triangular norms position paper ii general constructions and parameterized families this second part out of three of a series of position papers on triangular norms for part i see triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press deals with general construction methods based on additive and multiplicative generators and on ordinal sums also included are some constructions leading to non continuous t norms and a presentation of some distinguished families of t norms

yager s new class of implications jf and some classical tautologies recently yager r yager on some new classes of implication operators and their role in approximate reasoning information sciences 167 2004 193 216 has introduced a new class of fuzzy implications denoted j f called the f generated implications and has discussed some of their desirable properties such as neutrality exchange principle etc in this work we discuss the class of j f implications with respect to three classical logic tautologies viz distributivity law of importation and contrapositive symmetry necessary and sufficient conditions under which j f implications are distributive over t norms and t conorms and satisfy the law of importation with respect to a t norm have been presented since the natural negations of j f implications given by n j f x j f x 0 in general are not strong we give sufficient conditions under which they become strong and possess contrapositive symmetry with respect to their natural negations when the natural negations of j f are not strong we discuss the contrapositivisation of j f along the lines of j f implications a new class of implications called h generated implications j h has been proposed and the interplay between these two types of implications has been discussed notably it is shown that while the natural negations of j f are non filling those of j h are non vanishing properties which determine the compatibility of a contrapositivisation technique

a refinement of matrosov s theorem for differential inclusions this note presents a refinement of matrosov s theorem for a class of differential inclusions whose set valued map is defined as a closed convex hull of finitely many vector fields this class of systems may arise in the analysis of switched nonlinear systems when stability with arbitrary switching between the given vector fields is considered assuming uniform global stability of a compact set it is shown that uniform global attractivity of the set can be verified by tailoring matrosov functions to individual vector fields this refinement of matrosov s theorem is an extension of the existing matrosov results which may be easier to apply to certain differential inclusions than existing results as demonstrated by an example

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

triangular norms position paper iii continuous t norms this third and last part of a series of position papers on triangular norms for parts i and ii see e p klement r mesiar e pap triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press e p klement r mesiar e pap triangular norms position paper ii general constructions and parameterized families submitted for publication presents the representation of continuous archimedean t norms by means of additive generators and the representation of continuous t norms by means of ordinal sums with archimedean summands both with full proofs finally some consequences of these representation theorems in the context of comparison and convergence of continuous t norms and of the determination of continuous t norms by their diagonal sections are mentioned

a nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems a new infinitesimal sufficient condition is given for uniform global asymptotic stability ugas for time varying nonlinear systems it is used to show that a certain relaxed persistency of excitation condition called uniform spl delta persistency of excitation u spl delta pe is sufficient for uniform global asymptotic stability in certain situations u spl delta pe of the right hand side of a time varying differential equation is also shown to be necessary under a uniform lipschitz condition the infinitesimal sufficient condition for ugas involves the inner products of the flow field with the gradients of a finite number of possibly sign indefinite locally lipschitz lyapunov like functions these inner products are supposed to be bounded by functions that have a certain nested or triangular negative semidefinite structure this idea is reminiscent of a previous idea of matrosov who supplemented a lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the lyapunov function is zero for this reason we call the main result a nested matrosov theorem the utility of our results on stability analysis is illustrated through the well known case study of the nonholonomic integrator

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

fuzzy reasoning and fuzzy relational equations l a zadeh first suggested a method of fuzzy reasoning called compositional rule of inference cri for short instead of sup min composition we use a more general composition called sup t composition in cri according to the criteria gmp and gmt the fuzzy implication r must satisfy the sup t composite fuzzy relational equations email protected r s and email protected r 1 q in this paper we discuss the solution of these equations and point out that email protected composition and the implication r n a used to cri will be more suitable under criteria gmp and gmt

compositional rule of inference as an analogical scheme we discuss fuzzy deductive reasoning achieved through the compositional rule of inference from the analogy point of view merging it into the analogical scheme necessary and sufficient conditions on operators involved are investigated to fit the compositional rule of inference into the analogical scheme several examples including well known models are included

