partial quasi metrics in this article we introduce and investigate the concept of a partial quasi metric and some of its applications we show that many important constructions studied in matthews s theory of partial metrics can still be used successfully in this more general setting in particular we consider the bicompletion of the quasi metric space that is associated with a partial quasi metric space and study its applications in groups and bck algebras

connecting fuzzy preorders fuzzy consequence operators and fuzzy closure and co closure systems abstract the relations between fuzzy preorders fuzzy consequence operators fuzzy closure systems and co closure systems lattice structures are studied given a continuous t norm we will prove that the complete lattice of all fuzzy preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators moreover we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices a solution to the following open problem when two families of fuzzy possible valuations produce the same fuzzy implication relation is given concept of interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through interior operator is shown

limit theorems for functionals of mixing processes with applications to u statistics and dimension estimation in this paper we develop a general approach for investigating the asymptotic distribution of functional xn f zn k k z of absolutely regular stochastic processes zn n z such functional occur naturally as orbits of chaotic dynamical systems and thus our results can be used to study probabilistic aspects of dynamical systems we first prove some moment inequalities that are analogous to those for mixing sequences with their help several limit theorems can be proved in a rather straightforward manner we illustrate this by re proving a central limit theorem of ibragimov and linnik then we apply our techniques to u statistics matrix equation with symmetric kernel h r r r we prove a law of large numbers extending results of aaronson burton dehling gilat hill and weiss for absolutely regular processes we also prove a central limit theorem under a different set of conditions than the known results of denker and keller as our main application we establish an invariance principle for u processes un h h indexed by some class of functions we finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system

fixed points of adjoint functors enriched in a quantaloid abstract representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid in a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis fca and rough set theory rst which not only extend the realm of formal contexts to multi typed and multi valued ones but also provide a general approach to construct various kinds of representation theorems besides incorporating several well known representation theorems in fca and rst as well as formulating new ones it is shown that concept lattices in rst can always be represented as those in fca through relative pseudo complements of the given contexts especially if the contexts are valued in a non girard quantaloid

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

some corollaries of the correspondence between partial metrics and multivalued equalities multivalued equalities originally arose in the context of the theory of sheaves while partial metrics originally arose in the context of the theory of domains for denotational semantics it turns out that multivalued equalities and partial metrics are closely related one can either consider them equivalent up to the choice of dual notation or one can formally capture the duality between logical and metric viewpoints by explicitly requiring logical values and distances to be represented by dual structures not only does this allow the transfer of the ideas and results between these two fields but the most interesting situations are those of interplay between logical and metric considerations the computational understanding of partial metrics as upper bounds and of multivalued equalities as lower bounds is discussed together with the lower bound counterparts to partial metrics and the corresponding upper bound counterparts to multivalued equalities it is shown that separated pre sheaves of sets and functions over complete heyting algebras can be understood as the pre sheaves of ultrametrics valued in the corresponding brouwerian algebras and non expansive maps of those ultrametrics it is proposed that the natural logical counterparts for partial metrics valued in non negative reals are multivalued equalities valued in the quantale of non positive reals the issue of canonical partial metrics on higher order and reflexive scott domains is revisited the intuition behind strong triangularity vickers form the computational version of strong triangularity and weighted quasi metrics are discussed in the context of quantaloid enrichment

categorical foundations of topology with applications to quantaloid enriched topological spaces based on ordered monads this paper uncovers the categorical basis of topology in terms of a categorical formulation of neighborhood axioms here dense subobjects lower separation axioms and regularity receive a purely categorical representation in the case of appropriate submonads of the double presheaf monad this theory is applied to quantaloid enriched topological spaces which form a common framework for many valued topology as well as for non commutative topology as an illumination of this situation two examples are given the first one is chosen from probability theory and has the following characteristics weak convergence of smooth probability measures is topological the hausdorff separation axiom is valid dirac measures form a dense subset the second example is related to operator theory and explains the topologization of spectra of non commutative c algebras

