an equivalence classes model of fuzzy relational databases abstract the technique of employing sets of values for tuple components to express imprecision in relational databases was proposed by buckles and petry in their classic works on fuzzy relational databases in addition to providing an intuitively appealing scheme for representing fuzzy information the model of buckles and petry possesses several key properties of the classical relational model by employing finite scalar domains with similarity relations and special fuzzy number domains buckles and petry have demonstrated that the classical properties of uniqueness of tuple interpretations and well definedness of the relational algebra can be retained in the fuzzy relational database model the key to the preservation of these properties is the fact that scalar domains with similarity relations and the fuzzy number domains can be partitioned into equivalence classes however since equivalence classes can be constructed without assuming the existence of similarity relations or special fuzzy number domains it is desirable to generalize the fuzzy relational database model to one based only on equivalence classes in this work we show that the important properties of classical relational databases and of fuzzy relational databases are preserved in a generalized model built on equivalence relations on finite database domains further we generalize the notion of a functional dependency to the fuzzy relational model

weighted minimum and maximun operations in fuzzy sets theory abstract this short note comments on the interpretation of weighted minimum and maximum operations introduced in a previous paper as medians and corrects a misleading statement about it

a complete axiomatization of fuzzy functional dependencies using fuzzy function the notion of functional dependency in classical relational databases corresponds to a partial function therefore it appears natural that the notion of fuzzy functional dependency should correspond to the notion of partial fuzzy function following this approach we introduce the notion of fuzzy functional dependency in fuzzy relational databases and show that the classical inference rules for functional dependencies form a sound and complete formal system in our case

a computational approach to approximate and plausible reasoning with applications to expert systems the intended purpose of this paper is twofold proposing a common basis for the modeling of uncertainty and imprecision and discussing various kinds of approximate and plausible reasoning schemes in this framework together with probability different kinds of uncertainty measures credibility and plausibility functions in the sense of shafer possibility measures in the sense of zadeh and the dual measures of necessity sugeno s g fuzzy measures are introduced in a unified way the modeling of imprecision in terms of possibility distribution is then presented and related questions such as the measure of the uncertainty of fuzzy events the probability and possibility qualification of statements the concept of a degree of truth and the truth qualification of propositions are discussed at length deductive inference from premises weighted by different kinds of measures by uncertainty or by truth values in the framework of various multivalued logics is fully investigated then deductive inferences from imprecise or fuzzy premises are dealt with patterns of reasoning where both uncertainty and imprecision are present are also addressed the last section is devoted to the combination of uncertain or imprecise pieces of information given by different sources on the whole this paper is a tentative survey of quantitative approaches in the modeling of uncertainty and imprecision including recent theoretical proposals as well as more empirical techniques such as the ones developed in expert systems such as mycin or prospector the management of uncertainty and imprecision in reasoning patterns being a key issue in artificial intelligence

data mining by attribute generalization with fuzzy hierarchies in fuzzy databases in this paper we examine the soft computing issues involving data generalization on fuzzy information the attribute induction process is formalized for these three cases crisp hierarchy fuzzy data fuzzy hierarchy crisp data fuzzy hierarchy fuzzy data each case is exemplified by applications of the algorithm to a prototype database

possibilistic functional dependencies and their relationship to possibility theory this paper introduces possibilistic functional dependencies these dependencies are associated with a particular possibility distribution over possible worlds of a classical database the possibility distribution reflects a layered view of the database the highest layer of the classical database consists of those tuples that certainly belong to it while the other layers add tuples that only possibly belong to the database with different levels of possibility the relation between the confidence levels associated with the tuples and the possibility distribution over possible database worlds is discussed in detail in the setting of possibility theory a possibilistic functional dependency is a classical functional dependency associated with a certainty level that reflects the highest confidence level where the functional dependency no longer holds in the layered database moreover the relationship between possibilistic functional dependencies and possibilistic logic formulas is established related work is reviewed and the intended use of possibilistic functional dependencies is discussed in the conclusion

an approach to solve division like queries in fuzzy object databases the division operator is well known in the context of fuzzy relational databases fuzzy approaches to this operator can be found in the literature to solve flexible queries these approaches try to compute to which extent each element of a candidate set is connected with a given fuzzy set of elements this kind of queries is also interesting in the framework of fuzzy object databases where the management of complex objects instead of plain tuples causes additional difficulties the presence of fuzzily described objects in the database makes necessary to use suitable operators that take into account the resemblance that governs the comparison in the underlying reference universe in this paper we propose a method to solve this type of queries founded on the use of fuzzy inclusion operators we study two different alternatives for the way resemblance is considered and we also analyze the role of cardinality in the process

functional dependencies with null values fuzzy values and crisp values based on the concepts of the semantic proximity we present a definition of the fuzzy functional dependency we show that the inference rules for fuzzy functional dependencies which are the same as armstrong s axioms for the crisp case are correct and complete we also show that dependent constraints with dull values constitute a lattice functional dependencies in classical relational databases and null functional dependencies can be viewed as a special case of fuzzy functional dependencies by applying the unified functional dependencies to the relational database design we can represent the data with fuzzy values null values and crisp values under relational database management systems by using fuzzy functional dependencies we can compress the range of a fuzzy value and make this fuzzy value clearer

data summarization in relational databases through fuzzy dependencies in this paper we deal with the problem of treating with dependencies in relational databases which do not hold in an exact manner as classical functional dependencies but in a weaker sense i e we face with relations which satisfy dependencies such that people with similar age and height have similar weight we model this relationship through the concept of fuzzy dependency we see that these dependencies imply some kind of fuzzy redundancy and in order to avoid it we propose to use a projection operator which leads us to partition a relation r into two projections say r1 and r2 with a less amount of information then we proceed to replace the original relation by these projections in this process we must guarantee that we can recover the data we had in the original relation this will be possible by using a special join operator applied to r1 and r2 we must also guarantee that we can test the fuzzy dependency for new entries to the database in the same way either if we consider the original relation r or if we work with the projections r1 and r2 we also show that this definition of dependency maintains the good properties of completeness of the classical case 1999 elsevier science inc all rights reserved

the dependency preserving decomposition and a testing algorithm in a fuzzy relational data model abstract this paper focuses on one of the fundamental design issues in fuzzy database modeling namely the dependency preserving decomposition in a fuzzy relational data model where imprecision is reflected by possibility distributions for attribute values as well as by closeness relations for domain elements a poor model design can be remedied in many cases by decomposing relation schemes in order to eliminate reduce data redundancy and update anomalies on the other hand the decomposition should guarantee that the semantic knowledge and integrity constraints expressed by fuzzy functional dependency ffd are satisfactorily preserved by the resultant relation schemes based on the concept of ffd transitive closure an algorithm has been developed to test whether a given decomposition is dependency preserving with respect to a given set of ffds finally two special ffd sets one composed of a x 1 to x 1 ffd loop and the other composed of a x 1 x m ffd and a x 1 to x m ffd chain are investigated

on prime vs prime power pairs in this paper we consider pairs of a prime and a prime power with a fixed difference we prove an average result on the distribution of such pairs this is a partial improvement of the result of bauer acta arith 85 99 118 1998

the set of fuzzy rational numbers and flexible querying a fuzzy bag is a bag in which each occurrence of an element is associated with a grade of membership this notion can be viewed as a generalization of the concepts of set fuzzy set and bag the set of fuzzy integers n f provides a general characterization in which all these different concepts are treated in a uniform way and can then be composed in the field of databases the use of fuzzy bags is motivated by their ability to manage both quantities and preferences however n f becomes too restricted a framework when dealing with queries based on difference or division operations so a more general structure based on the set of fuzzy relative integers z f in which exact differences can be performed has been first developed in this paper we carry on with this approach and we extend z f to the set of fuzzy rational numbers q f this context leads to define a closed system of multiplicative operations and allows to perform exact divisions applied to flexible querying of databases q f and the notion of division on fuzzy numbers allow to generalize the relational division they define a sound basis for queries involving ratios between quantities

superposition laws for solutions of differential matrix riccati equations arising in control theory representation formulas are given for the general solution of the n times n matrix riccati equation dot w a wb cw wdw using n known solutions with n 1 5 n representations the 5 representation is a superposition formula in that it expresses the general solution explicitly as a function of five particular solutions and n2arbitrary constants n geq 2 using no further information the representation formulas can be used in numerical calculations the 4 and 5 representations are specially useful when a solution w t has a singularity for some finite t t 0 they also clarify the properties of the solution space the matrix elements of w t are meromorphic functions of t having simple poles as the only possible singularities the relation between the representation formulas and previously known results is discussed

fuzzy loss less decompositions in databases in this paper we deal with the problem of defining projection and join operators in the framework of a fuzzy relational database we study the properties of such operators relating to the concept of fuzzy functional dependency there are two main properties to achieve on the one hand if a relation r satisfies a dependency between two sets of attributes x and y then we want to preserve this dependency in the fuzzy projection over xy of r on the other hand we want to test a fuzzy dependency just by looking at this projection therefore such a projection operator must allow us to store the information conveyed in r into a separate relation with fewer tuples in order to define the criterion for fuzzy dependency we shall need to extend a resemblance relation defined for crisp data to the fuzzy case finally we shall introduce a fuzzy join operator which will allow us to recover the information given in a relation from its projections

evaluating flexible criteria on uncertain data abstract modern information management systems and databases are rapidly becoming better equipped for handling data imperfections a common imperfection is uncertainty indicating that a property s exact value is not known ideally such systems can be queried uniformly using flexible criteria regardless of whether the underlying data are uncertain or not the result thereof should always be informative and intuitive and should reflect to what degree the data satisfy the criteria and to which degree this is uncertain in this work we present a novel way to evaluate flexible criteria on uncertain data the result thereof is a distribution of uncertainty over degrees of satisfaction these so called suitability distributions are first constructed for possibilistic data it is shown that they can be used in all scenarios going from regular crisp criteria on certain data to flexible criteria on uncertain data and that they seamlessly generalize other alternatives importantly their interpretation is always the same so they can be used without needing to have prior knowledge regarding the quality of the data afterwards their properties and supported operations are given next it is shown that they can also be applied more broadly for example for probabilistic data examples illustrate their rich semantics ease of use and broad applicability

the study of joins in fuzzy relational databases abstract it is often necessary to join naturally two or more fuzzy relations to answer a user s query however the result of a join may not be what one expects in this paper we study the decomposition problem of fuzzy relation schemes we show that the algorithm proposed by aho et al 1 to test lossless join decomposition for classical relations can be applied in the fuzzy domain when particularizations are induced by functional dependencies

generalizing database relational algebra for the treatment of incomplete or uncertain information and vague queries abstract this paper deals with relational databases which are extended in the sense that fuzzily known values are allowed for attributes precise as well as partial imprecise uncertain knowledge concerning the value of the attributes are represented by means of 0 1 valued possibility distributions in zadeh s sense thus we have to manipulate ordinary relations on cartesian products of sets of fuzzy subsets rather than fuzzy relations besides vague queries whose contents are also represented by possibility distributions can be taken into account the basic operations of relational algebra union intersection cartesian product projection and selection are extended in order to deal with partial information and vague queries approximate equalities and inequalities modeled by fuzzy relations can also be taken into account in the selection operation then the main features of a query language based on the extended relational algebra are presented an illustrative example is provided this approach which enables a very general treatment of relational databases with fuzzy attribute values makes an extensive use of dual possibility and necessity measures

a fuzzy datalog deductive database system this paper describes a proposal for a deductive database system with fuzzy small mathsf datalog as its query language concepts supporting the fuzzy logic programming system small mathsf bousi sim small mathsf prolog are tailored to the needs of the deductive database system small mathsf des we develop a version of fuzzy small mathsf datalog where programs and queries are compiled to the small mathsf des core small mathsf datalog language weak unification and weak sld resolution are adapted for this setting and extended to allow rules with truth degree annotations we provide a public implementation in small mathsf prolog which is open source multiplatform portable and in memory featuring a graphical user interface a distinctive feature of this system is that unlike others we have formally demonstrated that our implementation techniques fit the proposed operational semantics we also study the efficiency of these implementation techniques through a series of detailed experiments moreover a database example for a recommender system is used to illustrate some of the features of the system and its usefulness

extensions of a resemblance relation abstract we consider an arbitrary crisp domain where a resemblance relation is defined and we are going to extend it to work with fuzzy values this will lead us to the concepts of weak and strong resemblance the extension may be carried out with or without the help of precision levels for the fuzzy values we show certain equivalence between both extensions and apply this study to the definition of fuzzy dependencies

a max plus based algorithm for a hamilton jacobi bellman equation of nonlinear filtering the hamilton jacobi bellman hjb equation associated with the robust hinfty filter as well as the mortensen filter is considered these filters employ a model where the disturbances have finite power the hjb equation for the filter information state is a first order equation with a term that is quadratic in the gradient yet the solution operator is linear in the max plus algebra this property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off line the precomputed solutions are stored as vectors of coefficients of the basis functions these coefficients are then used directly in the real time computations

eigenfactor solution of the matrix riccati equation a continuous square root algorithm this paper introduces a new algorithm for solving the matrix riccati equation differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix the solution of these equations yields then the time behavior of the eigenvalues and eigenvectors of the solution matrix a reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation this algorithm serves two purposes first being a square root solution it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy secondly it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors the algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples through these examples it is seen that the algorithm is indeed more accurate than the ordinary one moreover it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed form solution cannot be computed as a result of numerical difficulties

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

measuring properties of fuzzy sets a general technique and its use in fuzzy query evaluation abstract a fuzzy set can be viewed as a weighted max combination or as a convex combination of a family of nested ordinary sets the latter point of view allows the interpretation of a fuzzy set as a consonant random set then any numerical set function applied to a fuzzy set leads to a result which is a random number the expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set function this approach is very general and can be used for various kinds of measurements cardinality confidence measures attached to the occurrence of events diameter or perimeter of a fuzzy region distance between two sets etc also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed the weighted max combination view which leads to another type of scalar evaluation is also briefly discussed in this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property this kind of query can be managed by languages like sql in case of a crisp set of items when this set is fuzzily specified a scalar answer can still be provided as well as a fuzzy valued or interval valued one if we want to keep track of the effect of the fuzziness of the query the case where the data base contains precise information but also the more general case of fuzzy information data bases are considered

on a fuzzy bipolar relational algebra this paper presents an extension of relational algebra suitable for the handling of bipolar concepts the type of queries considered involves two parts a first one which expresses a possibly flexible constraint and a second one that corresponds to a possibly flexible wish the framework considered is that of bipolar fuzzy relations where each tuple is associated with a pair of satisfaction degrees

unfair coins and necessity measures towards a possibilistic interpretation of histograms this short note defines a bijective mapping which turns a probability measure into a possibility measure this transformation has intuitive grounds especially the converse mapping corresponds to the usual assignment of identical probability values to elementary events of a given set when no information is available

semantics of quotient operators in fuzzy relational databases abstract the quotient operator plays an important role for query evaluation in relational databases its extension to fuzzy databases raises the question of the intended interpretation of the intermediary degrees attached to the items in the fuzzy relations this note points out that the fuzzy quotient operator should be defined in slightly different ways depending on the possible interpretations of the degrees levels of satisfaction of a gradual property levels of importance of a required property uncertainty pertaining to the membership of an element to a subset

tolerant division queries and possibilistic database querying this article is situated in the framework of the relational model of data and it focuses on two quite orthogonal issues even if one may reasonably think that they can be mixed together the first topic is concerned with non traditional queries called flexible or fuzzy queries the key idea here is to introduce preferences inside queries addressed to a database thanks to fuzzy predicates so doing the user can express that some values are preferred to others but also that some criteria are more important than others this idea of preferences is especially illustrated with several extensions of the division operation the second question tackled is about the handling and querying of possibilistic databases basically a possibilistic database is similar to a regular one except that some attribute values are ill known and represented as possibility distributions in such a case it turns out that the set of queries that can be posed to the database with acceptable performances is significantly reduced with respect to the usual relational algebra however it is shown that diverse types of queries of interest against possibilistic databases can be envisaged

almost analytic representation for the solution of the differential matrix riccati equation an almost analytic representation is presented for the solution of the nonhomogeneous and homogeneous time invariant and time variant differential matrix riccati equation this representation gives a new insight into the behavior of the optimal estimator on a finite time interval for the time invariant case this solution is a basis for an exact true iterative algorithm for the time invariant case an approximate iterative algorithm is derived

fuzzy division in fuzzy relational databases an approach an approach to calculate the relational division in fuzzy databases is presented starting with the gefred model the tuples of the division are obtained with a possibility degree that indicates the extent to which they comply with the relational division requisites the results obtained are those expected a priori furthermore in order to formalize this method of fuzzy relational division two new operators are defined which may be useful in other operations the qualified fuzzy intersection and the generalized fuzzy projection with group functions some examples are included in order to clarify the method proposed here and the defined operators a comparison with other fuzzy division approaches that appear in the literature is included too

an extension of classical functional dependency dynamic fuzzy functional dependency abstract relational data model has constituted an incontestable success in database history in this context a lot of attention has been paid to functional dependencies due to their paramount importance in the design of relational database for about fifteen years several attempts to formalize soft real world constraints imposed on the data has been made leading to the emergence of the concept of fuzzy functional dependency in this paper an overview of the different proposals of the fuzzy functional dependency is presented a new extension of classical functional dependency based on the lukasiewicz implication is presented and called dynamic fuzzy functional dependency the associated axiomatic system is introduced and proved to be sound

fuzzy relational compositions abstract this paper investigates the fuzzy relational compositions defined by bandler and kohout it is indicated that these compositions have to be modified in order to capture the desired interpretation several alternative definitions are suggested and discussed the paper concludes with an overview of the properties of these improved fuzzy relational compositions

fuzzy functional dependencies and independencies in extended fuzzy relational database models abstract in this paper we have considered data dependencies more specifically restricted fuzzy functional dependencies in the framework of extended fuzzy relational database model a formal system is presented for the derivation of fuzzy functional dependencies which is both sound and complete the problem of lossless join decomposition is discussed in the framework of extended fuzzy relational model then we introduce a formal statement of fuzzy functional independency which expresses the fact that the corresponding restricted fuzzy functional dependency is not a legitimate constraint of the real world a formal system is presented for the derivation of restricted fuzzy functional dependencies and independencies the various problems such as the membership problem the contradiction problem and the problem of canonical represents are discussed a cover called minimal cover of restricted fuzzy functional independencies is shown to be unique

a flexible bipolar querying approach with imprecise data and guaranteed results in this paper we propose an approach to query a database when the user preferences are bipolar i e express both constraints and wishes about the desired result and the data stored in the database are imprecise results are then completely ordered with respect to these bipolar preferences giving priority to constraints over wishes additionally we propose a treatment that allows us to guarantee that any query will return a result even if no element satisfies all constraints specified by the user such a treatment may be useful when user s constraints are unrealistic i e cannot be all satisfied simultaneously and when the user desires a guaranteed result the approach is illustrated on a real world problem concerning the selection of optimal packaging for fresh fruits and vegetables

