comparison of fuzzy numbers based on the probability measure of fuzzy events abstract most approaches for ranking fuzzy numbers proposed in the literature are based on fuzzy sets theory only and suffer from lack of discrimination and occasionally conflict with intuition it is true that fuzzy numbers are frequently partial order and cannot be compared however this does not alleviate the need for comparison in practical applications in this paper the order of fuzzy numbers are determined based on the concept of probability measure of fuzzy events due to zadeh it considers both the mean and dispersion of alternatives and gives two groups of indices based on the uniform and the proportional probability distributions the approach is also extended to the comparison of random fuzzy numbers by means of a mean fuzzy number it is shown that several comparison indices in the literature can be obtained based on the probability present measure approach finally some typical examples are used to compare the various different approaches the different interpretations of the dispersion index under different physical situations are emphasized

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

capturing complexity classes by fragments of second order logic abstract we investigate the expressive power of certain fragments of second order logic on finite structures the fragments are second order horn logic second order krom logic as well as a symmetric and a deterministic version of the latter it is shown that all these logics collapse to their existential fragments in the presence of successor relation they provide characterizations of polynomial time deterministic and nondeterministic logspace and of the complement of symmetric logspace without successor relation these logics still can express certain problems that are complete in the corresponding complexity classes but on the other hand they are strictly weaker than previously known logics for these classes and fail to express some very simple properties

hereditarily finite sets data bases and polynomial time computability abstract the first informal result and the aim of this paper which is the extended version of sazonov 1988 is a step towards a somewhat more practically oriented version of set theoretic programming language sazonov 1985 1987 which may be considered as a resource bounded language for specifying data bases and corresponding queries in fact instead of the ordinary universe hf u of hereditarily finite sets over any given class u of urelements we consider following the ideas of red ko and basarab 1987 a more general universe hfa u of hereditarily finite sets of named elements i e sets of data qualified by some attributes the second result is more mathematical it generalizes a theorem of sazonov 1985 1987 to the case of hfa u makes it more precise and implies that language is sufficiently complete theorem cf 42 43 if language involves a linear order on u then its terms define exactly all operations hfa u hfa u which 1 preserve supports of their arguments 2 transform isomorphic arguments to isomorphic ones respecting their supports and 3 are computable in polynomial time under graphical representations of sets in hfa u

asymptotic conditional probabilities the unary case motivated by problems that arise in computing degrees of belief we consider the problem of computing asymptotic conditional probabilities for first order sentences given first order sentences phi and theta we consider the structures with domain 1 ldots n that satisfy theta and compute the fraction of them in which phi is true we then consider what happens to this fraction as n gets large this extends the work on 0 1 laws that considers the limiting probability of first order sentences by considering asymptotic conditional probabilities as shown by liogon kii math notes acad ussr 6 1969 pp 856 861 and by grove halpern and koller res rep rj 9564 ibm almaden research center san jose ca 1993 in the general case asymptotic conditional probabilities do not always exist and most questions relating to this issue are highly undecidable these results however all depend on the assumption that theta can use a nonunary predicate symbol liogonkii math notes acad ussr 6 1969 pp 856 861 shows that if we condition on formulas theta involving unary predicate symbols only but no equality or constant symbols then the asymptotic conditional probability does exist and can be effectively computed this is the case even if we place no corresponding restrictions on phi we extend this result here to the case where theta involves equality and constants we show that the complexity of computing the limit depends on various factors such as the depth of quantifier nesting or whether the vocabulary is finite or infinite we completely characterize the complexity of the problem in the different cases and show related results for the associated approximation problem

multi objective design optimization using cascade evolutionary computations the consideration of uncertainties in conjunction with the probability of violation of the constraints imposed by the design codes is examined in the framework of structural optimization the optimum design achieved based on a deterministic formulation is compared in terms of the optimum weight the probability of violation of the constraints and the probability of failure with the optimum designs achieved through a robust design formulation where the variance of the response is considered as an additional criterion the stochastic finite element problem is solved using the monte carlo simulation method combined with the latin hypercube sampling technique for improving its computational efficiency a non dominant cascade evolutionary algorithm based methodology is adopted for the solution of the multi objective optimization problem encountered in order to obtain the global pareto front curve

counting modulo quantifiers on finite structures we give a combinatorial method for proving elementary equivalence in first order logic fo with counting modulo n quantifiers dn inexpressibility results for fo dn with built in linear order are also considered for instance the class of linear orders of length divisible by n 1 cannot be expressed in fo dn using this result we prove that comparing cardinalities or connectivity of ordered graphs are not definable in fo dn we also show that the height of complete n ary trees cannot be expressed in fo dn with linear order interpreting the predicate y nx as a complete n ary tree we show that the predicate y px cannot be defined in fo dn with linear order whenever p has a prime factor that does not divide n this solves the problem raised by niwi ski and stolboushkin lics 93 we also discuss a connection between our results and the well known open problem in circuit complexity theory whether acc nc1

