the quintuple implication principle of fuzzy reasoning we propose the quintuple implication principle qip for fuzzy reasoning we applies qip to solve fmp and fmt for residual implications the mamdani type fuzzy reasoning is logical interpreted from the viewpoint of qip the qip is applied to multiple fuzzy rules fuzzy modus ponens fmp and fuzzy modus tollens fmt are two fundamental patterns of approximate reasoning suppose a and b are fuzzy predicates and if a then b is a fuzzy rule approximate reasoning often requires to derive an approximation b of b from a given approximation a of a or vice versa to solve these problems zadeh introduces the well known compositional rule of inference cri which models fuzzy rule by implication and computes b a resp by composing a b resp with a b wang argues that the use of the compositional operation is logically not sufficiently justified and proposes the triple implication principle tip instead both cri and tip do not explicitly use the closeness of a and a or that of b and b in the process of calculating the consequence which makes the thus computed approximation sometimes useless or misleading in this paper we propose the quintuple implication principle qip for fuzzy reasoning which characterizes the approximation b of b a of a resp as the formula which is best supported by a b a a and a a b b b and b resp based upon monoidal t norm logic mtl this paper applies qip to solve fmp and fmt for four important implications most importantly we show that qip when using godel implication computes exactly the same approximation as mamdani type fuzzy inference does this is surprising as mamdani interprets fuzzy rules in terms of the minimum operation while cri tip and qip all interpret fuzzy rules in terms of implication

some kinds of view the fuzzy filters of bl algebras the concepts of view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras are introduced and some related properties are investigated some characterizations of these generalized fuzzy filters are derived in particular we describe the relationships among ordinary fuzzy implicative positive implicative and fantastic filters view the mathml source fuzzy implicative positive implicative and fantastic filters and view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras finally we prove that a fuzzy set f of a bl algebra l is an view the mathml source fuzzy implicative filter of l if and only if it is both an view the mathml source fuzzy positive implicative filter and an view the mathml source fuzzy fantastic filter

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy equation

interpretations of various uncertainty theories using models of modal logic a summary abstract this paper summarizes our efforts to establish the usual semantics of propositional modal logic as a unifying framework for various uncertainty theories interpretations for fuzzy set theory dempster shafer theory probability theory and possibility theory are discussed some properties of these interpretations are also presented as well as directions for future research

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

on the combination and normalization of interval valued belief structures this paper investigates the issues of combination and normalization of interval valued belief structures within the framework of dempster shafer theory dst of evidence existing approaches are reviewed examined and critically analysed they either ignore the normalization or separate it from the combination process leading to irrational or suboptimal interval valued belief structures a new logically correct optimality approach is developed where the combination and the normalization are optimised together rather than separately numerical examples are provided throughout the paper

quasiperiodic solutions for dissipative boussinesq systems in this paper we analyze the behavior of the solution of the dissipative boussinesq systems partial t u partial x v alpha partial xxx v beta partial xxt u partial x uv partial t v partial x u c partial xxx u beta partial xxt v v partial x v where c 0 are parameters those systems model two dimensional small amplitude long wavelength water waves for 1 this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every c and almost every 1 it admits solutions that are quasiperiodic in time the proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the boussinesq system to the center manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem and studying the birkhoff normal form

new types of fuzzy filters of bl algebras this paper introduces the concepts of a b fuzzy implicative positive implicative fantastic filters and b a fuzzy implicative positive implicative fantastic filters of bl algebras where a b c q d c q d c q d a b c q d c q d c q d a c q d and b c q d and some related properties are investigated special attention is paid to c c q d fuzzy implicative positive implicative fantastic filters and c c q d fuzzy implicative positive implicative fantastic filters which are generalizations of q fuzzy implicative positive implicative fantastic filters and q fuzzy implicative positive implicative fantastic filters studied in ma et al 2008 2009 34 35 it also points out that examples 3 1 2 3 1 9 3 2 2 3 2 9 and 3 3 9 in ma et al 2009 35 are incorrect

error estimation of perturbations under cri the analysis of stability and robustness of fuzzy reasoning is an important issue in areas like intelligent systems and fuzzy control an interesting aspect is to what extent the perturbation of input in a fuzzy reasoning scheme causes the oscillation of the output in particular when the error limits restrictions of the input values are given what the error limits of the output values are in this correspondence we estimate the upper and lower bounds of the output error affected by the perturbation parameters of the input and obtain the limits of the output values when the input values range over some interval in many fuzzy reasoning schemes under compositional rule of fuzzy inference cri

interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference information systems which contain only crisp data precise and unique attribute values for all objects have been widely investigated due to the fact that in realworld applications imprecise data are abundant uncertainty is inherent in real information systems in this paper information systems are called fuzzy information systems and formalized by objects attributes f in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values to interpret and extract fuzzy decision rules from fuzzy information systems the meta theory based on modal logic proposed by resconi et al is modified the modified meta theory not only expresses uncertainty between objects and their attributes but also uncertainty in the process of recognizing fuzzy information systems in addition according to perception computing proposed by zadeh granules of fuzzy information systems can be represented by fuzzy decision rules so that fuzzy inference methods can be used to obtain the decision attribute of a new object finally a novel way of combining evidences based on the modified meta theory is introduced which extends the concept of combining evidences based on dempster shafer theory

