permutable indistinguishability operators perfect vague groups and fuzzy subgroups permutability between t indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures it will be shown that the sup t product email protected f of two t indistinguishability operators is also a t indistinguishability operator if and only if e and f are permutable t indistinguishability operators i e email protected email protected e this property will be related to the study of fuzzy subgroups fuzzy normal subgroups and vague groups the aggregation of fuzzy subgroups will also be analyzed

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

a collapse theorem for holographic algorithms with matchgates on domain size at most 4 holographic algorithms with matchgates are a novel approach to design polynomial time computation they use kasteleyn s algorithm for perfect matchings and more importantly a holographic reduction the two fundamental parameters of a holographic reduction are the domain size k of the underlying problem and the basis size a holographic reduction transforms the computation to matchgates by a linear transformation that maps to a tensor product space of a linear space of dimension 2 we prove a sharp basis collapse theorem which shows that for domain size 3 and 4 all non trivial holographic reductions can be expressed with a basis of size 1 or 2 respectively the main proof techniques are matchgate identities and a group property of matchgate signatures

modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator this work shows a method to obtain a hierarchy of partitions on the universe 0 1 in such a way that each of them is compatible with a refinement of lukasiewicz indistinguishability operator the classes of the partition at a given level present a relation of antonymy between them moreover the partition at a certain level can be seen as the refinement of a previous level by means of a class of linguistic modifiers due to this fact they seem appropriate for modeling linguistic labels of a linguistic variable the associated indistinguishability operators show the increasing granularity when the number of classes rises up

aggregation of fuzzy subgroups abstract given an aggregation function and two fuzzy subgroups of a group this study investigates whether the aggregation of them is also a fuzzy subgroup it is proven that if the cardinal of the group is a prime power then it is always a fuzzy subgroup for a general group it is proven that the existence of a binary relation between the fuzzy subgroups determines if their aggregation is a fuzzy subgroup moreover given two fuzzy subgroups if the aggregation of them is a fuzzy subgroup for some strictly monotone aggregation function then it is a fuzzy subgroup for any aggregation function the present study also examines conditions on the aggregation function that characterise the case where the aggregation of two arbitrary fuzzy subgroups is also a fuzzy subgroup

an effective dichotomy for the counting constraint satisfaction problem bulatov proceedings of the 35 th international colloquium on automata languages and programming part 1 lecture notes in comput sci 5125 springer new york 2008 pp 646 661 gave a dichotomy for the counting constraint satisfaction problem csp a problem from csp is characterized by a constraint language gamma a fixed finite set of relations over a finite domain d an instance of the problem uses these relations to constrain an arbitrarily large finite set of variables bulatov showed that the problem of counting the satisfying assignments of instances of any problem from csp is either in polynomial time fp or is p complete his proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field we give an elementary proof of bulatov s dichotomy based on succinct representations which we call frames of a class of highly structured relations which we call strongly rectangular we show that these are precisely the relation

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

a dichotomy theorem for constraint satisfaction problems on a 3 element set the constraint satisfaction problem csp provides a common framework for many combinatorial problems the general csp is known to be np complete however certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes there is therefore a fundamental research direction aiming to separate those subclasses of the csp that are tractable and those which remain np complete schaefer gave an exhaustive solution of this problem for the csp on a 2 element domain in this article we generalise this result to a classification of the complexity of the csp on a 3 element domain the main result states that every subproblem of the csp is either tractable or np complete and the criterion separating them is that conjectured in bulatov et al 2005 and bulatov and jeavons 2001b we also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure finally we exhibit a polynomial time algorithm which for a given set of allowed constraints outputs if this set gives rise to a tractable problem class to obtain the main result and the algorithm we extensively use the algebraic technique for the csp developed in jeavons 1998b bulatov et al 2005 and bulatov and jeavons 2001b