dialectic proof procedures for assumption based admissible argumentation we present a family of dialectic proof procedures for the admissibility semantics of assumption based argumentation these proof procedures are defined for any conventional logic formulated as a collection of inference rules and show how any such logic can be extended to a dialectic argumentation system the proof procedures find a set of assumptions to defend a given belief by starting from an initial set of assumptions that supports an argument for the belief and adding defending assumptions incrementally to counter attack all attacks the proof procedures share the same notion of winning strategy for a dispute and differ only in the search strategy they use for finding it the novelty of our approach lies mainly in its use of backward reasoning to construct arguments and potential arguments and the fact that the proponent and opponent can attack one another before an argument is completed the definition of winning strategy can be implemented directly as a non deterministic program whose search strategy implements the search for defences

categorical approaches to non commutative fuzzy logic in this paper we consider what it means for a logic to be non commutative how to generate examples of structures with a non commutative operation which have enough nice properties to serve as the truth values for a logic inference in the propositional logic is gotten from the categorical properties products coproducts monoidal and closed structures adjoint functors of the categories of truth values we then show how to extend this view of propositional logic to a predicate logic using categories of propositions about a type a with functors giving change of type and adjoints giving quantifiers in the case where the semantics takes place in set l goguen s category of l fuzzy sets the categories of predicates about a can be represented as internal category objects with the quantifiers as internal functors

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

the complexity of logic based abduction abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations when the application domain is described by a logical theory we speak about logic based abduction candidates for abductive explanations are usually subjected to minimality criteria such as subset minimality minimal cardinality minimal weight or minimality under prioritization of individual hypotheses this paper presents a comprehensive complexity analysis of relevant decision and search problems related to abduction on propositional theories our results indicate that abduction is harder than deduction in particular we show that with the most basic forms of abduction the relevant decision problems are complete for complexity classes at the second level of the polynomial hierarchy while the use of prioritization raises the complexity to the third level in certain cases

fuzzy presubsets as non idempotent and non commutative classifications of subalgebras in the framework of quantale preordered sets regular q fuzzy presubsets are identified with classifiable subalgebras this identification is based on the monad t of quasi presingletons which is an extension of the monad associated with the cauchy completion of quantale preordered sets even though not every t subalgebra is classifiable the classifiable hull of t subalgebras exists always an application of these constructions is the internalization of the negation of the underlying quantale as truth arrow in the category of t algebras

lattice valued preordered sets as lattice valued topological systems this paper provides variable basis lattice valued analogues of the well known results that the construct prost of preordered sets firstly is concretely isomorphic to a full concretely coreflective subcategory of the category top of topological spaces which employs the concept of the dual of the specialization preorder and secondly is non concretely isomorphic to a full coreflective subcategory of the category topsys of topological systems of s vickers which employs the spatialization procedure for topological systems dualizing these results one arrives at the similar properties of quasi pseudo metric spaces built over locales

continuity spaces reconciling domains and metric spaces abstract we use continuity spaces a common refinement of posets and metric spaces to develop a general theory of semantic domains which includes metric spaces and domains of cpo s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming our proposal for a general notion of semantic domain is a symmetrically compact continuity space where is a value quantale we show that the category of symmetrically compact continuity spaces with continuous maps has many of the key properties required of a category of domains and that it captures in a natural way the traditional examples in general the category will not be cartesian closed however powerdomains do exist and by adapting a construction of suenderhauf to continuity spaces we show that they define a computational monad in the sense of moggi

testing independence for multivariate time series via the auto distance correlation matrix we introduce the matrix multivariate auto distance covariance and correlation functions for time series discuss their interpretation and develop consistent estimators for practical implementation we also develop a test of the independent and identically distributed hypothesis for multivariate time series data and show that it performs better than the multivariate ljung box test we discuss computational aspects and present a data example to illustrate the method

semitopological groups semiclosure semigroups and quantales abstract as a valued domain quantales have been applied to the study of enriched category fuzzy algebra fuzzy topology and fuzzy domain one purpose of this paper is to investigate semitopological groups and semiclosure semigroups from the viewpoint of quantales we first show that a semitopological group is completely determined by the topological ideal conuclei of the power set quantale over a group next we prove that the concepts of semitopological groups and semiclosure groups are dually equivalent another purpose of this paper is to investigate ordered semigroups conditionally complete quantales and quantales from the viewpoint of semiclosure semigroups as an important result of semiclosure semigroups we prove that the category of complete semiclosure semigroups is isomorphic to the category of quantales