expressiveness of concept expressions in first order description logics we introduce a method for characterizing the expressive power of concept expressions in first order description logics the method is essentially model theoretic in nature in that it gives preservation results uniquely identifying a wide range of description logics as fragments of first order logic the languages studied in the paper all belong to the well known email protected email protected and plane1d 49c hierarchies

logical hierarchies in ptime we consider the problem of finding a characterization for polynomial time computable queries on finite structures in terms of logical definability it is well known that fixpoint logic provides such a characterization in the presence of a built in linear order but without linear order even very simple polynomial time queries involving counting are not expressible in fixpoint logic our approach to the problem is based on generalized quantifiers a generalized quantifier isn ary if it binds any number of formulas but at mostnvariables in each formula we prove that for each natural numbern there is a query on finite structures which is expressible in fixpoint logic but not in the extension of first order logic by any set ofn ary quantifiers it follows that the expressive power of fixpoint logic cannot be captured by adding finitely many quantifiers to first order logic furthermore we prove that for each natural numbern there is a polynomial time computable query which is not definable in any extension of fixpoint logic byn ary quantifiers in particular this rules out the possibility of characterizing ptime in terms of definability in fixpoint logic extended by a finite set of generalized quantifiers

on nonlinear equations for fuzzy mappings in probabilistic normed spaces in this paper we introduce the concept of probabilistic contractor couple which simplifies and weakens that of probabilistic g contractor given by cho et al fuzzy sets and systems 110 2000 115 122 by using this concept we study the existence and iteration convergence of solutions for set valued nonlinear operator equations and nonlinear equations of fuzzy mappings in menger pn spaces the results presented in this paper improve and extend the corresponding results of cho et al in a sense

random ambiguous point of random k ω set contractive operator in this paper we introduce a new concept of random ambiguous point of random operator and investigate some problems of random ambiguous point of random k set contractive operator and obtain some new results of random k set contractive operator

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

a restricted second order logic for finite structures we introduce a restricted version of second order logic so in which the second order quantifiers range over relations that are closed under the equivalence relation k of k variable equivalence for some k this restricted second order logic is an effective fragment of the infinitary logic l which differs from other such fragments in that it is not based on a fixpoint logic we explore the relationship of so with fixpoint logics showing that its inclusion relations with these logics are equivalent to problems in complexity theory we also look at the expressibility of np complete problems in this logic

a logical approach to asymptotic combinatorics i first order properties we shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics this framework provides methods for determining the probability that a large finite structure ran domly chosen from a given class will have a given property our main concern is the asymptotic probability the limiting value as the size of the structure increases for example a common problem in elementary probability texts is to show that the asymptotic probabity that a per mutation will have no fixed point is l e we shall say nothing about the closely related problem of determining rates of convergence although the methods presented here may extend to such problems to develop a general approach we must fix a language for specifying properties of structures thus our approach is logical logic is the branch of mathematics that deals with problems of language in this paper we con sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties in the sequel to this paper we consider properties expressible in the more general language of monadic second order logic clearly we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant therefore we choose to consider only classes closed under disjoint unions and components see sect 1 for definitions this condition is general enough to include the classes of graphs permutations unary functions and many other important examples the main theorems of the paper are about classes satisfying this condition theorems 5 7 and 5 8 state that the existence of asymptotic probabilities for a special set of sentences called component bounded sentences defined in sect 5 is equivalent to a con dition on the growth of such a class theorem 5 9 gives necessary and suf ficient conditions for every first order sentence to have an asymptotic probability of 0 or 1 in nonfast growing classes in section 8 we present a long list of examples from the literature

describing parameterized complexity classes we describe parameterized complexity classes by means of classical complexity theory and descriptive complexity theory for every classical complexity class we introduce a parameterized analogue in a natural way in particular the analogue of polynomial time is the class of all fixed parameter tractable problems we develop a basic complexity theory for the parameterized analogues of classical complexity classes and give among other things complete problems and logical descriptions we then show that most of the well known intractable parameterized complexity classes are not analogues of classical classes nevertheless for all these classes we can provide natural logical descriptions

bounded linear logic a modular approach to polynomial time computability typing is a way of describing the interactive behavior of algorithms usual typing systems are mainly concerned with input output specifications e g given terms f a b and a a the computation of f a by normalization yields a result of type b however one can dream of more refined typings that would not only ensure ethereal termination but would for instance yield feasible resource bounds it seems that time complexity does not lend itself naturally to modular manipulation we seek something more primitive

relational queries over interpreted structures we rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries we identify parts of the classical theory that go through as before when interpreted structure is present parts that go through only for classes of nicely behaved structures and parts that only arise in the interpreted case the first category include a number of results on language equivalence and expressive power characterizations for the active domain semantics for a variety of logics under this semantics quantifiers range over elements of a relational database the main kind of results we prove here are generic collapse results for generic queries adding operations beyond order does not give us extra power the second category includes results on the natural semantics under which quantifiers range over the entire interpreted structure we prove for a variety of structures natural active collapse results showing that using unrestricted quantification does not give us any extra power moreover for a variety of structures including the real field we give a set of algorithms for eliminating unbounded quantifications in favor of bounded ones furthermore we extend these collapse results to a new class of higher order logics that mix unbounded and bounded quantification we give a set of normal forms for these logics under special conditions on the interpreted structures as a by product we obtain an elementary proof of the fact that parity test is not definable in the relational calculus with polynomial inequality constraints we also give examples of structures with nice model theoretic properties over which the natural active collapse fails

perception based granular probabilities in risk modeling and decision making our focus is on the issue of decision making in risky situations we discuss the need for using decision functions to aid in capturing the decision maker s preference among these types of uncertain alternatives the use of fuzzy rule based formulations to model these functions is investigated we discuss the role of perception based granular probability distributions as a means of modeling the uncertainty profiles of the alternatives various properties of this method of describing uncertainty are provided tools for evaluating rule based decision functions in the face of perception based uncertainty profiles are presented consideration is given to the situation in which our uncertainty profiles are expressed in terms of a cumulative distribution function we introduce the idea of a perception based granular cumulative distribution and describe its representation in terms of a fuzzy rule based model

on the relative expressiveness of description logics and predicate logics abstract it is natural to view concept and role definitions in description logics as expressing monadic and dyadic predicates in predicate calculus we show that the descriptions built using the constructors usually considered in the dl literature are characterized exactly as the predicates definable by formulas in tl3 the subset of first order predicate calculus with monadic and dyadic predicates which allows only three variable symbols in order to handle number bounds we allow numeric quantifiers and for transitive closure of roles we use infinitary disjunction using previous results in the literature concerning languages with limited numbers of variables we get as corollaries the existence of formulas of fopc which cannot be expressed as descriptions we also show that by omitting role composition descriptions express exactly the formulas in tl2 which is known to be decidable

locality from circuit lower bounds we study the locality of an extension of first order logic that captures graph queries computable in ac 0 i e by families of polynomial size constant depth circuits the extension considers first order formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the particular interpretation of the numerical predicates we refer to such formulas as arb invariant first order we consider the two standard notions of locality gaifman and hanf locality our main result gives a gaifman locality theorem an arb invariant first order formula cannot distinguish between two tuples that have the same neighborhood up to distance log n c where n represents the number of elements in the structure and c is a constant depending on the formula when restricting attention to string structures we achieve the same quantitative strength for hanf locality in both cases we show that our bounds are tight we also present an applicati

research on some problems for nonlinear operators abstract in this paper we discuss some problems for nonlinear operators in the menger pn space meanwhile we prove existence theorems for the solution of an operator equation and obtain some new results by means of the theory of topological degree for the z p s space

inductive definitions over finite structures inductive definitions have played a central role in the foundations of mathematics for over a century they were used in the 1970s as the backbone of one major generalization of recursive function theory moschovakis 1974 aczel 1977 in recent years the relevance of inductive definitions in particular over finite structures to database theory to descriptive computational complexity and to logics of programs has been recognized a seminal paper on inductive definitions in database theory is chandra and hare1 1982 where they define a hierarchy of queries over finite structures within which minor steps successor ordinals correspond to first order quantifier alternations and major steps limit ordinals correspond to uses of lixpoints they left open the question of whether the hierarchy remains strict above the first major step level w this problem was answered in the negative by immerman 1986 since the collection of first order lixpoint queries over finite structures is closed under composition and first order operations other than negation moschovakis 1974 the main component of immerman s solution was

the expressiveness of a family of finite set languages abstract in this paper we characterize exactly the complexity of a set based database language called srl which presents a unified framework for queries and updates by imposing simple syntactic restrictions on it we are able to express exactly the classes p and logspace we also discuss the role of ordering in database query languages and show that the hom operator of machiavelli language in ohori et al 1989 does not capture all the order independent properties