on counting homomorphisms to directed acyclic graphs it is known that if p and np are different then there is an infinite hierarchy of different complexity classes that lie strictly between them thus if p np it is not possible to classify np using any finite collection of complexity classes this situation has led to attempts to identify smaller classes of problems within np where dichotomy results may hold every problem is either in p or is np complete a similar situation exists for counting problems if p p there is an infinite hierarchy in between and it is important to identify subclasses of p where dichotomy results hold graph homomorphism problems are a fertile setting in which to explore dichotomy theorems indeed feder and vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their long standing dichotomy conjecture for all constraint satisfaction problems in this article we give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs let h be a fixed directed acyclic graph the problem is given an input digraph g determine how many homomorphisms there are from g to h we give a graph theoretic classification showing that for some digraphs h the problem is in p and for the rest of the digraphs h the problem is p complete an interesting feature of the dichotomy which is absent from previously known dichotomy results is that there is a rich supply of tractable graphs h with complex structure

on the construction of point processes in statistical mechanics we present a new approach to the construction of point processes of classical statistical mechanics as well as processes related to the ginibre bose gas of brownian loops and to the dissolution in rd of ginibre s fermi dirac gas of such loops this approach is based on the cluster expansion method we obtain the existence of gibbs perturbations of a large class of point processes moreover it is shown that certain limiting gibbs processes are gibbs in the sense of dobrushin lanford and ruelle if the underlying potential is positive finally gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive

algebras of fuzzy sets in this paper we investigate two kinds of algebras of fuzzy sets which are obtained by using zadeh s extension principle we give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets we prove that if the structure of truth values is a complete residuated lattice the induced algebra of a subalgebra of an algebra a can be embedded into the induced algebra of fuzzy sets of a for direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product in the case of homomorphisms and direct products the two kinds of algebras of fuzzy sets behave in different ways

the generalized associative law in vague groups and its applications ii as a continuation of part i vague integral powers of elements in vague groups and their representation properties are introduced in this paper thereafter some rudimentary algebraic properties of vague integral powers of elements obtained from the generalized vague associative law formulated in part i are established the present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic for this reason special attention is also paid to the calculation of integral powers and multiples of real numbers in vague arithmetic and some practical applications related to the discrete structure of measurement instruments are also given

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

fuzzy groups fuzzy functions and fuzzy equivalence relations to each fuzzy subgroup a fuzzy equivalence relation is associated and it is proved that a fuzzy subgroup is normal if and only if the operation of the group is compatible with its associated fuzzy equivalence relation in the definition of fuzzy subgroup only the subset is fuzzy whilst the group operation remains crisp in an opposite direction another structure can be constructed maintaining crisp the basic set x of the group and fuzzifying the group operation when in addition it is imposed to the fuzzy operation to be compatible with a given fuzzy equivalence relation this structure is called a vague group the results of the paper allow us to associate a vague group to every fuzzy subgroup in such a way that it can be interpreted as the fuzzy quotient group x special attention is paid to normal fuzzy groups and vague groups of the additive group r of the real numbers two examples related to triangular numbers illustrating the results of the paper are provided

computation of the greatest right and left invariant fuzzy quasi orders and fuzzy equivalences abstract right invariant fuzzy quasi orders for fuzzy automata are broadly studied in the recent literature as they arise as solutions to particular systems of fuzzy relation equations and inequalities some of their applications include determinization and state reduction procedures as well as simulations and bisimulations for fuzzy automata in this paper we provide a procedure for computing the greatest right invariant fuzzy quasi order for a given fuzzy automaton over a complete residuated lattice the proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite when the previous condition is not satisfied we show that the greatest right invariant fuzzy quasi order can be obtained by taking the limit value of the convergent array of fuzzy quasi orders for fuzzy automata over bl algebras on the real unit interval 0 1 analogous procedures for computing the greatest left invariant fuzzy quasi order as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented in addition the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented

the complexity of approximating bounded degree boolean csp the degree of a csp instance is the maximum number of times that any variable appears in the scopes of constraints we consider the approximate counting problem for boolean csp with bounded degree instances for constraint languages containing the two unary constant relations 0 and 1 when the maximum allowed degree is large enough at least 6 we obtain a complete classification of the complexity of this problem it is exactly solvable in polynomial time if every relation in the constraint language is affine it is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of 0 1 and binary implication otherwise there is no fpras unless np rp for lower degree bounds additional cases arise where the complexity is related to the complexity of approximately counting independent sets in hypergraphs

construction of point processes for classical and quantum gases we propose a construction of point processes via the method of cluster expansion the important role of the class of infinitely divisible point processes is noted examples are permanental and determinantal processes as well as the classical gibbs point processes where the interaction is given by a stable and regular pair potential

on the relationship between fuzzy subgroups and indistinguishability operators abstract fuzzy subgroups are revisited considering their close relationship with indistinguishability operators fuzzy equivalences invariant under translations different ways to obtain new fuzzy subgroups from a given one are provided and different ways to characterize normal fuzzy subgroups are obtained the idea of double coset of two crisp subgroups allows us to relate them via their equivalence classes this is generalized to the fuzzy framework the conditions in which a fuzzy relation r on a group g can be considered a fuzzy subgroup of g g are obtained

construction of finite l groups abstract in this paper we give some necessary and sufficient conditions for a fuzzy relation to be a compatible fuzzy equality of some l group in the sense of b lohlavek these characterizations enable practical applications for construction of l groups with a given skeleton the construction has two steps determination of the general form of fuzzy pre equalities on a group and then solving a system of inequalities in the given residuated lattice some concrete examples of construction of finite l groups with the given skeleton are presented

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

fuzzy submonoids fuzzy preorders and quasi metrics let l be a residuated lattice then given a monoid m we define a galois connection from the lattice of the compatible l preorders in m and the lattice of l submonoids of m given a set s we define a galois connection between the lattice of the l preorders in s and the lattice of l submonoids of the monoid ss i a link with the notion of quasi metric is also established

on a monadic np vs monadic co np abstract it is a well known result of fagin that the complexity class np coincides with the class of problems expressible in existential second order logic 11 monadic np is the class of problems expressible in monadic 11 i e 11 with the restriction that the second order quantifiers range only over sets as opposed to ranging over say binary relations we prove that connectivity of finite graphs is not in monadic np even in the presence of arbitrary built in relations of moderate degree that is degree log n o 1 this extends earlier results of fagin and de rougemont our proof uses a combination of three techniques 1 an old technique of hanf for showing that two infinite structures agree on all first order sentences under certain conditions 2 a recent new approach to second order ehrenfeucht fraisse games by ajtai and fagin and 3 playing ehrenfeucht fraisse games over random structures this was also used by ajtai and fagin regarding 1 we give a version of hanf s result that is better suited for use as a tool in inexpressibility proofs for classes of finite structures the power of these techniques is further demonstrated by using them actually using just the first two techniques to give a very simple proof of the separation of monadic np from monadic co np without the presence of built in relations

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

t fuzzy mapping abstract we introduce the concepts of t fuzzy mapping surjective t fuzzy mapping injective t fuzzy mapping t fuzzy monomorphism t fuzzy epimorphism and t fuzzy partion and we prove some of their properties

fuzzy functions and their fundamental properties in this paper it is noticed that the definition of fuzzy equality of sasaki 1993 is not concordant with that of es and coker 1995 and sostak 1985 1988 furthermore the construction of fuzzy equality and fuzzy function is far from the way of natural fuzzification for this purpose the fuzzy equality of two points of a crisp set which is more convenient than that in es and coker 1995 and sostak 1985 1988 and the fuzzy function having weaker conditions than that is given in sasaki 1993 are redefined in much more natural way then various types of fuzzy functions are introduced and their fundamental properties are established

vague groups and generalized vague subgroups on the basis of 0 1 le in this paper various elementary properties of vague groups and some properties of vague binary operations related with their associativity aspects are obtained furthermore the concept of vague isomorphism is defined and some basic properties of this concept are studied the concept of external direct product of vague groups is established later the definition of generalized vague subgroup which is a generalization of the vague subgroup defined by demirci is introduced and the validity of some classical results in this setting is investigated on the basis of the particular integral commutative complete quasi monoidal lattice 0 1

constant query testability of assignments to constraint satisfaction problems for each finite relational structure a let csp a denote the csp instances whose constraint relations are taken from a the resulting family of problems csp a has been considered heavily