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

mutually exchangeable fuzzy implications recently vemuri and jayaram 2012 have proposed a novel generating method of fuzzy implications called the composition further they have also proposed mutual exchangeability me a generalization of the exchange principle ep to a pair of fuzzy implications and have shown that me plays a central role in the preservation of basic properties functional equations and families of fuzzy implications w r t the composition vemuri and jayaram 2014 due to the important role played by me in this work we investigate the pairs i j of fuzzy implications that satisfy me towards this we show first that there exist pairs i j of fuzzy implications that satisfy me and determine some necessary conditions on such fuzzy implications following this for a given i i we find the set j i of all fuzzy implications j such that the pair i j satisfies me keeping in view the variety of fuzzy implications and the complexity of the functional equation me we restrict our investigations to four important families of fuzzy implications namely s n r f and g implications further we discuss a generalization of the cauchy multiplicative equation whose solutions help us in obtaining the set j i for some families of fuzzy implications

on the law of importation x wedge y longrightarrow z equiv x longrightarrow y longrightarrow z in fuzzy logic the law of importation given by the equivalence x lambda y rarr z equiv xrarr y rarr z is a tautology in classical logic in a implications defined by turksen et al the above equivalence is taken as an axiom in this paper we investigate the general form of the law of importation j t x y z j x j y z where t is a t norm and j is a fuzzy implication for the three main classes of fuzzy implications i e r s and ql implications and also for the recently proposed yager s classes of fuzzy implications i e f and g implications we give necessary and sufficient conditions under which the law of importation holds for r s f and g implications in the case of ql implications we investigate some specific families of ql implications also we investigate the general form of the law of importation in the more general setting of uninorms and t operators for the above classes of fuzzy implications following this we propose a novel modified scheme of compositional rule of inference cri inferencing called the hierarchical cri which has some advantages over the classical cri following this we give some sufficient conditions on the operators employed under which the inference obtained from the classical cri and the hierarchical cri become identical highlighting the significant role played by the law of importation

on locally internal monotonic operations this paper deals with monotonic binary operations f 0 1 2 0 1 with the property called locally internal property that the value at any point x y is always one of its arguments x y after stating a theorem that characterizes this kind of operations some special cases are studied in detail by considering additional properties of the operation commutativity existence of a neutral element and associativity in case of locally internal associative monotonic operations with neutral element a characterization theorem gives an improvement of a well known theorem of czogala and drewniak on idempotent associative and increasing operations with neutral element as well as an improvement of a characterization theorem for left and right continuous idempotent uninorms

nested matrosov function theorem for nonlinear delayed systems abstract in this paper the stability problem for time varying nonlinear delayed systems is investigated an infinite dimensional version of the nested matrosov function theorem is proposed to conclude uniform global asymptotic stability for time varying nonlinear delayed systems such a stability theorem requires a weak lyapunov functional and some auxiliary functionals and the upper bounds of the derivatives on these functionals must satisfy some nested conditions the proposed results can be used to analyze uniform asymptotic stability of time varying delayed systems where lasalle s invariance principle cannot be applied and barbalat s lemma is not sufficient to conclude the uniform attractivity property an example is given to illustrate the effectiveness of the proposed stability theorem

on computation of the compositional rule of inference under triangular norms abstract this paper is devoted to the derivation of exact calculation formulas for the compositional rule of inference under archimedean t norms when both the observation and the relation parts are given by hellendoorn s function 6

fuzzy controllers with conditionally firing rules mamdani 1975 controller was successfully used in many applications one of its interpretations is that it uses a fuzzy relation as an approximation of the desirable input output correspondence we analyze mathematical properties of mamdani controller and notice that it has lower computational complexity when compared to the residuum based controller however we show that in standard situations both these fuzzy controllers do not represent the rule base properly in the sense of finding a solution to the related system of fuzzy relational equations first we consider the premises and consequents as typical inputs and outputs and we want their correspondence to be kept next we require that each normal input produces an output that bears nontrivial information these two conditions appear to be almost contradictory to the previous controllers we suggest a generalization of mamdani controller which allows us to satisfy these requirements the theory and experiments suggest that it performs better without any change of rule base and without a substantial increase of complexity

a recurrence principle for stochastic difference inclusions the invariance principle is extended to a recurrence principle and is developed for stochastic difference inclusions for these systems random solutions are not unique under appropriate lyapunov like conditions it is established for every random solution that almost every complete sample path converges to the largest weakly totally recurrent set contained in a level set of the lyapunov like function such a set is not larger and is sometimes smaller than the largest weakly invariant set contained in the level set the principle is useful for establishing robust uniform asymptotic stability in probability or robust uniform strong recurrence under weak lyapunov conditions for stochastic discrete time control systems that employ discontinuous feedback laws examples demonstrate the achieved results

asymptotic stability in hybrid systems via nested matrosov functions a theorem on nested matrosov functions is extended to time varying hybrid systems it provides sufficient conditions for uniform global asymptotic stability of a compact set an application to parameter identification with state resets is made and illustrated on an example

similarity and dissimilarity measures between fuzzy sets a formal relational study the paper deals with the well known notion of dis similarity measures between fuzzy sets we provide three separate lists of axioms that fit with the respective notions of general comparison measure similarity measure and dissimilarity measure then we review some of the most important axiomatic definitions of dis similarity measures in the literature by referring to the axioms in those lists satisfied by each specific definition this common framework will make our study about the formal relationships among different axiomatic definitions much easier some of them which are apparently different do in fact share many commonalities we provide a self contained picture of these relationships by providing formal results and counterexamples that reflect which of the dis similarity definitions in the literature are connected by implication relations and which of them are not we finalize the paper with an in depth study about the notion of duality between similarity and dissimilarity measures as well as with some concluding remarks

roughness bounds in rough set operations this paper presents some roughness bounds for rough set operations the results show that a bound of the set operation can be determined from their operand s roughnesses we prove also that this bound is not determined under some special operations

comments on combinatorial rule explosion eliminated by a fuzzy rule configuration with reply in the original paper ieee trans fuzzy syst vol 6 p 1 11 1998 combs and andrews proved the following logical equivalence stated here for two antecedents p and q and one consequent r but easily generalize to an arbitrary number of antecedents and consequents p spl and q spl rarr r spl harr p spl rarr r v q spl rarr r this is a very significant result because it suggests that we can replace multi antecedent rules with an interconnection of single antecedent rules which eliminates the rule explosion that is associated with multi antecedent rules combs and andrews refer to the left hand side of this equivalence as an intersection rule configuration irc and to its right hand side as a union rule configuration urc their result gives rise to two distinctly different paths for the design of fuzzy logic systems irc which leads to rule explosion and urc which does not the authors discuss four points about the irc spl harr urc relation the original authors reply acknowledging some of the points and stating that they would present their results differently if starting now

measuring roughness of generalized rough sets induced by a covering in this paper we propose new lower and upper approximations and obtain some important properties in generalized rough set induced by a covering especially these properties are compared with ones of pawlak s rough sets and bonikowski s covering generalized rough sets respectively moreover we define a measure of roughness based on generalized rough sets with the new approximations and discuss some significant properties of the measure

rough approximation quality revisited in rough set theory the approximation quality is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes in this paper we re interpret the classical in terms of a classic measure based on sets the marczewski steinhaus metric and also in terms of proportional reduction of errors pre measures we also exhibit infinitely many possibilities to define like statistics which are meaningful in situations different from the classical one and provide tools to ascertain the statistical significance of the proposed measures which are valid for any kind of sample copyright 2001 elsevier science b v

a converse lyapunov theorem and robustness for asymptotic stability in probability a converse lyapunov theorem is established for discrete time stochastic systems with non unique solutions in particular it is shown that global asymptotic stability in probability implies the existence of a continuous lyapunov function smooth outside of the attractor that decreases in expected value along solutions the keys to this result are mild regularity conditions imposed on the set valued mapping that characterizes the update of the system state and the ensuing robustness of global asymptotic stability in probability to sufficiently small state dependent perturbations

a converse lyapunov theorem for strong global recurrence abstract a converse lyapunov theorem is established for discrete time stochastic systems modeled by a set valued mapping under mild regularity conditions for this class of systems it is shown that strong global recurrence is a necessary and sufficient condition for the existence of a smooth lyapunov function that decreases in expected value along solutions outside of the set that is strongly globally recurrent robustness of strong global recurrence to sufficiently small perturbations is also established

a matrosov theorem for adversarial markov decision processes matrosov s relaxation of lyapunov conditions for uniform global asymptotic stability in time varying systems is extended to stochastic set valued discrete time systems nested matrosov functions are used to give conditions for stability that complement invariance principles for time invariant systems unlike invariance principles matrosov functions also can be applied to general time varying systems

a lyapunov theorem certifying global weak reachability for stochastic difference inclusions with random inputs abstract in this paper a lyapunov condition certifying global weak reachability of a compact set is given for stochastic set valued discrete time systems with respect to such systems under mild regularity assumptions it is shown that the existence of a lower semicontinuous function that decreases in expected value for at least one measurable selection from the set valued mapping outside a compact set is a sufficient condition for global weak reachability of such a set

monotonicity of siso fuzzy relational inference with an implicative rule base a fuzzy relational inference fri mechanism is appraised based on the different desirable properties it possesses among these properties monotonicity of an fri has not received much attention in this work we investigate the monotonicity of a single input single output fri with an implicative form of the rule base in all the previous works that deal with monotonicity of an fri with the implicative form of rule base the employed fuzzy implications come from a residuated lattice it can be noticed that this rich underlying structure plays a major role in proving the results further we also modify the given monotone rule base this work differs from the previous works in that 1 the fuzzy implications employed in it do not come from any known residuated structure on text 0 1 and 2 the original rule base is employed without any alteration we determine conditions under which monotonicity of an fri where the rule base is modeled by a strict fuzzy implication can be ensured without transforming the original rule base thus the results in this work further augment the case for considering fuzzy implications other than those from the residuated setting to be used in applications

the notion of roughness of a fuzzy set abstract the roughness of a set according to the notion introduced by pawlak in 1991 can be regarded as the mz distance between its upper and the lower approximations with this idea in mind we have generalized pawlak s definition by replacing the mz distance by a general distance measure we also generalize the notion of roughness of fuzzy sets introduced by huyhn and nakamori in 2005

asymptotic stability and smooth lyapunov functions abstract we establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth lyapunov function since well known concepts of generalized solutions of differential equations with discontinuous right hand side can be described in terms of solutions of certain related differential inclusions involving upper semicontinuous multifunctions this result gives a lyapunov characterization of asymptotic stability of either filippov or krasovskii solutions for differential equations with discontinuous right hand side in the study of weak as opposed to strong asymptotic stability the existence of a smooth lyapunov function is rather exceptional however the methods employed in treating the strong case of asymptotic stability are applied to yield a necessary condition for the existence of a smooth lyapunov function for weakly asymptotically stable differential inclusions this is an extension to the context of lyapunov functons of brockett s celebrated covering condition from continuous feedback stabilization theory

stability theorems for delay differential inclusions this technical note addresses stability problems of delay differential inclusions a chain rule is proposed for a wide class of lyapunov krasovskii functionals which are not necessarily invariantly differentiable we also obtain an invariance like theorem where the derivative of the candidate functional is bounded above by a continuous negative semidefinite function several examples are given to show the effectiveness of the results

