interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

generating an interpretable family of fuzzy partitions from data in this paper we propose a new method to construct fuzzy partitions from data the procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets the maximum size n is determined according to the data distribution and corresponds to the finest resolution level we use an ascending method for which a merging criterion is needed this criterion is based on the definition of a special metric distance suitable for fuzzy partitioning and the merging is done under semantic constraints the distance we define does not handle the point coordinates but directly their membership degrees to the fuzzy sets of the partition this leads to the introduction of the notions of internal and external distances the hierarchical fuzzy partitioning is carried independently over each dimension and to demonstrate the partition potential they are used to build fuzzy inference system using a simple selection mechanism due to the merging technique all the fuzzy sets in the various partitions are interpretable as linguistic labels the tradeoff between accuracy and interpretability constitutes the most promising aspect in our approach well known data sets are investigated and the results are compared with those obtained by other authors using different techniques the method is also applied to real world agricultural data the results are analyzed and weighed against those achieved by other methods such as fuzzy clustering or discriminant analysis

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

oscillation theorems for nth order functional differential equations abstract two oscillation theorems for even order equations x n t p t x t x h t 0 are established these results are the extensions of those reported by hamedani for the second order equation x t p t x t x h t 0

linguistic summarization of time series using a fuzzy quantifier driven aggregation we propose new types of linguistic summaries of time series data that extend those proposed in our previous papers the proposed summaries of time series refer to the summaries of trends identified here with straight line segments of a piecewise linear approximation of time series we first show how to construct such an approximation then we employ a set of features attributes to characterize the trends such as the slope of the line segment the goodness of approximation and the length of the trend the derivation of a linguistic summary of a time series is then related to a linguistic quantifier driven aggregation of trends for this purpose we employ the classic zadeh s calculus of linguistically quantified propositions but extending our previous works with different t norms in addition to the basic minimum we show an application to the analysis of time series data on daily quotations of an investment fund over an eight year period present some interesting linguistic summaries obtained and show results for different t norms the results are very promising

extended tsukamoto s inference method for solving multi objective linguistic optimization problems abstract multi objective linguistic optimization problems molops are quite useful in modeling a number of real life problems the decision variables used to define the problem constraints and the objective functions take linguistic values in molops in those molops where the values of objective functions are unknown at all the points of the variable space the link between the values of decision variables and objective functions is defined using if then rules molops formulated in this way were often solved using the tsukamoto s inference method however the tsukamoto s method can solve molops if the consequents in the if then rules are defined as monotonic functions also the solution obtained from tsukamoto s method is a precise number so in the present work we propose a novel extension of tsukamoto s inference method the extended tsukamoto s inference method can overcome both the above limitations that is it can solve molops where the consequents are defined as non monotonic functions as well as monotonic functions also the extended tsukamoto s method generates intervals instead of unique numbers in the solution from these intervals we then generate type 1 fuzzy set membership functions and compare them using two of their fuzzy set uncertainty measures viz centroid and fuzziness furthermore we generate linguistic recommendations for these generated type 1 fuzzy set membership functions as humans understand linguistic information naturally we have also demonstrated the applicability of the proposed extended tsukamoto s inference method to the case study for student s performance evaluation we strongly believe that our proposed method is a novel one and will provide directions to future research

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

oscillation criteria for second order functional differential equations in this paper oscillation criteria for general second order functional differential equations are given integral criterion for oscillation using recent nth primitive method introduced by cheh chih yeh is considered

softer optimization and control models via fuzzy linguistic quantifiers a softening of a basic formulation of multicriterion optimization and control multistage decision making is presented for optimization instead of seeking an optimal solution that best satisfies all the fuzzy objectives as has been done so far we seek an optimal solution that best satisfies most much more than 50 etc a linguistic quantifier in general of the fuzzy objectives for control we seek in turn an optimal sequence of controls that best satisfies the fuzzy constraints and fuzzy goals at most much more than 50 etc of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used some applications to softer evidence aggregation in expert systems are also indicated

on oscillation of second order neutral type delay differential equations oscillation criteria are obtained by using the so called h method for the second order neutral type delay differential equations of the form r t j x t z t q t f x s t 0 t t 0 where z t x t p t x t t r p q t s c t 0 r and f email protected c r r the results of the paper contains several results obtained previously as special cases furthermore we are also able to fix an error in a recent paper related to the oscillation of second order nonneutral delay differential equations

a distance metric for a space of linguistic summaries producing linguistic summaries of large databases or temporal sequences of measurements is an endeavor that is receiving increasing attention these summaries can be used in a continuous monitoring situation like eldercare where it is important to ascertain if the current summaries represent an abnormal condition it is therefore necessary to compute the distance between summaries as a basis for such a determination in this paper we propose a dissimilarity measure between summaries based on fuzzy protoforms and prove that this measure is a metric we take into account not only the linguistic meaning of the summaries but also two quality evaluations namely the truth values and the degrees of focus we present examples of how the distance metric behaves and show that it corresponds with intuition

recursively defined metric spaces without contraction in this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1 bounded complete metric spaces and nonexpansive functions in contrast to previous approaches we do not assume that the endofunctors are locally contractive and our results do not depend on banach s fixed point theorem our approach is particularly suitable for constructing models of systems that feature quantitative data for instance using the kantorovich metric on probability measures we construct a quantitative model for probabilistic transition systems the metric in our model can reasonably be seen as measuring the behavioural distance between states of the system it depends exclusively on the transition probabilities and not on an arbitrary discount factor

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

terminal coalgebras in well founded set theory abstract this paper shows that in order to obtain the theorem of aczel and mendler on the existence of terminal coalgebras for an endofunctor on the category of sets it is entirely unnecessary to delve into such exotica as non well founded set theory in addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology by way of example we calculate the terminal coalgebra for various simple endofunctors

a framework for fuzzy quantification models analysis a framework for description of fuzzy quantification models is presented within this framework the fuzzy quantified statements evaluation problem is described as the compatibility between the fuzzy quantifier and a fuzzy cardinality or a fuzzy aggregation measure a list of desirable properties for quantification models is presented and those models that fit the framework are confronted with it

terminal metric spaces of finitely branching and image finite linear processes well known metric spaces for modelling finitely branching and image finite systems are shown to be the carrier of terminal coalgebras

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

interpretability of fuzzy linguistic summaries this paper investigates the question of the interpretability of fuzzy linguistic summaries both at the sentence level and at the summary level seen as a set of sentences the individual sentence interpretability is examined as depending both on its representativity measured by a quality degree and on its linguistic expression different properties at the summary level are also discussed namely their consistency their non redundancy and the information they convey

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

oscillation and nonoscillation criteria for second order nonlinear differential equations i abstract a second order nonlinear differential equation is considered oscillation and nonoscillation criteria are given