fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

reducing nfas by invariant equivalences we give new general methods for constructing small non deterministic finite automata nfa from arbitrary ones given an nfa we compute the largest right invariant equivalence on the set of states and then merge the equivalent states to obtain a smaller automaton when applying this method to position automata we get a way to convert regular expressions into nfas which are always smaller than or equal to the position partial derivative and follow automata it can be arbitrarily smaller the construction can be dually made for left invariant equivalences and then the two can be combined for even better results

fuzzy homomorphisms of algebras in this paper we consider fuzzy relations compatible with algebraic operations which are called fuzzy relational morphisms in particular we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations called uniform fuzzy relational morphisms and those which are partially uniform f functions called fuzzy homomorphisms both uniform fuzzy relations and partially uniform f functions were introduced in a recent paper by us uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras we give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms we establish certain relationships between them and fuzzy congruences and we prove homomorphism and isomorphism theorems concerning them we also point to some applications of uniform fuzzy relational morphisms

image enhancement using the hypothesis selection filter theory and application to jpeg decoding we introduce the hypothesis selection filter hsf as a new approach for image quality enhancement we assume that a set of filters has been selected a priori to improve the quality of a distorted image containing regions with different characteristics at each pixel hsf uses a locally computed feature vector to predict the relative performance of the filters in estimating the corresponding pixel intensity in the original undistorted image the prediction result then determines the proportion of each filter used to obtain the final processed output in this way the hsf serves as a framework for combining the outputs of a number of different user selected filters each best suited for a different region of an image we formulate our scheme in a probabilistic framework where the hsf output is obtained as the bayesian minimum mean square error estimate of the original image maximum likelihood estimates of the model parameters are determined from an offline fully unsupervised training procedure that is derived from the expectation maximization algorithm to illustrate how to apply the hsf and to demonstrate its potential we apply our scheme as a post processing step to improve the decoding quality of jpeg encoded document images the scheme consistently improves the quality of the decoded image over a variety of image content with different characteristics we show that our scheme results in quantitative improvements over several other state of the art jpeg decoding methods

computing behavior of finite fuzzy machines algorithm and its application to reduction and minimization we define finite fuzzy machines and investigate their behavior algorithm and software are proposed for computing behavior for establishing equivalence and redundancy of states and for solving reduction and minimization problems computational complexity of the algorithm is discussed testing examples are supplied the results are valid for finite max min min max and max product fuzzy machines

ccs expressions finite state processes and three problems of equivalence abstract we examine the computational complexity of testing finite state processes for equivalence in milner s calculus of communicating systems ccs the equivalence problems in ccs are presented as refinements of the familiar problem of testing whether two nondeterministic finite automata nfa are equivalent i e accept the same language three notions of equivalence proposed for ccs are investigated namely observational equivalence strong observational equivalence and failure equivalence we show that observational equivalence can be tested in polynomial time as defined in ccs observational equivalence is the limit of a sequence of successively finer equivalence relations k where 1 is nondeterministic finite automaton equivalence we prove that for each fixed k deciding k is pspace complete we show that strong observational equivalence can be decided in polynomial time by reducing it to generalized partitioning a new combinatorial problem of independent interest finally we demonstrate that testing for failure equivalence is pspace complete even for a very restricted type of process

algebraic properties of la languages in this study we introduce the concepts of l valued regular substitution la substitution deterministic l valued regular substitution dla substitution l valued fuzzy homomorphism and its inverse images homomorphism and its inverse images for a lattice ordered monoid l we also study the properties of la languages and dla languages under the above mentioned algebraic operations the algebraic characterization of the l valued regular language is given

minimization algorithm of fuzzy finite automata in this paper a class of fuzzy finite automata corresponding to the mealy type of ordinary automata is formulated and also two types of statewise equivalence relations are introduced from the equivalence relations a minimal form is defined and a minimization algorithm of the mealy type of fuzzy finite automata is obtained

on the generating sequences of regular languages on k symbols the main result is a characterization of the generating sequences of the length of words in a regular language on k symbols we say that a sequence s of integers is regular if there is a finite graph g with two vertices i t such that sn is the number of paths of length n from i to t in g thus the generating sequence of a regular language is regular we prove that a sequence s is the generating sequence of a regular language on k symbols if and only if both sequences s sn n 0 and t kn sn n 0 are regular

