optimisation environment genetic algorithms gas in particular to construct ordinal sums and show how the ga mechanism helps optimize subintervals and to allocate individual local t norms the application of the genetically designed ordinal sums is shown in case of zimmermann zysno logic operator data we also demonstrate the use of the ordinal sum to the construction of neuro fuzzy systems in this case we quantify the performance of ordinal sums to standard logic operators used in such models

a logical approach to fuzzy truth hedges the starting point of this paper are the works of hajek and vychodil on the axiomatization of truth stressing and depressing hedges as expansions of hajek s bl logic by new unary connectives they showed that their logics are chain complete but standard completeness was only proved for the expansions over godel logic we propose weaker axiomatizations over an arbitrary core fuzzy logic which have two main advantages i they preserve the standard completeness properties of the original logic and ii any subdiagonal resp superdiagonal non decreasing function on 0 1 preserving 0 and 1 is a sound interpretation of the truth stresser resp depresser connectives hence these logics accommodate most of the truth hedge functions used in the literature about of fuzzy logic in a broader sense

adjoint negations more than residuated negations negation operators are useful operators which have extensively been studied and used in different fuzzy settings such as in fuzzy logic this paper introduces a new and flexible kind of negation operator based on adjoint triples and which is called adjoint negation relative to a fixed element besides proving several interesting properties the comparison with weak negations studied by trillas esteva et al and pairs of weak negations introduced by georgescu and popescu is presented furthermore adjoint negations are used in order to define the corresponding dual disjunctive to an adjoint conjunctor

achieving higher order convergence rates for density estimation with binned data abstract a method is proposed for achieving fast o n 8 9 and faster asymptotic mean integrated squared error amise convergence rates in density estimation using data condensed to standard histogram bin counts and edges the method involves the two elements of b splines and the correct mass proportions property in which the probability mass for the histogram bins equals exactly the fraction of the data found in those bins it is shown that the asymptotic bias of such an estimate using a spline of order p and bin width h will be o hp resulting in optimal amise of o n 2p 2p 1 form theoretical and numeric comparisons to kernel based and other estimators show that the histospline estimator is usually comparable and often superior an extension to the multivariate density estimation setting is described

parameterizing the semantics of fuzzy attribute implications by systems of isotone galois connections we study the semantics of fuzzy if then rules called fuzzy attribute implications parameterized by systems of isotone galois connections the rules express dependences between fuzzy attributes in object attribute incidence data the proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches we formalize the general parameterizations propose bivalent and graded notions of semantic entailment of fuzzy attribute implications show their characterization in terms of least models and complete axiomatization and provide characterization of bases of fuzzy attribute implications derived from data

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

fuzzy logic programming in this paper we consider the theory of fuzzy logic programming without negation our results cover logical systems with a wide variety of connectives ranging from t norm and conorms through conjunctors and disjunctors and their residuals to aggregation operators rules of our programs are many valued implications we emphasize that in contrast to other approaches our logic is truth functional i e according to p hajek we work in fuzzy logic in narrow sense we prove the soundness and the completeness of our formal model we deal with applications to threshold computation abduction fuzzy unification based on similarity we show that fuzzy unification based on similarities has applications to fuzzy databases and flexible querying

computing sets of graded attribute implications with witnessed non redundancy in this paper we extend our previous results on sets of graded attribute implications with witnessed non redundancy we assume finite residuated lattices as structures of truth degrees and use arbitrary idempotent truth stressing linguistic hedges as parameters which influence the semantics of graded attribute implications in this setting we introduce algorithm which transforms any set of graded attribute implications into an equivalent non redundant set of graded attribute implications with saturated consequents whose non redundancy is witnessed by antecedents of the formulas as a consequence we solve the open problem regarding the existence of general systems of pseudo intents which appear in formal concept analysis of object attribute data with graded attributes and linguistic hedges furthermore we show a polynomial time procedure for determining bases given by general systems of pseudo intents from sets of graded attribute implications which are complete in data

representations of generalized inverses of partitioned matrix involving schur complement in this article we consider some representations of 1 3 1 4 1 2 3 and 1 2 4 inverses of the partitioned matrix m which are equivalent to some rank additivity conditions we present the application of these results to generalizations of the sherman morrison woodbury type formulae

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests