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

mtl algebras arising from partially ordered groups after dropping the implication like operation and reversing the order an mtl algebra becomes a partially ordered structure l 0 1 based on the single addition like operation furthermore may be restricted to a partial but cancellative addition without loss of information we deal in this paper with the case that the resulting partial algebra l 0 1 is embeddable into the positive cone of a partially ordered group it turns out that most of the examples of mtl algebras known at present can be represented in this way so the systematisation of a considerable class of mtl algebras is achieved and the methods of their construction are brought onto a common line

solving discrete zero point problems in this paper we present two theorems on the existence of a discrete zero point of a function from the n dimensional integer lattice n to the n dimensional euclidean space n the theorems differ in their boundary conditions for both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps

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

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

solving discrete systems of nonlinear equations abstract we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice z n of the n dimensional euclidean space r n it is assumed that the set is integrally convex which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of z n and each simplex of the triangulation lies in an n dimensional cube of size one with respect to this triangulation we assume that the function satisfies some property that replaces continuity under this property and some boundary condition the function has a zero point to prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations the standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied because the continuity property is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself we apply the main existence result to prove the existence of a pure cournot nash equilibrium in a cournot oligopoly model we further obtain a discrete analogue of the well known borsuk ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem

a variable dimension fixed point algorithm and the orientation of simplices a variable dimension algorithm with integer labelling is proposed for solving systems ofn equations inn variables the algorithm is an integer labelling version of the 2 ray algorithm proposed by the author the orientation of lower dimensional simplices is studied and is shown to be preserved along a sequence of adjacent simplices

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

consistency degrees of finite theories in łukasiewicz propositional fuzzy logic abstract the concept of divergence degrees of theories in mathematical logic can be used to grade the extent of consistency of theories based on the analysis of relationships among the properties of consistent inconsistent and fully divergent theories the present paper proposes a membership function for reflecting the consistency degrees of finite theories in ukasiewicz propositional fuzzy logic it is proved that the consistency degrees of consistent finite theories are in between 1 2 and 1 and that of inconsistent theories are equal to 0 and that of completely consistent finite theories are equal to 1 finally a sufficient and necessary condition for theories being inconsistent is also proposed

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

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

computational complexity of a piecewise linear homotopy algorithm we give a bound on the number of steps required by the piecewise linear algorithm based on component wise homotopy proposed by the author for structured problems when solving a linear problem when the coefficient matrix is symmetric and positive definite this bound is polynomial inn and linear in the condition number of the matrix we also investigate the expected value of the bound for a particular distribution of such matrices

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

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

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

superimposed codes in the hamming space we consider the problem of simultaneous identification and transmission of information by means of superimposed codes it is assumed that a total population of t users share a common channel at each time at most in of the t users are active the output from the channel is the modulo 2 sum of the binary codewords transmitted by the active users possibly disturbed by errors a superimposed code is actually a set of codes one for each user designed in such a way that both the at most m active users and their messages can be identified even in the presence of a permissible number of errors in the channel we give several explicit constructions for superimposed codes and analyze in detail their performance in several of the cases under study we offer codes that are proved to be best possible

superimposed codes in r sup n the authors introduce the concept of superimposed codes in euclidean n space r sup n an asymptotic existence bound is derived for such codes the proof uses the idea of random coding in particular the asymptotic properties of long codes are studied it is shown that the derived existence bound differs only by a factor of four from a nonexistence bound obtained by a simple sphere packing argument

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

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

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

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

an early approach toward graded identity and graded membership in set theory the paper considers an early approach toward a fuzzy set theory with a graded membership predicate and a graded equality relation which had been developed by the german mathematician klaua in 1965 in the context of the mathematical fuzzy logic mtl of left continuous t norms we discuss some properties of these graded relations we compare the simultaneous recursive definitions of these relations with the very similar approach toward boolean algebra valued interpretations of membership and equality presented in 1967 by scott and r solovay in the context of independence proofs for zf set theory finally we speculate about possible reasons why klaua soon abandoned this approach

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

consistency degrees of theories in some systems of propositional fuzzy logic based on the analysis of relationships among the properties of consistent inconsistent and fully divergent theories and by means of the concept of divergence degrees the present paper proposes a new membership function for reflecting the consistency degrees of general theories in four different types of fuzzy logic systems i e lukasiewicz fuzzy logic system luk godel fuzzy logic system god product fuzzy logic system pro and the r 0 fuzzy logic system l

a discrete multivariate mean value theorem with applications we establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions a constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling moreover we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

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

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

implication operators in fuzzy logic the choice of fuzzy implication as well as other connectives is an important problem in the theoretical development of fuzzy logic and at the same time it is significant for the performance of the systems in which fuzzy logic technique is employed there are mainly two ways in fuzzy logic to define implication operators 1 an implication operator is considered as the residuation of conjunction operator and 2 it is directly defined in terms of negation conjunction and disjunction operators the purpose of this paper is to determine the number of implication operators defined in the second way for some usual negation conjunction and disjunction operators in fuzzy logic

some types of filters in mtl algebras the aim of this work is to introduce the concepts of imtl filters and strong mtl filters in mtl algebras and show that these filters are related to imtl algebras and strong mtl algebras respectively we also introduce the concepts of eimtl and associative filters in mtl algebras the relationships between these filters and quotient algebras that are constructed via these filters are described we clarify that eimtl imtl and fantastic filters coincide in bl algebras whereas they have different properties in mtl algebras moreover the relations between them are delineated we prove that strong filters include some current filters such as implicative positive implicative and fantastic filters

fuzzy sets and formal logics the paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics our focus is on the historical side of these developments

efficiency and implementation of simplicial zero point algorithms in this paper we compare the efficiency of several simplicial variable dimension algorithms to do so we first treat the issues of degeneracy and accelerating we present a device for solving degeneracy furthermore we compare several accelerating techniques the technique of iterated quasi newton steps after each major cycle of the simplicial algorithm is implemented in a computer code which is used to compare the efficiency of the n 1 ray 2n ray 2n ray and 3n 1 ray algorithms except for the n 1 ray algorithm the number of function evaluations does not differ very much between the various algorithms it appeared however that the 2n algorithm needs considerably less computation time