fixpoint logics relational machines and computational complexity we establish a general connection between fixpoint logic and complexity on one side we have fixpoint logic parameterized by the choices of 1st order operators inflationary or noninflationary and iteration constructs deterministic nondeterministic or alternating on the other side we have the complexity classes between p and exptime our parameterized fixpoint logics capture the complexity classes p np pspace and exptime but equally is achieved only over ordered structures there is however an inherent mismatch between complexity and logic while computational devices work on encodings of problems logic is applied directly to the underlying mathematical structures to overcome this mismatch we use a theory of relational complexity which bridges the gap between standard complexity and fixpoint logic on one hand we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complexity classes on the other hand the expressive power of fixpoint logic can be precisely characterized in terms of relational complexity classes this tight three way relationship among fixpoint logics relational complexity and standard complexity yields in a uniform way logical analogs to all containments among the complexity classes p np pspace and exptime the logical formulation shows that some of the most tantalizing questions in complexity theory boil down to a single question the relative power of inflationary vs noninflationary 1st order operators

complexity of the first order theory of almost all finite structures a first order sentence of a relational type l is true almost everywhere if the proportion of its models among the structures of type l and cardinality m tends to 1 when m tends to it is shown that th l the set of sentences of type l true almost everywhere is complete in pspace further various upper and lower bounds of the complexity of this theory are obtained for example if the arity of the relation symbols of l is d 2 and if pr th l is the set of prenex sentences of th l then pr th l dspace n log n d and pr th l ntime o n log n d if r is a binary relation symbol and l r th l is the theory of almost all graphs then pr th l nspace o n log n these results are optimal modulo open problems in complexity such as ntime t dspace t and nspace s dspace s2

infinitary logic and inductive definability over finite structures the extensions of first order logic with a least fixed point operator fo lfp and with a partial fixed point operator fo pfp are known to capture the complexity classes p and pspace respectively in the presence of an ordering relation over finite structures recently abiteboul and vianu in proceedings of the 23rd acm symposium on the theory of computing 1991 investigated the relationship of these two logics in the absence of an ordering using a machine model of generic computation in particular they showed that the two languages have equivalent expressive power if and only if p pspace these languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables l see for instance kolaitis and vardi in proceedings of the 5th ieee symposium on logic in computer science pp 156 167 1990 we investigate this logic on finite structures and provide a normal form for it we also present a treatment of abiteboul and vianu s results from this point of view in particular we show that we can write a formula of fo lfp that defines an ordering of the lk types uniformly over all finite structures one consequence of this is a generalization of the equivalence of fo lfp and p from ordered structures to classes of structures where every element is definable we also settle a conjecture mentioned by abiteboul and vianu by showing that fo lfp is properly contained in the polynomial time computable fragment of l raising the question of whether the latter fragment is a recursively enumerable class

a new fuzzy operator and its application to topology design of distributed local area networks topology design of distributed local area networks can be classified as a hard combinatorial optimization problem the problem has several conflicting objectives such as cost reliability network delay and number of hops between source and destination these objectives can conveniently be expressed in linguistic terms a key component of fuzzy logic this paper presents an approach based on fuzzy logic that combines the conflicting objectives into a single optimization function a new fuzzy operator namely the unified and or uao operator is also proposed and a decision making approach based on fuzzy rules and preference rules is introduced the uao operator is empirically compared with the well known ordered weighted averaging owa operator through application to an evolutionary algorithm results show that the uao operator exhibits comparatively better performance than the owa operator

existential second order logic over strings existential second order logic eso and monadic second order logic mso have attracted much interest in logic and computer science eso is a much expressive logic over successor structures than mso however little was known about the relationship between msoand syntatic fragments of eso we shed light on this issue by completely characterizing this relationship for the prefix classes of eso over strings i e finite successor structures moreover we determine the complexity of model checking over strings for all eso prefix classes let eso q denote the prefix class containing all sentences of the shape r q 4 where r is a list of predicate variables q is a first order predicate qualifier from the prefix set q and 4 is quantifier free we show that eso and eso are the maximal standard eso prefix classes contained in mso thus expressing only regular languages we further prove the following dichotomy theorem an eso prefix class either expresses only regular languages and is thus in mso or it expresses some np complete languages we also give a precise characterization of those eso prefix classes that are equivalent to mso over strings and of the eso prefix classes which are closed under complementation on strings

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

an analysis of fixed point queries on binary trees lindell s an analysis of fixed point queries on binary trees theoretical computer science 85 1991 75 95 the presence of ordering appears to play an essential role in the logical expressibility of polynomialtime queries on finite structures by examining the expressibility and complexity of inductive queries on the class of complete unordered binary trees we are able to show that the ability to calculate cardinality is strictly less powerful than the assumption of order

from almost optimal algorithms to logics for complexity classes via listings and a halting problem let c denote one of the complexity classes polynomial time logspace or nondeterministic logspace we introduce a logic l c inv and show generalizations and variants of the equivalence l c inv captures c if and only if there is an almost c optimal algorithm in c for the set taut of tautologies of propositional logic these statements are also equivalent to the existence of a listing of subsets in c of taut by corresponding turing machines and equivalent to the fact that a certain parameterized halting problem is in the parameterized complexity class xcuni

d languages for sets and logspace computable graph transformers abstract we discuss several versions of a set theoretic language as a reasonable prototype for nested database query language where database states and queries are considered respectively as hereditarily finite sets and set theoretic operations in a previous work such a language exactly corresponding to ptime computability was introduced it is supposed that hf sets are naturally presented by vertices of acyclic graphs here we consider a number of languages for sub ptime computable set operations via corresponding graph transformers two such languages lead to a notion of nlogspace and respectively dlogspace computable queries over hf which appear the most natural at our present knowledge among others considered here unlike the flat relational databases the problem of finding sufficiently good corresponding approach for hf proves to be more intricate and furthermore gives rise to some interesting questions in finite model theory cf section 13

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

random fixed points of set valued maps some random fixed point theorems for set valued random operators under very mild conditions are established some recent results of o regan proc amer math soc 126 1998 3045 3053 and computers math applic 35 1998 27 34 are improved significantly the discussion in this paper underlines in addition to generality the unifying aspects of our result

equivalence and normal forms for the restricted and bounded fixpoint in the nested algebra the nested model is an extension of the traditional flat relational model in which relations can also have relation valued entries its default query language the nested algebra is rather weak unfortunately since it is only a conservative extension of the traditional flat relational algebra and thus can express only a small fraction of the polynomial time queries therefore it was proposed to extend the nested algebra with a fixpoint construct but the resulting language turned out to be too powerful many inherently exponential queries could also be expressed two polynomial time restrictions of the fixpoint closure of the nested algebra were proposed the restricted fixpoint closure by gyssens and van gucht and the bounded fixpoint closure by suciu here we prove two results first we show that both restrictions are equivalent in expressive power the proof technique relies on known encodings of nested relations into flat ones and on a novel technique called type substitution by which we reduce the equivalence of the two restrictions to its obvious counterpart in the flat relational model second we prove that both the bounded fixpoint queries and the restricted fixpoint queries admit normal forms in which the fixpoint occurs exactly once the proof technique relies on a novel encoding method of nested relations into flat ones

the computational complexity of asymptotic problems 1 partial orders the class of partial orders is shown to have ol laws for first order logic and for inductive fixed point logic a logic which properly contains first order logic this means that for every sentence in one of these logics the proportion of labeled or unlabeled partial orders of size n satisfying the sentence has a limit of either 0 or 1 as n goes to co this limit called the asymptotic probability of the sentence is the same for labeled and unlabeled structures the computational complexity of the set of sentences with asymptotic probability 1 is determined for first order logic it is pspace complete for inductive fixed point logic it is exptime complete

calculations of random fixed point index in this paper we prove an important inequality and investigate some new calculating problems of random fixed point index and generalize famous theorem by means of the theory of random fixed point index

random graphs and the parity quantifier the classical zero one law for first order logic on random graphs says that for every first order property p in the theory of graphs and every p 0 1 the probability that the random graph g n p satisfies p approaches either 0 or 1 as n approaches infinity it is well known that this law fails to hold for any formalism that can express the parity quantifier for certain properties the probability that g n p satisfies the property need not converge and for others the limit may be strictly between 0 and 1 in this work we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier foltr over g n p thus eluding the above hurdles specifically we establish the following modular convergence law for every foltr sentence p there are two explicitly computable rational numbers a0 a1 such that for i 0 1 as n approaches infinity the probability that the random graph g 2npi p satisfies p approaches ai our results also extend appropriately to fo equipped with modq quantifiers for prime q in the process of deriving this theorem we explore a new question that may be of interest in its own right specifically we study the joint distribution of the subgraph statistics modulo 2 of g n p namely the number of copies mod 2 of a fixed number of graphs f1 fe of bounded size in g n p we first show that every foltr property p is almost surely determined by subgraph statistics modulo 2 of the above type next we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n mod 2 and we determine this limiting distribution completely interestingly both these steps are based on a common technique using multivariate polynomials over finite fields and in particular on a new generalization of the gowers norm the first step is analogous to the razborov smolensky method for lower bounds for ac0 with parity gates yet stronger in certain ways for instance it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every foltr sentence which is something that is not known for ac0ltr

some new theorems for cyclic contractions in gb metric spaces and some applications abstract in this paper some new theorems for various cyclic contractions are established in gb metric spaces to discuss the existence and uniqueness of the solutions for a class of the operator equations which generalize many known results in corresponding literatures on the other hand we introduce the notion of the cyclic contractive mappings in gb metric spaces and establish a new fixed point theorem which is applied to consider the existence of the solutions of the integral equations and ordinary differential equations moreover some examples are given to support our main results

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

on the complexity of single rule datalog queries datalog programs containing a unique rule and possibly some facts are known as single rule programs or sirups we study the complexity of evaluating sirups over variable and fixed databases respectively as well as the descriptive complexity of sirups i e their expressive power in all cases it turns out that even very restricted classes of sirups have the same complexity and essentially the same expressive power as general datalog programs in particular the evaluation of single clause programs is exptime complete combined complexity and if restricted to linear recursive rules pspace complete moreover sirups with one recursive rule and one fact capture ptime on ordered structures if a certain data representation is assumed and certain predefined relations are provided we also prove that the datalog clause implication problem i e deciding whether a datalog clause implies another one is exptime complete our main technical tool is a product construction which maps a datalog programs to an essentially equivalent sirup

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

the kolmogorov expression complexity of logics the purpose of the paper is to propose a completely new notion of complexity of logics in finite model theory it is the kolmogorov variant of the vardi sexpression complexity we define it by considering the value of the kolmogorov complexityc la of the infinite stringla of all truth values of sentences oflin a the higher is this value the more expressive is the logiclin a if d is a class of finite models then the value ofc la over all a d is a measure of expressive power oflin d unboundedness ofc la c l a for a d implies nonexistence of a recursive interpretation oflinl a version of this statement with complexities modulo oracles implies the nonexistence of any interpretation oflinl thus the valuesc la modulo oracles constitute an invariant of the expressive power of logics over finite models depending on their real absolute expressive power and not on the syntax we investigate our notion for fragments of the infinitary logic l least fixed point logic lfp and partial fixed point logic pfp we prove a precise characterization of 0 1 laws for these logics in terms of a certain boundedness condition placed onc la we get an extension of the notion of a 0 1 law by imposing an upper bound on the value ofc la growing not too fast with cardinality of a which still implies inexpressibility results similar to those implied by 0 1 laws we also discuss classes d in whichc pfpka is very high it appears that then pfp or its simple extension can define all the pspace subsets of d

circumscribing datalog expressive power and complexity abstract in this paper we study a generalization of datalog the language of function free definite clauses it is known that standard datalog semantics i e least herbrand model semantics can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized the extension proposed here called datalog circ consists in considering the general form of circumscription where some predicates are minimized some predicates are fixed and some vary we study the complexity and the expressive power of the language thus obtained we show that this language and actually its non recursive fragment is capable of expressing all the queries in db co np and as such is much more powerful than standard datalog whose expressive power is limited to a strict subset of ptime queries both data and combined complexities of answering datalog circ queries are studied data complexity is proved to be co np complete combined complexity is shown to be in general hard for co ne and complete for co ne in the case of herbrand bases containing k distinct constant symbols where k is bounded

a comparison between algebraic query languages for flat and nested databases abstract recently much attention has been paid to query languages for nested relations in the present paper we consider the nested algebra and the powerset algebra and compare them both mutually as well as to the traditional flat algebra we show that either nest or difference can be removed as a primitive operator in the powerset algebra while the redundancy of the nest operator might have been expected the same cannot be said of the difference basically this result shows that the presence of one nonmonotonic operator suffices in the powerset algebra as an interesting consequence of this result the nested algebra without the difference remains complete in the sense of bancilhon and paredaens finally we show there are both similarities and fundamental differences between the expressiveness of query languages for nested relations and that of their counterparts for flat relations

granular risk based design optimization optimization considering uncertainty is an increasingly important and continuously developing uncertainty mitigation technique for modern design compared with its well established branches i e reliability based design optimization rbdo and robust design optimization risk based design optimization rdo is just regarded as an extension of rbdo by incorporating future cost hence it has not received much deep theoretical study based on the generalized theory of uncertainty we introduce different levels of probability granulation into rdo and propose a granular risk based design optimization grdo methodology the risks are modeled as granular probabilities their mean values and standard deviations two multiobjective optimization mo formulations of grdo are proposed and solved by multiobjective evolutionary algorithm based on decomposition aided with solution filtering criterion based on the results of a structural design example the capability of grdo on uncertainty management is validated by comparing the performances of different mo formulations while uncertainty mitigation using grdo is achieved by controlling the risks associated with fixed level of uncertainty this way grdo is approved as a general design frame rather than just an uncertainty mitigation technique like conventional rdo

a zero one law for logic with a fixed point operator the logic obtained by adding the least fixed point operator to first order logic was proposed as a query language by aho and ullman in proc 6th acm sympos on principles of programming languages 1979 pp 110 120 and has been studied particularly in connection with finite models by numerous authors we extend to this logic and to the logic containing the more powerful iterative fixed point operator the zero one law proved for first order logic in glebskii kogan liogonki and talanov 1969 kibernetika 2 31 42 fagin 1976 j symbolic logic 41 50 58 for any sentence of the extend logic the proportion of models of among all structures with universe 1 2 n approaches 0 or 1 as n tends to infinity we also show that the problem of deciding for any whether this proportion approaches 1 is complete for exponential time if we consider only s with a fixed finite vocabulary or vocabularies of bounded arity and complete for double exponential time if is unrestricted in addition we establish some related results

linear time and the power of one first order universal quantifier the aim of this paper is to give two new logical characterizations of nlin nondeterministic linear time improving significantly a previous characterization of e grandjean 1994 siam j comput 23 573 597 and f olive 1998 comput complexity 7 54 97 it is known that nlin coincides with the class of problems definable by formulas of the prenex form f1 fk where the fi are unary function symbols and is first order prenex with only one universal quantifier we show that the characterization remains true in the two following cases a unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h b when only ordered structures are considered unary functions are restricted to be bijective

lower bounds for invariant queries in logics with counting we study the expressive power of counting logics in the presence of auxiliary relations such as orders and preorders the simplest such logic is the first order logic with counting this logic captures the complexity class tc0 over ordered structures we also consider first order logic with arbitrary unary quantifiers and with infinitary extensions we start by giving a simple direct proof that first order logic with counting in the presence of pre orders that are almost everywhere linear orders cannot express the transitive closure of a binary relation the proof is based on locality of formulae we then show that the technique cannot be extended to linear orders we further show that this result does not say anything about the power of invariant queries in first order logic with counting vs the class tc0 in the presence of these preorders in the second part of the paper we prove a separation result showing that for all the counting logics above a linear order is more powerful than a preorder that is a linear order almost everywhere in fact we prove that the expressive power of invariant queries in the presence of such preorders can be characterized by a property normally associated with first order definability over unordered structures we do this by using locality techniques from finite model theory however as some standard notions of locality fail in this setting we have to modify them to prove the main result

on winning strategies in ehrenfeucht fraisse games we present a powerful and versatile new sufficient condition for the second player the duplicator to have a winning strategy in an ehrenfeucht fra isst game on graphs we accomplish two things with this technique first we give a simpler and much easier to understand proof of ajtai and fagin s result that reachability in directed finite graphs is not in monadic np monadic np otherwise known as monadic xi corresponds to existential second order logic with the restriction that the second order quantifiers range only over sets and not over relations of higher arity such as binary relations second we show that this result holds in the presence of a larger class of built in relations than was known before

hypothetical datalog complexity and expressibility abstract we present an extension of horn clause logic which can hypothetically add and delete tuples from a database such logics have been discussed in the literature but their complexities and expressibilities have remained an open question this paper examines two such logics in the function free predicate case it is shown in particular that augmenting horn clause logic with hypothetical addition increases its data complexity from ptime to pspace when deletions are added as well complexity increases again to exptime we then augment the logic with negation as failure and develop the notion of stratified hypothetical rulebases it is shown that negation does not increase complexity to establish expressibility we view the logic as a query language for relational databases it is shown that any typed generic query that is computable in pspace can be expressed as a stratified rulebase of hypothetical additions similarly any typed generic query that is computable in exptime can be expressed as a stratified rulebase of hypothetical additions and deletions neither of these results assumes the data domain is linearly ordered in this way we establish the expressive completeness of our logics for queries in pspace and exptime respectively

machine independent characterizations and complete problems for deterministic linear time this article presents two algebraic characterizations and two related complete problems for the complexity class dlin that was introduced in e grandjean ann math artif intell 16 1996 pp 183 236 dlin is essentially the class of all functions that can be computed in linear time on a random access machine which uses only numbers of linear value during its computations the algebraic characterizations are in terms of recursion schemes that define unary functions one of these schemes defines several functions simultaneously while the other one defines only one function from the algebraic characterizations we derive two complete problems for dlin under new very strict and machine independent affine reductions