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

triple i method of fuzzy reasoning abstract the theory of the triple i method with total inference rules of fuzzy reasoning is investigated by using zadeh s implication operator rz the computational formulae for both fuzzy modus ponens fmp and fuzzy modus tollens fmt are obtained the reversibility properties for fmp and fmt are analyzed and the reversibility criteria are given we also investigated the generalized problem of the triple i method and obtained the formulae for the triple i fmp and the triple i fmt

approximation theorem of the fuzzy transform in fuzzy reasoning and its application to the scheduling problem in this paper the theoretical foundation of fuzzy reasoning is analyzed and the idea that the fuzzy transform given in the fuzzy reasoning method should be continuous with respect to a certain fuzzy distance is proposed also the fuzzy transforms given in the two fuzzy reasoning methods the mamdani method and the iii method are proved to be continuous based on the continuity of the fuzzy transform the approximation theorem of the continuous fuzzy number transform is proven then on the basis of the approximation theorem a simple fuzzy number transform is constructed in order to implement the fuzzy reasoning based on multiple rules at last the fuzzy reasoning based on multiple rules implemented by a simple fuzzy number transform is applied to machine scheduling problems and numerical computational results of different scale scheduling problems with the objective of minimizing the total number of tardy jobs show that it is more effective than usual heuristics based on rules and in the computational time it has the obvious advantage over the reasoning by fuzzy rules directly

domain semantics of possibility computations the purpose of this paper is to introduce a kind of computational model called possibility computation to deal with non determinism both its denotational and logical semantics in the framework of domain theory are established and their equivalence is verified in this approach the denotational semantics will be set up by assigning to programs possibility state transformers i e scott continuous functions from the domain of input states to the possibility powerdomain of the codomain of final states this possibility powerdomain of a dcpo will consist of all possibility valuations ordered pointwise of this dcpo the logical semantics will be given by fuzzy predicate transformers introduced by the first author and jung following dijkstra s predicate transformers two semantics equivalence will be verified in terms of the integration of fuzzy predicates with respect to possibility valuations the categorical monad of the possibility powerdomain will be investigated

on the dempster shafer framework and new combination rules abstract we discuss the basic concepts of the dempster shafer approach basic probability assignments belief functions and probability functions we discuss how to represent various types of knowledge in this framework we discuss measures of entropy and specificity for belief structures we discuss the combination and extension of belief structures we introduce some concerns associated with the dempster rule of combination inherent in the normalization due to conflict we introduce two alternative techniques for combining belief structures the first uses dempster s rule while the second is based upon a modification of this rule we discuss the issue of credibility of a witness

stable and unstable manifolds for hyperbolic bi semigroups we show the existence of local lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bi semigroups we do not assume backward nor forward uniqueness of solutions we do not use cut off functions because we do not assume global smallness conditions on the nonlinearities we introduce what we call dichotomous flows which recovers the symmetry between the past and the future thus we need to prove only a stable manifold theorem we modify the conley mcgehee moeckel approach to avoid appealing to wazewski principle and brouwer degree theory hence we allow both the stable and unstable directions to be infinite dimensional we apply our theorem to the elliptic system u u g u u in an infinite cylinder r

on the logic foundation of fuzzy reasoning abstract fuzzy reasoning and fuzzy formal deduction theory are two closely related subjects and each of them has been investigated by many researchers unfortunately it seems that the two branches have not been successfully connected the aim of this paper is to establish a so called quasi propositional deductive system syntactically as well as semantically and then set a logic foundation for developing the theory of fuzzy reasoning therefrom

some types of generalized fuzzy filters of bl algebras the aim of this paper is to introduce the notions of interval valued q fuzzy boolean implicative positive implicative and fantastic filters in bl algebras and to investigate some related properties some characterizations of these generalized fuzzy filters are derived finally the relations among these generalized fuzzy filters are discussed it is proved that an interval valued q fuzzy filter in bl algebras is an interval valued q fuzzy boolean implicative filter if and only if it is both an interval valued q fuzzy positive implicative filter and an interval valued q fuzzy fantastic filter

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

stability of some waves in the dissipative boussinesq system abstract in this paper we study analytically a class of waves in the variant of the classical dissipative boussinesq system given by t u x v x x x v x x t u x u v t v x u c x x x u x x t v v x v where c 0 is a small parameter and 0 1 this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every c and almost every 1 it contains solutions that are defined for large values of time and they are very close of order o to a linear torus for long times of order o 1 the proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates

three and two valued łukasiewicz theories in the formal deductive system l nm logic in order to offer a logic foundation for fuzzy reasoning and to reduce the gap between fuzzy reasoning and artificial intelligence a new formal deductive system l was introduced in 1996 it was proved that the prominent nilpotent minimum logic is a logic system equivalent to l as is well known in classical propositional logic there is a one to one correspondence between deductively closed theories and topologically closed subsets of the stone space of all maximally consistent theories recently by means of the strong completeness of l we obtained the structural and topological characterizations of maximally consistent theories in this logic it was proved that the set of all maximally consistent theories over l with an induced topology is a cantor space in order to establish a similar connection between theories over l and closed subsets of this topological space to the classical case we introduce in this paper the notion of three valued lukasiewicz theory and the notion of two valued lukasiewicz boolean theory we prove that there is also a one to one correspondence between deductively closed three valued lukasiewicz theories and closed subsets of the space and boolean theories correspond to the closed subsets of the subspace of all maximally consistent boolean theories lastly we give an alternative proof for the structural characterization of the maximality of consistent theories over l by induction on the complexity of formulas which is independent of the strong completeness of l and also independent of the perfectness of the standard r 0 algebra

existence of traveling waves of invasion for ginzburg landau type problems in infinite cylinders we study a class of systems of reaction diffusion equations in infinite cylinders which arise within the context of ginzburg landau theories and describe the kinetics of phase transformation in second order or weakly first order phase transitions with non conserved order parameters we use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non linearities we also prove boundedness regularity and some other properties of the obtained solutions as well as several sufficient conditions for existence or non existence of such traveling waves and give rigorous upper and lower bounds for their speed in addition we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized we give a sample application of our results using a computer assisted approach

simplification and independence of axioms of fuzzy logic systems imtl and nm monoidal t norm based logic mtl is the common formalization of all fuzzy logic of left continuous t norms and their residua this system has two important schematic extensions involutive monoidal t norm based logic imtl and nilpotent minimum logic nm in this paper we discuss simplification and independence of axioms of imtl and nm the shorter axioms of both logics and an independent axioms of the system imtl are given

characterizations of maximal consistent theories in the formal deductive system l nm logic and cantor space a maximal consistent theory is a maximal theory with respect to its consistency the present paper is divided into two parts the first one is devoted to characterize the maximality of a consistent theory in the formal deductive system l which is a logic system equivalent to the nilpotent minimum logic it is proved that each maximal consistent theory in this logic must be the deductive closure of a collection of simple compound formulas hence it follows that there is a one to one correspondence between the set of all maximal consistent theories and the set of evaluations e assigning to each propositional variable p its truth degree e p 0 12 1 the satisfiability theorem and compactness theorem of l are obtained the second part is to investigate the topological structure of the set of all maximal consistent theories over l and the results show that this topological space is a cantor space

integrated semantics and logic metric spaces the concept of tautology degree of an abstract proposition is proposed in this paper by means of integral of the corresponding function with respect to certain semantics over 0 1 then a pseudo metric among abstract propositions is introduced and the r logic metric space is proposed where r is a suitable implication operator finally a new theoretical framework for approximate reasoning in the r logic metric space is provided

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

maximal confidence intervals of the interval valued belief structure and applications an unsolved problem in the interval valued belief structure is the existence of maximal confidence intervals referring to the credible and maximal interval valued beliefs assigned to focal elements in which every point can be reached by a combination rule in this study the existence of maximal confidence intervals of the interval valued belief structure was investigated and validated in addition some applications of the maximal confidence interval in constructing valid and normalized interval valued belief structures are discussed a numerical experiment was conducted to illustrate the existence of maximal confidence intervals several propositions and a theorem were further proposed to demonstrate the validity of the maximal confidence interval by using the concept of the maximal confidence interval a procedure was proposed to construct valid and normalized interval valued belief structures and some meaningful guidance for assigning interval valued beliefs was also suggested to the experts finally a series of well designed examples was presented as applications of the maximal confidence interval the proposed procedure and the suggestions they indicated that valid and normalized interval valued belief structures can be achieved accurately and efficiently in practical applications with the proposed procedure and guidance on the basis of the concept of the maximal confidence interval