products of t generalized state machines and t generalized transformation semigroups abstract we introduce the concepts of t generalized state machines t generalized transformation semigroups and coverings and direct products of t generalized state machines and t generalized transformation semigroups and we investigate some of algebraic properties of them

determinization of fuzzy automata with membership values in complete residuated lattices in this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices in comparison with the previous methods developed by belohlavek r belohlavek determinism and fuzzy automata information sciences 143 2002 205 209 and li and pedrycz y m li w pedrycz fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids fuzzy sets and systems 156 2005 68 92 our method always gives a smaller automaton and in some cases when the previous methods result in infinite automata our method can result in a finite one we also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them and in particular to the concept of the nerode s fuzzy right congruence of a fuzzy automaton which we introduce and study here

finite l fuzzy machines abstract for finite l fuzzy machines we develop behaviour reduction and minimization theory here l stands for l 0 1 where l is a totally ordered set with universal bounds 0 and 1 and the operations are join and meet the most essential results include investigating behaviour equivalence reduction and minimization problems and their algorithmical decidability

pumping lemma in context free grammar theory based on complete residuated lattice valued logic residuated lattices are important algebras and have close links with various important algebras automata theory based on complete residuated lattice valued logic called l valued automata has been established by the second author in 2001 and 2002 as a continuation of automata theory based on complete residuated lattice valued logic in this paper we mainly deal with the problem concerning pumping lemma in l valued context free languages l cfls as a generalization of the notion in the theory of formal grammars the definition of l valued context free grammars l cfgs is introduced we also discuss a special case of l cfgs l right or left linear grammars and show the equivalence between l linear grammars and l regular grammars this result shows that we generalize the pumping lemma in l valued regular languages l rls more recently established by the second author

generalized morphisms a new tool for comparative evaluation of performance of fuzzy implications t norms and co norms in relational knowledge elicitation this paper describes a technique for identification of the knowledge structures implicit in scientific data and further develops the supporting methodology for analysis of fuzzy relational structures using the mathematical theory of fuzzy relations and generalized morphism initial knowledge elicited using repertory grid with semiotic descriptors is transformed to fuzzy relational knowledge structures which capture semantic relationships in a problem domain by means of bk product in analysis the fuzzy relational structures computed in the different sized domains or those computed by means of various fuzzy logics can be analyzed using generalized morphism namely global structure embedding or structure preserving as well as meta analysis of fuzzy relational structures the result of meta analysis is represented as the hasse diagram structure of meta relations by classifying equivalent classes of fuzzy relational structures the paper is concluded with a case study using real life engineering data from the aviation industry

equivalence in automata theory based on complete residuated lattice valued logic automata theory based on complete residuated lattice valued logic called l valued finite automata abbr l vfas was introduced by the second author in 2001 in this paper we deal with the problems of equivalence between l valued sequential machines abbr l vsms and l vfas we define l vsms and particularly present a method for deciding the equivalence between l vsms as well an algorithm procedure for deciding the equivalence between l vsms is constructed we analyze the complexity and efficiency of the algorithm procedure and obtain the relative results to l vfas moreover the definitions of l valued languages abbr l vls and l valued regular languages abbr l vrls recognized by l vfas are given and some related properties are also discussed we show an equivalent relation between l vrls and conventional regular languages by using l valued pumping lemma we get a necessary and sufficient condition for an l vl to be nonconstant

decompositions of t generalized transformation semigroups we introduce several decompositions of t generalized transformation semigroups and investigate some of algebraic properties of them

a note on trillas chc models trillas et al e trillas s cubillo e castineira on conjectures in orthocomplemented lattices artificial intelligence 117 2000 255 275 recently proposed a mathematical model for conjectures hypotheses and consequences abbr chcs and with this model we can execute certain mathematical reasoning and reformulate some important theorems in classical logic we demonstrate that the orthomodular condition is not necessary for holding watanabe s structure theorem of hypotheses and indeed in some orthocomplemented but not orthomodular lattices this theorem is still valid we use the chc operators to describe the theorem of deduction the theorem of contradiction and the lindenbaum theorem of classical logic and clarify their existence in the chc models a number of examples is presented and we re define the chc operators in residuated lattices and particularly reveal the essential differences between the chc operators in orthocomplemented lattices and residuated lattices