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

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

an algebraic approach to linguistic hedges in zadeh s fuzzy logic the paper addresses the mathematical modelling of domains of linguistic variables i e term sets of linguistic variables in order to obtain a suitable algebraic structure for the set of truth values of zadeh s fuzzy logic we shall give a unified algebraic approach to the natural structure of domains of linguistic variables which was proposed by ho and wechler fuzzy sets and systems 35 1990 281 and then by ho and nam proc ncst vietnam 9 1 1997 15 logic algebra and computer science vol 46 banach center publications pwn warsaw 1999 p 63 in this approach every linguistic domain can be considered as an algebraic structure called hedge algebra because properties of its unary operations reflect semantic characteristics of linguistic hedges many fundamental properties of refined hedge algebras rh algebras are examined especially it is shown that every rh algebra of a linguistic variable with a chain of the primary terms is a distributive lattice rh algebras with exactly two distinct primary terms one being an antonym of the other will also be investigated and they will be called symmetrical rh algebras it is shown that a class of finite symmetrical rh algebras has a rich enough algebraic structure

simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds ideas of a simplicial variable dimension restart algorithm to approximate zero points onrn developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds the algorithm can be considered as a modification of the2n ray zero point finding algorithm onrn it appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n ray algorithm applied to an equivalent zero point problem this is caused by the fact that the algorithm utilizes the complementarity conditions on the variables

continuous weakly cancellative triangular subnorms i their web geometric properties abstract the paper studies web geometric properties of continuous weakly cancellative t subnorms particularly it studies the reidemeister closure condition which is known from web geometry as a visual characterization of those loops that are associative as a result the paper delimits a subset of the domain of a continuous weakly cancellative t subnorm in which the reidemeister closure condition is necessarily satisfied

a new variable dimension algorithm for the fixed point problem we propose a new variable dimension algorithm for solving a system ofn equations inn variables the algorithm solves a sequence of subproblems consisting of the first several equations the dimension of subproblems does not however increase monotonically a convergence condition is also presented

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

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

a unifying model based on retraction for fixed point algorithms we present a unifying model based on retraction for several restart fixed point algorithms the model embraces the interpretation of the algorithms in terms of stationary point problem by van der laan and talman and fully explains the 2 ray method

a homotopy for solving large sparse and structured fixed point problems we consider here the problem of solving a system of n nonlinear equations in n variables when n is large but the underlying mapping has a sparse jacobian and is also structured we present a homotopy having a variable dimension feature whose implementation in a pl algorithm effectively exploits the sparsity of the jacobian and separability of the mapping the implementation given here uses the cholesky factorization and is thus stable an application to a large system is also discussed

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

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

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

a new theory consistency index based on deduction theorems in several logic systems based on deduction theorems completeness theorems and by means of the theory of truth degrees of formulas and the concept of divergence degrees of theories the present paper proposes another new index reflecting the extent to which a general theory is consistent in five different logic systems i e classical two valued logic system c 2 lukasiewicz fuzzy logic system luk godel fuzzy logic system god product fuzzy logic system p and the r 0 fuzzy logic system l the concepts of normal consistency and almost inconsistency of theories in the above mentioned logic systems are introduced and sufficient and necessary conditions for theories being normal consistent or almost inconsistent are given finally comparison with the existing concept of consistency degrees of theories is analyzed

a better triangulation for wright s 2n ray algorithm abstract we give a triangulation for wright s 2n ray algorithm to compute solutions of nonlinear equations according to measures of efficiency of triangulations it is better than any other available triangulation for the 2n ray algorithm

term satisfiability in flew algebras fl ew algebras form the algebraic semantics of the full lambek calculus with exchange and weakening we investigate two relations called satisfiability and positive satisfiability between fl ew terms and fl ew algebras for each fl ew algebra the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory we identify and investigate the complements as fragments of its universal theory we offer characterizations of those algebras that positively satisfy just those terms that are satisfiable in the two element boolean algebra providing its semantics to classical propositional logic in case of positive satisfiability these algebras are just the nontrivial weakly contractive fl ew algebras in case of satisfiability we give a characterization by means of another property of the algebra the existence of a two element congruence further we argue that positive satisfiability problems in fl ew algebras are computationally hard some previous results in the area of term satisfiability in mv algebras or bl algebras are thus brought to a common footing with known facts on satisfiability in heyting algebras

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

a graded notion of functionality abstract we will introduce the notion of functionality as a graded property of a fuzzy relation we will extend known results relating to functional fuzzy relations to their graded counterparts and provide new properties w r t various kinds of relational compositions as a contribution to the theory of fuzzy control we will moreover provide an example of use of graded properties in establishing implicative models of fuzzy rule bases where the non graded property fails