some methods of reasoning for fuzzy conditional propositions abstract we consider inference for propositions of the form if then else in this paper we propose four methods of inference applying logical constructs developed by mizumoto for these propositions which do not fit the zadeh and mamdani methods of inference we show how these methods satisfy many of our intuitions under several criteria

existence of quasiperiodic solutions of elliptic equations on rn 1 via center manifold and kam theorems abstract we consider elliptic equations on r n 1 of the form 1 x u u y y g x u 0 x y r n r where g x u is a sufficiently regular function with g 0 0 we give sufficient conditions for the existence of solutions of 1 which are quasiperiodic in y and decaying as x uniformly in y such solutions are found using a center manifold reduction and results from the kam theory we discuss several classes of nonlinearities g to which our results apply

primitive ideal space of higher rank graph c algebras and decomposability abstract in this paper we describe primitive ideal space of the c algebra c associated to any locally convex row finite k graph to do this we will apply the farthing s desourcifying method on a recent result of carlsen kang shotwell and sims we also characterize certain maximal ideals of c furthermore we study the decomposability of c we apply the description of primitive ideals to show that if i is a direct summand of c then it is gauge invariant and isomorphic to a certain k graph c algebra so we may characterize decomposable higher rank c algebras by giving necessary and sufficient conditions for the underlying k graphs moreover we determine all such c algebras which can be decomposed into a direct sum of finitely many indecomposable c algebras

approximation theory of fuzzy systems based upon genuine many valued implications siso cases it is proved that the single input and single output siso fuzzy systems based upon genuine many valued implications are universal approximators it is shown theoretically that fuzzy control systems based upon genuine many valued implications are equivalent to those based upon t norm implications the general approach to construct fuzzy systems is given it is also shown that defuzzifier based upon center of areas is not appropriate to the fuzzy systems based upon genuine many valued implications

variable universe stable adaptive fuzzy control of a nonlinear system abstract a kind of stable adaptive fuzzy control of a nonlinear system is implemented based on the variable universe method proposed first in 1 first of all the basic structure of variable universe adaptive fuzzy controllers is briefly introduced then the contraction expansion factor which is a key tool of the variable universe method is defined by means of the integral regulation idea and then a kind of adaptive fuzzy controller is designed by using such a contraction expansion factor the simulation on the first order nonlinear system is done and as a result its simulation effect is quite good in comparison with the corresponding result in 2 3 second it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of liapunov theory the simulation on a second order nonlinear system shows that its simulation effect is also quite good in comparison with the corresponding result in 2 besides a useful tool called symbolic factor is proposed which may be of universal significance it can greatly reduce the setting time and enhance the robustness of the system

linear fractional multi objective optimization problems subject to fuzzy relational equations with a continuous archimedean triangular norm in this paper linear fractional multi objective optimization problems subject to a system of fuzzy relational equations fres using the max archimedean triangular norm composition are considered first some theorems and results are presented to simplify systems of fuzzy relational equations next the feasible set of the optimization problem is reduced then the linear fractional multi objective optimization problem is converted to a linear one using nykowski and zolkiewski s approach furthermore the efficient solutions are obtained by applying the improved e constraint method in addition we design an algorithm to generate consistent systems of fuzzy relational equations randomly using the covering problem finally the proposed method is effectively tested by solving a randomly generated consistent test problem

consistency measure inclusion degree and fuzzy measure in decision tables classical consistency degree has some limitations for measuring the consistency of a decision table in which the lower approximation of a target decision is only taken into consideration in this paper we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory for three types of decision tables complete incomplete and maximal consistent blocks the membership functions of an object are defined through using the equivalence class tolerance class and maximal consistent blocks including itself respectively based on these membership functions we introduce consistency measures to assess the consistencies of a target set and a decision table and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables in addition the relationships among the consistency inclusion degree and fuzzy measure are established as well these results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues

formalized theory of general fuzzy reasoning first a rigorous mathematical theory for calculating the conclusion of general fuzzy reasoning is proposed second a formalized non fuzzy version of the theory of general fuzzy reasoning is founded in classical logics

the generalized truth degree of quantitative logic in the logic system l n n valued nm logic system the concept of the truth degree of a formula is the crucial tool and the building block in quantitative logic from which the concept of logic metric in quantitative logic is derived logic metric takes an important role in quantitative logic related to which are other concepts in quantitative logic such as divergence consistency etc in the present paper having combined the theory of generalized tautologies with the theory of truth degrees in quantitative logic we have proposed the theory of s c truth degrees of formulas related to theory c in the logic system l n n valued nm logic system and discussed some of its properties with the help of the properties of s c truth degrees t c a email protected c a b email protected c b we have obtained the c logic metric on the set f s of formulas in the propositional logic system l n n valued nm logic system by the work of this paper we can generalize the theory of quantitative logic in all round way and establish an approximate reasoning s framework related to theory c in the logic system l n n valued nm logic system