products of fuzzy finite state machines abstract we introduce the concepts of fuzzy transformation semigroups coverings cascade and wreath products for fuzzy finite state machines in order to overcome some of the difficulties which arise from the fuzzification of these concepts we also introduce the notions of polysemigroups and weak coverings

formal power series and regular operations on fuzzy languages in this paper we study formal power series over a quantale with coefficients in the algebra of all languages over a given alphabet and representation of fuzzy languages by these formal power series this representation generalizes the well known representation of fuzzy languages by their cut and kernel languages we show that regular operations on fuzzy languages can be represented by regular operations on power series which are defined by means of operations on ordinary languages we use power series in study of fuzzy languages which are recognized by fuzzy finite automata and deterministic finite automata and we study closure properties of the set of polynomials and the set of polynomials with regular coefficients under regular operations on power series

on covering of products of fuzzy finite state machines in this paper we have given several ways of constructing products of fuzzy finite state machines and their mutual relationship through isomorphism and coverings products called direct sum and sum of fuzzy finite state machines are introduced some algebraic properties of these products are investigated it is proved that the wreath product of two fuzzy finite state machines covers their cascade products similar relationship holds in direct sum and sum of two fuzzy finite state machines an interesting distributive property of cascade product over the sum of fuzzy finite state machines in relation to covering of fuzzy finite state machine is established

finite automata theory with membership values in lattices in this paper we study finite automata with membership values in a lattice which are called lattice valued finite automata the extended subset construction of lattice valued finite automata is introduced then the equivalences between lattice valued finite automata lattice valued deterministic finite automata and lattice valued finite automata with e moves are proved a simple characterization of lattice valued languages recognized by lattice valued finite automata is given then it is proved that the kleene theorem holds in the frame of lattice setting a minimization algorithm of lattice valued deterministic finite automata is presented in particular the role of the distributive law for the truth valued domain of finite automata is analyzed the distributive law is not necessary to many constructions of lattice valued finite automata but it indeed provides some convenience in simply processing lattice valued finite automata

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

myhill nerode type theory for fuzzy languages and automata the myhill nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid in this paper we develop a general myhill nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1 which are needed to take crisp languages in consideration we establish connections between extensionality of fuzzy languages w r t right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids and we prove the myhill nerode type theorem for fuzzy languages we also prove that each fuzzy language possess a minimal deterministic automaton recognizing it we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers in the second part of the paper we introduce and study nerode s and myhill s automata assigned to a fuzzy automaton with membership values in a complete residuated lattice the obtained results establish nice relationships between fuzzy languages fuzzy automata and deterministic automata

weighted finite automata over strong bimonoids we investigate weighted finite automata over strings and strong bimonoids such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold we define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions moreover we show that in this case the given weighted automata can be determinized

on quotient machines of a fuzzy automaton and the minimal machine in this paper the concept of substitution property sp partition for a finite fuzzy automaton is formulated and the quotient automaton with respect to an sp partition is defined the state minimization problem for such automata by using the method of quotient machines is solved

a theory of vague lattices based on many valued equivalence relations i general representation results the present work introduces a new and general theory of ordering relations and lattices based on many valued equivalence relations under the name vague ordering relations and vague lattices respectively representations and constructions of vague ordering relations and vague lattices are the main subjects of this paper and various desirable results in this direction are established furthermore algebraic characterizations of vague lattices are studied

dunkl spectral multipliers with values in umd lattices we show a h ormander spectral multiplier theorem for a a 0 otimes mathrm id y acting on the bochner space l p mathbb r d h 2 kappa y where a 0 is the dunkl laplacian h 2 kappa a weight function invariant under the action of a reflection group and y is a umd banach lattice we follow hereby a transference method developed by bonami clerc and dai xu passing through a marcinkiewicz multiplier theorem on the sphere we hereby generalize works for a 0 delta acting on l p mathbb r d dx by girardi weis hyt onen and others before we apply our main result to maximal regularity for cauchy problems involving a