strong laws for l and u statistics strong laws of large numbers are given for l statistics linear combinations of order statistics and for u statistics averages of kernels of random samples for ergodic stationary processes extending classical theorems of hoeffding and of helmers for lid sequences examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied and an application is given to estimation of correlation dimension of invariant measures

preordered sets valued in a gl monoid let m l be a gl monoid an m valued preordered set is an l subset endowed with a reflexive and m transitive l relation it is essentially a category enriched in a quantaloid generated by m this paper presents a study of m valued preordered sets with emphasis on symmetrization and the cauchy completion the main result states that symmetrization and the cauchy completion of m valued preordered sets commute up to a natural isomorphism

numerical representation of transitive fuzzy relations we show that there exists a continuous fuzzy quasi order r on the set of real numbers r such that for any fuzzy quasiorder r on a finite set x there is a mapping f x r such that r x y r f x f y for all x y x

free q algebras abstract the main purpose of this paper is to consider some methods for constructing q algebras and to give the concrete form of a free unital q algebra over a set a semigroup an ordered semigroup

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

rank tests for short memory stationarity abstract we propose a rank test of the null hypothesis of short memory stationarity possibly after linear detrending for the level stationarity hypothesis the test statistic we propose is a modified version of the popular kpss statistic in which ranks substitute the original observations we prove that the rank kpss statistic shares the same limiting distribution as the standard kpss statistic under the null and diverges under i 1 alternatives for the trend stationarity hypothesis we apply the same rank kpss statistic to the residuals of a theil sen regression on a linear trend we derive the asymptotic distribution of the theil sen estimator under short memory errors and prove that the theil sen detrended rank kpss statistic shares the same weak limit as the least squares detrended kpss we study the asymptotic relative efficiency of our test compared to the kpss and prove that it may have unbounded efficiency gains under fat tailed distributions compensated by very moderate efficiency losses under thin tailed distributions for this and other reasons discussed in the body of the article our rank kpss test turns out to be a valuable alternative to the kpss for most real world economic and financial applications the weak convergence results and asymptotic representations proved in this article should interest a wider audience than that concerned with stationarity testing as they extend to ranks analogous invariance principles widely used in unit root econometrics

the double power monad is the composite power monad in the context of quantaloid enriched categories we rely essentially on the classifying property of presheaf categories to give a conceptual proof of a theorem due to hhle the double power monad and the composite power monad on the category of quantaloid enriched categories are the same via the theory of distributive laws we identify the algebras of this monad to be the completely codistributive complete categories and the homomorphisms between such algebras are the bicontinuous functors with these results we hope to contribute to the further development of a theory of q valued preorders in the sense of pu and zhang

fuzzy galois connections on fuzzy sets abstract in fairly elementary terms this paper presents how the theory of preordered fuzzy sets more precisely quantale valued preorders on quantale valued fuzzy sets is established under the guidance of enriched category theory motivated by several key results from the theory of quantaloid enriched categories this paper develops all needed ingredients purely in order theoretic languages for the readership of fuzzy set theorists with particular attention paid to fuzzy galois connections between preordered fuzzy sets

simple involutive quantales abstract involutive quantales were introduced in 7 as complete lattices equipped with a multiplication and an involution such structures are well known from the calculus of relations the set r el x of binary relations on any setxforms an involutive quantale however the motivating example of an involutive quantale is the spectrum max aof a non commutativec algebraa where max ais the involutive quantale consisting of all closed linear subspaces ofa in 8 it was indeed shown thatacan be reconstructed from the involutive quantale max a demonstrating that involutive quantales do provide a convenient algebraic invariant of non commutativec algebras the aim of this paper is to study involutive quantales from the algebraic point of view our main result is the characterization of simple involutive quantales that is involutive quantales on which the only surjections are isomorphisms or constant morphisms we also show that this class contains all quantales q s of sup preserving endomorphisms on a complete latticeshaving a duality a fortiori all hilbert quantales as defined in 7 a special case of this is r el x which can be seen to be q 2x our investigation of the class of simple involutive quantales is also motivated byc algebras it is proved in 8 that for anyc algebraa max ahas enough homomorphisms into the hilbert quantales q p h where p h is the complete lattice of closed linear subspaces of a hilbert spaceh in this spirit viewing involutive quantales as duals of generalized non commutative topological spaces simple involutive quantales correspond to spaces having only trivial subspaces that is to points we begin a study of those involutive quantales which have enough homomorphisms into simple involutive quantales noting their spatial character such a study has algebraic antecedents in the early work 4 and the more recent work 1 which studied certain lattices noncomplete involutive quantales having enough homomorphisms into simple involutive quantales of the form r el x

on rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions we consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as hausdorff dimension box dimension etc we also introduce different ways to define and study the generalized spectrum for dimensions a one parameter family of characteristics of dimension type

sequential monte carlo methods for diffusion processes in this paper we show how to use sequential monte carlo methods to compute expectations of functionals of diffusions at a given time and the gradients of these quantities w r t the initial condition of the process in some cases via the exact simulation of the diffusion there is no time discretization error otherwise the methods use euler discretization we illustrate our approach on both high and low dimensional problems from optimal control and establish that our approach substantially outperforms standard monte carlo methods typically adopted in the literature the methods developed here are appropriate for solving a certain class of partial differential equations as well as for option pricing and hedging

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

default reasoning using classical logic abstract in this paper we show how propositional default theories can be characterized by classical propositional theories for each finite default theory we show a classical propositional theory such that there is a one to one correspondence between models for the latter and extensions of the former this means that computing extensions and answering queries about coherence set membership and set entailment are reducible to propositional satisfiability the general transformation is exponential but tractable for a subset which we call 2 dt a superset of network default theories and disjunction free default theories consequently coherence and set membership for the class 2 dt is np complete and set entailment is co np complete this work paves the way for the application of decades of research on efficient algorithms for the satisfiability problem to default reasoning for example since prepositional satisfiability can be regarded as a constraint satisfaction problem csp this work enables us to use csp techniques for default reasoning to illustrate this point we use the taxonomy of tractable csps to identify new tractable subsets for reiter s default logic our procedures allow also for computing stable models of extended logic programs

the order theoretic duality and relations between partial metrics and local equalities for a gl monoid l hohle introduced l valued sets to formalize the mathematical theory of identity and existence in monoidal logic for the purpose of analyzing deadlock situations in data flow networks the notion of partial metric space was proposed by matthews later on a lattice theoretic generalization of pseudo partial metric spaces extending the range of pseudo partial metrics to a value lattice v were studied under the name of v pseudo pmetric spaces referring to a gl monoid l and a dual gl monoid v we show that l valued sets are order theoretically dual to v pseudopmetric spaces and apply this duality to the representation of v pseudopmetric spaces and the determination of their various categorical properties in addition to this the present paper provides not only some new results about the representation of l valued sets and their categories but also some non trivial categorical connections between l valued sets and v pseudopmetric spaces

a formal study of linearity axioms for fuzzy orderings abstract this contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings different existing concepts are evaluated with respect to three fundamental correspondences from the classical case linearizability of partial orderings intersection representation and one to one correspondence between linearity and maximality as a main result we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case if we do not require a one to one correspondence between linearity and maximality however we obtain that an implication based definition appears to constitute a sound compromise in particular if ukasiewicz type logics are considered

on the injective hulls of quantum b algebras abstract quantum b algebras as implicational subreducts of quantales were introduced by rump and yang which cover the majority of implicational algebras and provide a unified semantics for non commutative algebraic logic due to the adjunction between multiplication and its residuals they can be regarded as the implicational counterpart of partially ordered semigroups by means of the injective hulls envelopes of quantum b algebras rump gave the injective hulls of partially ordered semigroups in this paper one part of our work is to give new examples of quantum b algebras where the lower sets do not form a quantale another part of our work is to consider the inverse problem of rump s way that is we can also give the injective hulls of quantum b algebras in terms of the injective hulls of partially ordered semigroups

sierpinski object for affine systems abstract motivated by the concept of sierpinski object for topological systems of s vickers presented recently by r noor and a k srivastava this paper introduces the sierpinski object for many valued topological systems and shows that it has three important properties of the crisp sierpinski space of general topology