reverse triple i method of fuzzy reasoning for the implication operator rl first the reverse triple i method with lukasiewicz s implication operator is studied the formulas of supremum for fuzzy modus ponens fmp and infimum for fuzzy modus tollens fmt of inverse triple i method are obtained respectively second the reductivity of reverse triple i method is considered lastly the theory of sustention degree is discussed and its properties are analysed the generalized problem of reverse triple i method is solved and the corresponding formulas of a reverse triple i method with sustention degree are also given

measures for evaluating the decision performance of a decision table in rough set theory as two classical measures approximation accuracy and consistency degree can be employed to evaluate the decision performance of a decision table however these two measures cannot give elaborate depictions of the certainty and consistency of a decision table when their values are equal to zero to overcome this shortcoming we first classify decision tables in rough set theory into three types according to their consistency and introduce three new measures for evaluating the decision performance of a decision rule set extracted from a decision table we then analyze how each of these three measures depends on the condition granulation and decision granulation of each of the three types of decision tables experimental analyses on three practical data sets show that the three new measures appear to be well suited for evaluating the decision performance of a decision rule set and are much better than the two classical measures

abundance of entire solutions to nonlinear elliptic equations by the variational method abstract we study entire bounded solutions to the equation u u u 3 0 in r 2 our approach is purely variational and is based on concentration arguments and symmetry considerations this method allows us to construct in an unified way several types of solutions with various symmetries radial breather type rectangular triangular hexagonal etc both positive and sign changing it is also applicable for more general equations in any dimension

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 reasoning based on a new fuzzy rough set and its application to scheduling problems in this paper we study the fuzzy reasoning based on a new fuzzy rough set first we define a broad family of new lower and upper approximation operators of fuzzy sets between different universes using a set of axioms then based on the approximation operators above we propose the fuzzy reasoning based on the new fuzzy rough set by means of the above fuzzy reasoning based on the new fuzzy rough set for a given premise we can obtain the fuzzy reasoning consequence expressed by the fuzzy interval constructed by the above two approximations of fuzzy sets furthermore through the defuzzification of the lower and upper approximations we can get the corresponding two values constructing the interval used as the fuzzy reasoning consequence after defuzzification then from the above interval a suitable value can be selected as the final reasoning consequence so that some special constraints are satisfied as possibly at last we apply the fuzzy reasoning based on the new fuzzy rough set to the scheduling problems and numerical computational results show that the fuzzy reasoning based on the new fuzzy rough set is more suitable for the scheduling problems compared with the fuzzy reasoning based on the cri method and the iii method

existence of quasiperiodic solutions for the second order approximation boussinesq system abstract in this paper we study analytically a class of waves in the variant of the classical second order approximation boussinesq system given by t u b x x t u c x x x x t u x v x u v 1 3 2 b x x x v b x x x u v b x u x x v a x x x x x v t v b x x t v c x x x x t v x u 1 2 x x x u v x v x u x x u b x v x x v d x x x x x u where a b c d are some real constants this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every a b c and d it admits solutions that are quasiperiodic in time

random sets and fuzzy interval analysis abstract a general expression of function with random set valued arguments is stated which encompasses zadeh s extension principle as well as functions of random variables and interval analysis a monotonicity property is derived for algebraic operations performed between random set valued variables this property extends the monotonicity of functions with set valued arguments in the sense of inclusion the application of this result to the approximate calculation of functions of random variables is suggested and illustrated by an example

soft bl algebras based on fuzzy sets by means of soft sets and q soft sets some characterizations of implicative positive implicative and fantastic filteristic soft bl algebras are investigated finally we prove that a soft set is an implicative filteristic soft bl algebra if and only if it is both a positive implicative filteristic soft bl algebra and a fantastic filteristic soft bl algebra