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

correlations of the von mangoldt and higher divisor functions i long shift ranges we show that the expected asymptotic for the sums x n 2x n n h x n 2x d k n d l n h and x n 2x n d k n h hold for almost all h h h provided that x8 33 e h x 1 e with an error term saving on average an arbitrary power of the logarithm over the trivial bound previous work of mikawa and baier browning marasingha zhao covered the range h x 1 3 e we also obtain an analogous result for n n n n our proof uses the circle method and some oscillatory integral estimates following a paper of zhan to reduce matters to establishing some mean value estimates for certain dirichlet polynomials associated to type d 3 and type d 4 sums as well as some other sums that are easier to treat after applying holder s inequality to the type d 3 sum one is left with two expressions one of which we can control using a short interval mean value theorem of jutila and the other we can control using exponential sum estimates of robert and sargos the type d 4 sum is treated similarly using the classical l2 mean value theorem and the classical van der corput exponential sum estimates

fuzzy information in extended fuzzy relational databases abstract the queries on type 2 fuzzy relational databases may get non reasonable answers and as pointed out by buckles a tuple s membership value is not static but is a measure of the appropriateness of that tuple to a given query here we use two supplementary attributes r and i e to model the concepts of fuzziness and uncertainty respectively we also use these two attribute values to compute the appropriateness of a tuple to a given query in this paper the proposed fuzzy relational model is defined by first order logical in our proposed model the classical properties of the database which are 1 database interpretation being redundant free being uniquely determined and 2 the well definedness of relational algebra are preserved for the fuzzy relational algebra therefore the proposed fuzzy relational model can capture more meaning than the conventional approaches and is more suitable to handle the real life application

flexible queries on relational databases using fuzzy logic and ontologies combines fuzzy logic and semantic techniques to perform flexible queries in a relational database alternative method to fuzzy logic to execute flexible queries on non scalar attributes answers are ordered according with the accomplishment degree to the query ontologies are used as an attribute domain frame to return semantically similar information to the query fuzzy logic are used on scalar attributes nowadays there are many proposals that allow users to perform fuzzy queries on relational databases regardless of these proposals fuzzy queries are really useful on scalar values where fuzzy sets can be adjusted to the user needs and domains but non scalar values are a more complex task here we extend non scalar attribute management in fuzzy queries with the use of ontologies thus we allow to compute this kind of queries not only with the similarity relationships defined explicitly on a fuzzy set but with semantically interrelated terms modeled as a domain ontology as well moreover we present the architecture of a novel system that combines both techniques to return an answer as much complete as possible and ordered by a degree of accomplishment finally a qualitative and quantitative study about the use of flexible queries on relational databases is included in this work as well

max plus representation for the fundamental solution of the time varying differential riccati equation using the tools of optimal control semiconvex duality and max plus algebra this work derives a unifying representation of the solution for the matrix differential riccati equation dre with time varying coefficients it is based upon a special case of the max plus fundamental solution first proposed in fleming and mceneaney 2000 such a fundamental solution can extend a particular solution of certain bivariate dres into the general solution and the dres can be analytically solved from any initial condition this paper also shows that under a fixed duality kernel the semiconvex dual of a dre solution satisfies another dual dre whose coefficients satisfy the matrix compatibility conditions involving hamiltonian and certain symplectic matrices for the time invariant dre this allows us to make dual dre linear and thereby solve the primal dre analytically this paper also derives various kernel duality relationships between the primal and time shifted dual dres which lead to an array of dre solution methods time invariant analogue of one of these methods was first proposed in mceneaney 2008

flexible querying in a relational framework supported by possibilistic logic traditionally flexible querying systems are supported by a logical framework that is based on satisfaction degrees alternatively certainty about query satisfaction can also be modelled by possibility and necessity measures this approach was extended by our group in order to handle the cases of missing information more adequately in this paper it is presented how our extended logical framework can be used to support flexible querying on conventional relational databases the handling of preferences both inside elementary query conditions and between query conditions is presented as well as the extension of the traditional selection projection and join operators motivated examples are provided to illustrate the used concepts