the kolmogorov expressive power of boolean query languages abstract we address the question how much of the information stored in a given database can be retrieved by all boolean queries in a given query language in order to answer it we develop a kolmogorov complexity based measure of expressive power of boolean query languages over finite structures this turns the above informal question into a precisely defined mathematical one this notion gives a meaningful definition of the expressive power of a boolean query language in a single finite database the notion of kolmogorov expressive power of a boolean query language l in a finite database a is defined by considering two values the kolmogorov complexity of the isomorphism type of a equal to the length of the shortest description of this type and the number of bits of this description that can be reconstructed from truth values of all queries from l in a the closer is the second value to the first the more expressive is the query language after giving the definitions and proving that they are correct we concentrate our efforts on first order logic and its powerful extensions inflationary fixpoint logic and partial fixpoint logic we explore some connections between the proposed kolmogorov expressive power of boolean queries in these languages and their standard expressive power in particular with the definability of order we show that except of being of interest on its own our notion may have important diagnostic value for database query optimization

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

bounded fixpoints for complex objects we study a query language for complex object databases which is designed to 1 express only tractable queries and 2 be as expressive over flat relations as first order logic with fixpoints the language is obtained by extending the nested relational algebra nra with a bounded fixpoint operator similar to results for flat relations all tractable queries over ordered databases are expressible in this language the main result consists in proving that this language is a conservative extension of the first order logic with fixpoints or of the while queries depending on the interpretation of the bounded fixpoint inflationary or partial that is a query from flat relations to flat relations is expressible in our language if and only if it is expressible in first order logic with fixpoints or in the while queries respectively the proof technique for this theorem uses indexes to encode complex objects into flat relations it can serve as basis for an implementation method of complex objects databases in terms of relational databases which works well for queries expressed both with fixpoints and with bounded fixpoint we also define a complex object logical calculus with fixpoints and prove that its range restricted fragment is equivalent to nra with bounded fixpoints

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

positive versions of polynomial time abstract we show that restricting a number of characterizations of the complexity class p to be positive in natural ways results in the same class of monotone problems which we denote by posp by a well known result of razborov posp is a proper subclass of the class of monotone problems in p we exhibit complete problems for posp via weak logical reductions as we do for other logically defined classes of problems our work is a continuation of research undertaken by grigni and sipser and subsequently stewart indeed we introduce the notion of a positive deterministic turing machine and consequently solve a problem posed by grigni and sipser

several nonlinear operator problems in the menger pn space abstract in this paper we investigate some nonlinear operator problems and the theory of topological degree meanwhile we obtain some new fixed point theorems in the m pn space

fixed parameter tractability definability and model checking in this article we study parameterized complexity theory from the perspective of logic or more specifically descriptive complexity theory we propose to consider parameterized model checking problems for various fragments of first order logic as generic parameterized problems and show how this approach can be useful in studying both fixed parameter tractability and intractability for example we establish the equivalence between the model checking for existential first order logic the homomorphism problem for relational structures and the substructure isomorphism problem our main tractability result shows that model checking for first order formulas is fixed parameter tractable when restricted to a class of input structures with an excluded minor on the intractability side for every t ge 0 we prove an equivalence between model checking for first order formulas with t quantifier alternations and the parameterized halting problem for alternating turing machines with t alternations we discuss the close connection between this alternation hierarchy and downey and fellows w hierarchy on a more abstract level we consider two forms of definability called fagin definability and slicewise definability that are appropriate for describing parameterized problems we give a characterization of the class fpt of all fixed parameter tractable problems in terms of slicewise definability in finite variable least fixed point logic which is reminiscent of the immerman vardi theorem characterizing the class ptime in terms of definability in least fixed point logic

fixed point definability and polynomial time on graphs with excluded minors we give a logical characterization of the polynomial time properties of graphs embeddable in some surface for every surface s a property p of graphs embeddable in s is decidable in polynomial time if and only if it is definable in fixed point logic with counting it is a consequence of this result that for every surface s there is a k such that a simple combinatorial algorithm namely the k dimensional weisfeiler lehman algorithm decides isomorphism of graphs embeddable in s in polynomial time we also present without proof generalizations of these results to arbitrary classes of graphs with excluded minors

mathematical logic and quantum finite state automata this paper is a review of the connection between formulas of logic and quantum finite state automata in respect to the language recognition and acceptance probability of quantum finite state automata as is well known logic has had a great impact on classical computation it is promising to study the relation between quantum finite state automata and mathematical logic after a brief introduction to the connection between classical computation and logic the required background of the logic and quantum finite state automata is provided and the results of the connection between quantum finite state automata and logic are presented