fuzzy reasoning as a control problem different from the dominant view of treating fuzzy reasoning as generalization of classical logical inference in this paper fuzzy reasoning is treated as a control problem a new fuzzy reasoning method is proposed that employs an explicit feedback mechanism to improve the robustness of fuzzy reasoning the fuzzy rule base given a priori serves as a controlled object and the fuzzy reasoning method serves as the corresponding controller the fuzzy rule base and the fuzzy reasoning method constitute a control system that may be open loop or closed loop depending on the underlying reasoning goals constraints the fuzzy rule base the fuzzy reasoning method and the corresponding reasoning goals constraints define the three distinct ingredients of fuzzy reasoning while various existing fuzzy reasoning methods are essentially a static mapping from the universe of single fuzzy premises to the universe of single fuzzy consequences the new fuzzy reasoning method maps sequences of fuzzy premises to sequences of fuzzy consequences and is a function of the underlying reasoning goals constraints the monte carlo simulation shows that the new fuzzy reasoning method is much more robust than the optimal fuzzy reasoning method proposed in our previous work the explicit feedback mechanism embedded in the fuzzy reasoning method does significantly improve the robustness of fuzzy reasoning which is concerned with the effects of perturbations associated with given fuzzy rule bases and or fuzzy premises on fuzzy consequences the work presented in this paper sets a new starting point for various principles of feedback control and optimization to be applied in fuzzy reasoning or logical inference and to explore new forms of reasoning including robust reasoning and adaptive reasoning it can be also expected that the new fuzzy reasoning method presented in this paper can be used for modeling and control of complex systems and for decision making under complex environments

decomposition and resolution of min implication fuzzy relation equations based on s implications the problem of solving min implication fuzzy relation equations based on s implications is studied in this paper decomposition of min implication fuzzy relation equations is given in a finite case the solution existence problem of min implication fuzzy relation equations is discussed and for ns implications a new solvability criterion based upon solution matrices introduced in this paper is studied it is shown that for each solution r of a min implication fuzzy relation equation there exists a maximal solution r of this equation such that r is greater than or equal to r whenever the solution set of this equation is nonempty the complete solution set of min implication fuzzy relation equation can be determined by the maximal solution set of this equation which is finite an effective method to solve min implication fuzzy relation equations based on s implication is proposed

on equivalent forms of fuzzy logic systems nm and imtl the nilpotent minima are important t norms in fuzzy logic and fuzzy reasoning two interesting formal systems nilpotent minimum logic nm and l based on nilpotent minima have been proposed and have been intensively studied in this paper their equivalence is pointed out moreover coincidence of two related algebraic structures nm and r0 algebras is also proved in addition we give an equivalent form of involutive monoidal t norm based logic imtl

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

a triangular norm based propositional fuzzy logic in 1997 wang introduced the formal system l for a type of fuzzy propositional calculus given by a particular t norm different from the three famous ones lukasiewicz godel product recently pei proved the completeness theorem for l with respect to w semantic by developing a theory of algebraic systems in this paper we aim to prove it by essentially metamathematical method furthermore we discussed originally the completeness for the schematic extension l n of l with respect to w n semantic

fuzzy implicative and boolean filters of r 0 algebras the notions of fuzzy filters fuzzy implicative filters and fuzzy boolean filters are introduced in r0 algebras respectively some of their properties are investigated the relations between fuzzy implicative filters and fuzzy boolean filters are discussed the characterizations of fuzzy implicative filters are studied and the following is proved to be equivalent 1 a fuzzy filter f is implicative 2 f is a fuzzy boolean filter 3 x y z l f x z f x z y f y z 4 x y l f x y f x y y 5 x l n 1 f nx f x

undefinability of min conjunction in mtl firstly a sufficient and necessary condition is given to decide whether the lattice operations of a wnm chain can be definable from its other operations secondly we show that it is indeed necessary to introduce the min conjunction into mtl as a primitive connective by constructing a left continuous t norm such that in the propositional calculus defined by this t norm the min conjunction and max disjunction are not definable from the strong conjunction implication and truth constant email protected

the extensions l n of the formal system l n and their completeness in this paper the method of well combined semantics and syntax proposed by pavelka is applied to the research of the propositional calculus formal system l the part of constant values are taken as formulas formulas are fuzzified in two manners semantically and syntactically and inferring processes are fuzzified a sequence of new extensions l n of the system l is proposed and the completeness of l n is proved

robustness of fuzzy reasoning via logically equivalence measure abstract in this paper we discuss robustness of fuzzy reasoning after proposing the definition of perturbation of fuzzy sets based on some logic oriented equivalence measure we present robustness results for various fuzzy logic connectives fuzzy implication operators inference rules and fuzzy reasoning machines and discuss the relations between the robustness of fuzzy reasoning and that of fuzzy conjunction and implication operators the robustness results are presented in terms of equalities of fuzzy sets based on some logic oriented equivalence measure and the maximum of which ensures the corresponding equality holds is derived

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

perturbation of fuzzy sets and fuzzy reasoning based on normalized minkowski distances in this paper we extend the concept of the perturbation of fuzzy sets based on normalized minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets these operations are induced by triangular norms and conorms furthermore we discuss the perturbation of fuzzy reasoning

non fuzzy versions of fuzzy reasoning in classical logics abstract fuzzy reasoning features fuzzy logic in its narrow sense fln and fuzzy modus ponens fmp is the core of fuzzy reasoning as is well known fmp is far more general and complicated than the conventional rule of modus ponens mp and takes the latter as special case under certain conditions this paper proves that there exist non fuzzy versions of fmp in classical 2 valued logic as well as in conventional multivalued systems and reveals the roots of these seemingly surprising results

generalized consistency degrees of theories w r t formulas in several standard complete logic systems abstract the present paper carries out a deep analysis on the inconsistency of theories by means of deduction theorems completeness theorems and satisfiability degrees of formulas and introduces the concept of the degree of entailment of a contradiction from a theory in classical two valued logic system ukasiewicz fuzzy logic system godel fuzzy logic system product fuzzy logic system and the r 0 fuzzy logic system it is proved that in classical two valued logic system ukasiewicz fuzzy logic system and the r 0 fuzzy logic system respectively the concept of consistency degrees of theories defined by the divergence degrees of theories in x n zhou g j wang consistency degrees of theories in some systems of propositional fuzzy logic fuzzy sets and systems 152 2005 321 331 h j zhou g j wang a new theory index based on deduction theorems in several logic systems fuzzy sets and systems 157 2006 427 443 is reasonable and can accurately measure the consistency degrees of theories the concept of the degree of entailment of a contradiction from a theory is generalized by replacing the contradiction with a general formula and then the generalized consistency degrees of theories w r t general formulas in the above mentioned logic systems is established

the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources abstract we catalogue the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph the primitive ideals in the cuntz krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group the cuntz krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic

canonical forms of fuzzy truthoods by meta theory based upon modal logic in this paper we redefine with the meta theory based on modal logic the operations between fuzzy sets of verity i e fuzzy sets of truthood with this new approach we can unify the different formulas for the operations and or and not between the fuzzy sets of verity i e truthood the operations between the fuzzy sets of verity become sensitive to the logic value true or false that agents persons sensors assign to the worlds contexts the operations are also sensitive to the difference of the worlds and time of synchronisation it should be pointed out that when we use the logic operations as and or and not we generally assume that the worlds are the same and change their truth value at the same time but it is known that there are cases where this synchronic situation and identity is not always valid that is there exist transformations that change one world for one proposition to another world for another proposition in conclusion the linguistic and or not operations become dependent on the particular truth value of a world on the synchronisation and on the worlds assigned to the two propositions via transformations thus all of these possible changes in the structure of the worlds in the modal logic cause the gradation of the linguistic operations e g and or and not an individual world person agent sensor assigns to an atomic sentence either a true or false value and uses the classical two value logic operations of and or not that is the crisp true or false responses assignments of worlds generate gradation of truthood value the uncertainty in a fuzzy set is represented with sets of worlds in a conflict situation i e the same proposition may be true in one world and false in another consonant or dissonant relations between sets of worlds that depends on the synchronisation and or transformation of worlds cause the generation of gradation by the linguistic operations and or and not that is the linguistic operations change for different concrete situations set of worlds caused by the generation of a representation in each world where the descriptive set membership assignment d may be two or infinite valued but the verity assignment v is two valued in a particular world representation whereas membership values of the fuzzy set of truth verifications associated with the set of worlds are in 0 1 furthermore the combination of fuzzy membership values generates a type ii fuzzy set that is captured by fdcf and fccf formulas in this paper we show that there exist a deeper connection between the set membership assignment and the verity assignment of truthood by modal logic with suitable extensions

unified full implication algorithms of fuzzy reasoning this paper discusses the full implication inference of fuzzy reasoning for all residuated implications induced by left continuous t norms unified a triple i algorithms are constructed to generalize the known results as the corollaries of the main results of this paper some special algorithms can be easily derived based on four important residuated implications these algorithms would be beneficial to applications of fuzzy reasoning based on properties of residuated implications the proofs of the many conclusions are greatly simplified

robustness of fuzzy reasoning and spl delta equalities of fuzzy sets fuzzy reasoning methods or approximate reasoning methods are extensively used in intelligent systems and fuzzy control in this paper the author discusses how errors in premises affect conclusions in fuzzy reasoning that is he discusses the robustness of fuzzy reasoning after reviewing his previous work 1996 he presents robustness results for various implication operators and inference rules all the robustness results are formulated in terms of spl delta equalities of fuzzy sets two fuzzy sets are said to be spl delta equal if they are equal to an extent of spl delta