flexible queries in relational databases the example of the division operator abstract allowing for flexible queries enables database users to express preferences inside minimal requirements and if necessary priorities inside compound queries in other words clear cut properties can be refined by ordering the interpretations compatible with them according to user s preferences often the representation of these preferences can be viewed as modelling linguistic like terms in requests in this paper the theoretical issues raised by the introduction of flexible queries are studied in the case of the division operator in the framework of fuzzy sets and possibility theory the notion of division is well known in the context of regular relations and the extension of this operation to fuzzy relations induced by the flexible queries is investigated several types of extended divisions can be envisaged depending on the meaning of the grades attached to the tuples of the fuzzy relations degree of fulfillment of gradual properties level of importance of components in a query or uncertainty pervading data we focus on the first two meanings which are associated with different multiple valued logic implications and we examine their properties and their expression in the framework of an extended sql like language where no specific construct for the division is available

incomplete conjunctive information abstract many information systems capable of handling incomplete or fuzzy information manipulate objects with single valued attributes information is then said to be disjunctive information is said to be conjunctive when pertaining to many valued attributes while a piece of incomplete disjunctive information is easily represented by means of a set of mutually exclusive possible values modeling incomplete conjunctive information theoretically leads to consider families of sets since attributes are then set valued under complete information some proposals are made in order to efficiently and rigorously represent incomplete conjunctive information and deal with query evaluation especially in the case where only upper and or lower bounds of the set of values of a many valued attribute are known applications of this approach can be expected for the processing of time intervals as well as spatial reasoning among other topics in knowledge base management

a new fundamental solution for differential riccati equations arising in control the matrix differential riccati equation dre is ubiquitous in control and systems theory the presence of the quadratic term implies that a simple linear systems fundamental solution does not exist of course it is well known that the bernoulli substitution may be applied to obtain a linear system of doubled size here however tools from max plus analysis and semiconvex duality are brought to bear on the dre we consider the dre as a finite dimensional solution to a deterministic linear quadratic control problem taking the semiconvex dual of the associated semigroup one obtains the solution operator as a max plus integral operator with quadratic kernel the kernel is equivalently represented as a matrix using the semigroup property of the dual operator one obtains a matrix operation whereby the kernel matrix propagates as a semigroup the propagation forward is through some simple matrix operations this time indexed family of matrices forms a new fundamental solution for the dre solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition in analogy with standard algebra linear systems the fundamental solution can be viewed as an exponential form over a certain idempotent semiring this fundamental solution has a particularly nice control interpretation and might lead to improved dre solution speeds

multivalued dependencies in fuzzy relational databases and lossless join decomposition abstract the fuzzy multivalued dependency is defined on the lines of classical multivalued dependency to resolve the redundancy and inconsistency in fuzzy databases the implication problem of fmvd s has been examined and a set of sound and complete inference axioms has been proposed we next examine lossless join problem of fuzzy relations in the presence of fuzzy functional depandencies and fuzzy multivalued dependencies and obtain decomposition of relation schemes

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

twofold fuzzy sets and rough sets some issues in knowledge representation abstract this paper deals with the representation of sets where the membership of some elements may be ill known rather than just a matter of degree as in a fuzzy set the notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete a twofold fuzzy set is made of a nested pair of fuzzy sets the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by pawlak set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined twofold fuzzy relations are also introduced finally various applications of twofold fuzzy sets in knowledge representation are briefly discussed

non transitive fuzzy dependencies i in this work we deal with the problem of defining projection and join operators in a fuzzy database so that they allow us to compress the information given in a relation r of such a database such a compression is possible when a relation satisfies one kind of fuzzy functional dependency f f d such projection and join operators allow us to decompose the original relation r into two new relations in such a way that the amount of information to store is less than originally with r in this process of decomposition there are two main properties to be achieved on the one hand we must be able to test the fuzzy dependency just by looking at one of the projections as it has less records than r this represents a computational advantage on the other hand the join of both projections is such that we can recover the information we had in r but with fuzzier values in order to measure the degree of increasing fuzziness the concept of granularity level is used the definition of fuzzy dependency does not verify the transitivity inference axiom