equivalent characterizations of input to state stability for stochastic discrete time systems input to state stability iss for stochastic difference inclusions is studied first iss in probability relative to a compact set is defined subsequently several equivalent characterizations are given for example iss in probability is shown to be equivalent to global asymptotic stability in probability when the disturbance takes values in a ball whose radius is determined by a sufficiently small but unbounded function of the distance of the state to the compact set in turn a recent converse lyapunov theorem for global asymptotic stability in probability provides an equivalent lyapunov characterization finally robust iss in probability is defined and is shown to give another equivalent characterization

discrete time stochastic control systems a continuous lyapunov function implies robustness to strictly causal perturbations abstract discrete time stochastic systems employing possibly discontinuous state feedback control laws are addressed allowing discontinuous feedbacks is fundamental for stochastic systems regulated for instance by optimization based control laws we introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations under basic regularity conditions the existence of a continuous stochastic lyapunov function is sufficient to establish that asymptotic stability in probability for the closed loop system is robust to sufficiently small state dependent strictly causal worst case perturbations robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state dependent perturbations and in a semiglobal practical sense in the case of persistent perturbations an example shows that a continuous stochastic lyapunov function is not sufficient for robustness to arbitrarily small worst case disturbances that are not strictly causal our positive results are also illustrated by examples

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

matrosov theorem for parameterized families of discrete time systems a version of matrosov s theorem for parameterized discrete time time varying systems is presented the theorem is a discrete time version of the continuous time result in loria et al 2002 d persistency of excitation a necessary and sufficient condition for uniform attractivity 2002 submitted for publication our result facilitates controller design for sampled data nonlinear systems via their approximate discrete time models an application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete time plant models is presented

a matrosov theorem for strong global recurrence a theorem on nested matrosov functions is presented for time varying stochastic set valued discrete time systems satisfying mild regularity conditions it establishes sufficient conditions for uniform strong global recurrence of an open bounded set in general matrosov functions are required to satisfy less rigid requirements than typical lyapunov functions that satisfy a strict decrease condition along trajectories

representations through a monoid on the set of fuzzy implications fuzzy implications are one of the most important fuzzy logic connectives in this work we conduct a systematic algebraic study on the set ii of all fuzzy implications to this end we propose a binary operation denoted by which makes i i a non idempotent monoid while this operation does not give a group structure we determine the largest subgroup ss of this monoid and using its representation define a group action of ss that partitions ii into equivalence classes based on these equivalence classes we obtain a hitherto unknown representations of the two main families of fuzzy implications viz the f and g implications

two new operators in rough set theory with applications to fuzzy sets in this paper two new operators are introduced for the rough set theory using them two inequalities well known in the rough set theory can now be modified to become equalities with this change no information will be lost in the new expressions hence many properties in rough set theory can be improved and in particular the union the intersection and the complement operations can be redefined based on the two equalities furthermore the collection of rough sets of an approximation space forms a boolean algebra under these new operators finally roughness properties of fuzzy sets are analyzed using the new operations

a roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

on the convex combination of td and continuous triangular norms the problem of when t l 1 l t d lt l 0 1 is a triangular norm where t d is the drastic product and t is a continuous triangular norm is studied it is shown that t l cannot be a triangular norm when t is nilpotent it is also shown that t l is a triangular norm if t is strict and its additive generator f satisfies f lx f x f l for all x 0 1 the cases that t t m and t is the ordinal sum of continuous archimedean summands are also discussed some left continuous t norms which can be combined with each other are given

on the use of words and fuzzy sets this paper only tries to stimulate some reflections by showing a possible way towards rethinking fuzzy sets from their roots a rethinking that does not change the view that fuzzy sets are mathematical entities giving extension to predicates in such a way that if the predicates are precise the corresponding new entities are nothing else than classical sets what perhaps is a key idea is that the use of a predicate organizes in some way the universe of discourse when this organization is a preorder the extension or l set appears once a degree numerical or not but consistent with the organization can be defined the ultimate goal of those above mentioned reflections provided they would be realized in the future is to extend the current theories of fuzzy sets to wider areas of both language and reasoning our objective is to reach a better knowledge of the links between language and its representations for the progress of computing with words and perceptions