unified forms of triple i method first the present paper provides unified forms of triple i method for fuzzy modus ponens and fuzzy modus tollens of which diverse implication operators can be employed second it is clarified that in a sense zadeh s cri method for fuzzy modus ponens can be brought into line with the unified form of the triple i method lastly a unified form of a triple i method is established as well and a duality result concerning a triple i solutions of fuzzy modus ponens and fuzzy modus tollens is obtained

fuzzy reasoning method by optimizing the similarity of truth tables this paper presents a new fuzzy reasoning method by optimizing the similarity of truth tables os method for short its basic idea is to find a fuzzy set such that the truth tables generated by the antecedent rule and the consequent rule are as similar as possible based on this idea the principle of os method and the fuzzy reasoning with os method are given and discussed and then the os methods with certain similarity measure and several fuzzy implications are investigated finally numerical examples are analyzed to compare the proposed method with compositional rule of inference cri method

a fuzzy logic for an ordinal sum t norm among the class of residuated fuzzy logics a few of them have been shown to have standard completeness both for propositional and predicate calculus like godel nm and monoidal t norm based logic systems in this paper a new residuated logic nmg which aims at capturing the tautologies of a class of ordinal sum t norms and their residua is introduced and its standard completeness both for propositional calculus and for predicate calculus are proved

some properties of fuzzy reasoning in propositional fuzzy logic systems in order to analyze the logical foundation of fuzzy reasoning this paper first introduces the concept of generalized roots of theories in lukasiewicz propositional fuzzy logic luk godel propositional fuzzy logic god product propositional fuzzy logic p and nilpotent minimum logic nm the r 0 propositional fuzzy logic l next it is proved that all consequences of a theory c named d c are completely determined by its generalized root whenever c has a generalized root moreover it is proved that every finite theory c has a generalized root which can be expressed by a specific formula finally we demonstrate the existence of a non fuzzy version of fuzzy modus ponens fmp in luk god p and nm l and we provide its numerical version as a new algorithm for solving fmp

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

the r0 type fuzzy logic metric space and an algorithm for solving fuzzy modus ponens fuzzy modus ponens briefly fmp is the most fundamental form of fuzzy reasoning and has been extensively discussed by diverse researchers the aim of the present paper is to propose a formalized form of fmp called generalized modus ponens in the fuzzy logic system l and solve it in l and then provide its numerical version as a new algorithm for solving fmp as a preparation some related questions such as what is a fuzzy logic metric space why is r 0 implication operator selected and what kind of merits does the fuzzy pseudo metric logic space f s r possess etc are analyzed

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

stone like representation theorems and three valued filters in r0 algebras nilpotent minimum algebras nilpotent minimum algebras nm algebras are algebraic counterpart of a formal deductive system where conjunction is modeled by the nilpotent minimum t norm a logic also independently introduced by guo jun wang in the mid 1990s such algebras are to this logic just what boolean algebras are to the classical propositional logic in this paper by introducing respectively the stone topology and a three valued fuzzy stone topology on the set of all maximal filters in an nm algebra we first establish two analogues for an nm algebra of the well known stone representation theorem for a boolean algebra which state that the boolean skeleton of an nm algebra is isomorphic to the algebra of all clopen subsets of its stone space and the three valued skeleton is isomorphic to the algebra of all clopen fuzzy subsets of its three valued fuzzy stone space respectively then we introduce the notions of boolean filter and of three valued filter in an nm algebra and finally we prove that three valued filters and closed subsets of the stone space of an nm algebra are in one to one correspondence and boolean filters uniquely correspond to closed subsets of the subspace consisting of all ultrafilters

stability of some solutions for elliptic equations on a cylindrical domain we study analytically a class of solutions for the elliptic equation formula 1 where 0 and e is a small parameter this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every 0 it contains solutions that are defined for large values of time and they are very close of order o e to a linear torus for long times of order o e 1 the proof uses the fact that the equation leaves invariant a smooth centre manifold and for the restriction of the system to the centre manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates 1 embed graphic 1 gif

a new approach to inference in approximate reasoning abstract we introduce a new approach to inference in approximate reasoning based on truth value restriction the degree to which the actual given value a of a variable x agrees with the antecedent value b in a production if x is b then y is c is represented as a fuzzy subset of a truth space we introduce a new form of implication based on the exponential operation a new compatibility relation based on the input value a and the antecedent value b is defined a fuzzy truth value true derived from the antecedent value b is also defined it is shown that by mapping this compatibility relation into the truth space a fuzzy truth value is generated which produces an exponential family of the fuzzy truth value true we show that the inferred consequent can be generated from this truth value theoretical results showing the validity of this method to 1 chain rules consisting of several single fuzzy conditional propositions and 2 conjunctive fuzzy conditional propositions are also presented theoretical and simulation results demonstrate that our method is superior to the existing methods when intuitively correct responses are required