gefred a generalized model of fuzzy relational databases abstract in this paper we present a fuzzy relational databases model whose main characteristics are the integration of previous models in the same framework representation capabilities for a wide series of fuzzy information and a coherent and flexible handling of it this model aims to solve each problem of representation and handling of fuzzy information taking into account its specific nature and hence it allows the user to choose the comparison operator and the fuzzy compatibility measure to be used in a query besides it permits the user to specify the precision with which the conditions involved in a query are satisfied

efficient refinement of uncertain data by fuzzy integrity constraints abstract fuzzy data models were proposed to handle uncertain data based on a set of fuzzy functional dependencies the fuzzy values in a fuzzy relation may be refined into clearer ones in this paper we develop efficient algorithms for the refinement process a dependency graph is defined for a set of fuzzy functional dependencies by this graph an efficient refinement sequence can be determined these algorithms are proved correct and efficient

on the impact of regular functional dependencies when moving to a possibilistic database framework this paper is situated in the area of fuzzy databases i e databases containing imprecise information more precisely it deals with the notion of a functional dependency when data involved in this type of property are possibly imprecise contrary to previous works the idea is not to fuzzify the concept of a functional dependency in the view suggested here we rather consider regular functional dependencies and we study the impact of the presence of such fds on the insertion and handling of imprecise data

equivalence and transformation of extended algebraic operators in fuzzy relational databases relational algebra serves as the theoretical basis for database design and queries this paper extends our efforts on fuzzy relational algebraic operations under a fuzzy relational data model namely extended possibility based model with a more general setting of fuzzy relational algebraic operators some properties of the operators are investigated including equivalent definitions and transformation rules of the operators these properties are important and desirable for processing queries in an effective and efficient manner

upper and lower images of a fuzzy set induced by a fuzzy relation applications to fuzzy inference and diagnosis in fuzzy set theory the image of a fuzzy set induced by a fuzzy relation is usually obtained by a sup t norm composition this corresponds to the upper image which gathers the elements in relation with at least one element of the fuzzy set however the dual point of view leading to the definition of the lower image as the fuzzy set of elements in relation with all the elements of the fuzzy set whose image is computed is not often considered fuzzy sets and fuzzy relations depending on the situations can be interpreted either in a conjunctive manner or as subsets of mutually exclusive possible values for variables whose precise values are ill known disjunctive view the applications of upper and lower images are investigated in both interpretations the generalized modus ponens used in fuzzy rule based systems corresponds to the disjunctive view the interest of upper and lower images is also emphasized for diagnosis problems where the conjunctive interpretation is encountered

a domain calculus for fuzzy relational databases abstract a complete relational calculus is introduced for similarity based fuzzy relational databases the ordinary domain calculus is chosen as the basis from which to develop the fuzzy counterpart because it tends to be closer to natural language than tuple calculus a constructive proof is used to show that the fuzzy relational algebra can be mapped into the calculus the proof is significant from the standpoint of demonstrating implementability of the calculus as a query language

some properties of extended fuzzy preference relations using modalities abstract a fuzzy preference relation is extended to six kinds of relations between fuzzy sets the properties of the six extended relations are investigated especially a sufficient condition for reducing the extended relations to four kinds is discussed when fuzzy sets are bounded and normal and some examples are given to illustrate it further a sufficient condition for the extended negative relations to coincide with the extended inverse relations is described

an initial approach to the evaluation of possibilistic queries addressed to possibilistic databases abstract it is now recognized that querying databases containing imprecise information raises several problems including that of the computational complexity of evaluating the query in this paper we consider a new kind of query called possibilistic query of the form to what extent is it possible that a given tuple t belongs to the answer of q where q is a regular relational query the aim is to show that a reasonable complexity can be expected for a specified subset of possibilistic queries