fuzzy t transitive relations eigenvectors and generators abstract after some preliminaries the set of generators of a generalized equality relation t indistinguishability operator is studied this set is identified with the set of the eigenvectors of the relation the relation between the fuzzy and metric topologies derived from these equalities is stablished the concept of basis is introduced and the construction of a procedure in order to calculate explicitly a basis of a t indistinguishability operator for t archimedean is proposed

preserving fuzzy subgroups and indistinguishability operators abstract in this paper we study the class of all fuzzy subgroups defined with respect to a given t norm on a fixed group we find necessary and sufficient conditions over two t norms to guarantee that the class of all fuzzy subgroups induced by the first t norm is contained in the class of all fuzzy subgroups with respect to by the second one we characterize the functions that transform fuzzy subgroups into fuzzy subgroups the close relationship between indistinguishability operators and fuzzy subgroups allows us to obtain similar results for some classes of indistinguishability operators

a simple algorithm for mal tsev constraints a mal tsev operation is a ternary operation varphi that satisfies the identities varphi x y y varphi y y x x constraint satisfaction problems involving constraints invariant under a mal tsev operation constitute an important class of constraint satisfaction problems which includes the affine satisfiability problem subgroup and near subgroup constraints and many others it is also known that any tractable case of the counting constraint satisfaction problem involves only mal tsev constraints the first algorithm solving the arbitrary constraint satisfaction problem with mal tsev constraints has been given by bulatov however this algorithm is very sophisticated and relies heavily on advanced algebraic machinery in this paper we give a different and much simpler algorithm for this type of constraint

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

beyond csp a dichotomy for counting weighted eulerian orientations with ars abstract we define and explore a notion of unique prime factorization for constraint functions and use this as a new tool to prove a complexity classification for counting weighted eulerian orientation problems with arrow reversal symmetry ars we prove that all such problems are either polynomial time computable or p hard we show that the class of weighted eulerian orientation problems subsumes all weighted counting constraint satisfaction problems csp on boolean variables more significantly we establish a novel connection between csp and counting weighted eulerian orientation problems that is global in nature this connection is based on a structural determination of all half weighted affine linear subspaces over z 2 which is proved using mobius inversion

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

the complexity of weighted boolean csp this paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted boolean constraint satisfaction problem the problem is parameterized by a finite set mathcal f of nonnegative functions that may be used to assign weights to the configurations feasible solutions of a problem instance classical constraint satisfaction problems correspond to the special case of 0 1 valued functions we show that computing the partition function i e the sum of the weights of all configurations is text sf fp text sf p complete unless either 1 every function in mathcal f is of product type or 2 every function in mathcal f is pure affine in the remaining cases computing the partition function is in p

complexity of counting csp with complex weights we give a complexity dichotomy theorem for the counting constraint satisfaction problem csp in short with algebraic complex weights to this end we give three conditions for its tractability let f be any finite set of algebraic complex valued functions defined on an arbitrary finite domain we show that csp f is solvable in polynomial time if all three conditions are satisfied and is p hard otherwise our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination a the problem of counting graph homomorphisms is the special case when f has a single symmetric binary function dyer and greenhill 2000 bulatov and grohe 2005 goldberg et al 2010 cai et al 2013 b the problem of counting directed graph homomorphisms is the special case when f has a single but not necessarily symmetric binary function dyer et al 2007 cai and chen 2010 c the unweighted form of csp is when all functions in f take values in 0 1 bulatov 2008 dyer and richerby 2013

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

equivalence operators in nilpotent systems a consistent connective system generated by nilpotent operators is not necessarily isomorphic to the ukasiewicz system using more than one generator function consistent nilpotent connective systems so called bounded systems can be obtained with the advantage of three naturally derived negation operators and thresholds in this paper equivalences in bounded systems are examined here three different types of operators are studied and a paradox of the equivalence i e there is no equivalence relation in a non boolean setting which fulfils x e x x 1 and e x n x 0 is resolved by aggregating the implication based equivalence and its dual operator we will also show that the aggregated equivalence has nice properties like associativity threshold transitivity and t transitiviy

holographic algorithm with matchgates is universal for planar csp over boolean domain we prove a complexity classification theorem that classifies all counting constraint satisfaction problems csp over boolean variables into exactly three classes 1 polynomial time solvable 2