limit theorems for von mises statistics of a measure preserving transformation for a measure preserving transformation t of a probability space x mathcal f mu and some d ge 1 we investigate almost sure and distributional convergence of random variables of the form begin aligned x rightarrow frac 1 c n sum 0 le i 1 ldots i d n f t i 1 x ldots t i d x n 1 2 ldots end aligned where c 1 c 2 ldots are normalizing constants and the kernel f belongs to an appropriate subspace in some l p x d mathcal f otimes d mu d we establish a form of the individual ergodic theorem for such sequences using a filtration compatible with t and the martingale approximation we prove a central limit theorem in the non degenerate case for a class of canonical totally degenerate kernels and d 2 we also show that the convergence holds in distribution towards a quadratic form sum m 1 infty lambda m eta 2 m in independent standard gaussian variables eta 1 eta 2 ldots

quantale valued preorders each divisible and unital quantale is associated with two quantaloids categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders this paper deals with the relationship between these two kinds of enriched categories this is a special case of the change base issue in enriched category theory two functors the forward globalization functor and the backward globalization functor that map a fuzzy set with a fuzzy preorder to a crisp set with a fuzzy preorder are considered it is shown that both of them preserve cocompleteness with respect to several classes of weights

a non commutative and non idempotent theory of quantale sets in fuzzy set theory non idempotency arises when the conjunction is interpreted by arbitrary t norms there are many instances in mathematics where set theory ought to be non commutative and or non idempotent the purpose of this paper is to combine both ideas and to present a theory of non commutative and non idempotent quantale sets among other things standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases more specifically the category q set is investigated where q is an arbitrary involutive quantale objects quantale sets are pairs consisting of a set and a q valued equality with a suitable symmetry axiom three important properties of q set are shown it is complete cocomplete and has the epi extremal mono factorization property its subcategory s q set of separated quantale sets is reflective in q set and shares the same fundamental properties with q set in particular s q set is also a complete and cocomplete epi extremal mono category the objects of the category q set are interesting categories in their own right two categorical frameworks for objects of q set are exhibited first it is shown that q valued equalities arise from q valued preorders with self adjoint extents by symmetrization which leaves q valued equalities invariant here sets with quantale preorders are shown to be b categories with base b being a specific quantaloid the second approach is based on involutive quantaloids a combination of two well known things quantaloids and ordered categories with involution in this context quantale sets are precisely symmetric b categories w r t appropriately chosen quantaloid b with involution further the cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames our non commutative generalization of frames and all left continuous t norms there exist at least two monads on q set the singleton monad and quasi singleton monad which are of special interest for fuzzy set theory the kleisli category associated with the singleton monad is the non commutative and non idempotent analogue of higgs topos while the eilenberg moore category of the quasi singleton monad permits the internalization of lukasiewicz negation as truth arrow finally an application of quantale sets to c algebras is given and the change of base is treated

a logic based theory of deductive arguments abstract we explore a framework for argumentation based on classical logic in which an argument is a pair where the first item in the pair is a minimal consistent set of formulae that proves the second item which is a formula we provide some basic definitions for arguments and various kinds of counter arguments defeaters this leads us to the definition of canonical undercuts which we argue are the only defeaters that we need to take into account we then motivate and formalise the notion of argument trees and argument structures which provide a way of exhaustively collating arguments and counter arguments we use argument structures as the basis of our general proposal for argument aggregation there are a number of frameworks for modelling argumentation in logic they incorporate formal representation of individual arguments and techniques for comparing conflicting arguments in these frameworks if there are a number of arguments for and against a particular conclusion an aggregation function determines whether the conclusion is taken to hold we propose a generalisation of these frameworks in particular our new framework makes it possible to define aggregation functions that are sensitive to the number of arguments for or against we compare our framework with a number of other types of argument systems and finally discuss an application in reasoning with structured news reports

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

stratified categorical fixed basis fuzzy topological spaces and their duality abstract for an abstract category c a class m of c monomorphisms and a fixed c object l we introduce stratified c m l spaces to be categorical counterparts of stratified fixed basis fuzzy topological spaces in c and consider their category sc m l top as two main results of this paper it is shown that sc m l top is dually adjoint to the comma category l c and this adjunction can be restricted to a dual equivalence between the full category of l c with comma spatial objects and the full category of sc m l top with comma sober objects the present paper also describes applications and relationships of these results to stratified fixed basis fuzzy topological spaces in this respect a considerable part of this paper is devoted to stratified l quasi topological spaces and their duality

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

towards probabilistic partial metric spaces diagonals between distance distributions abstract the quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories motivated by the categorical interpretation of partial metric spaces we are led to investigate the quantaloid of diagonals between distance distributions which is expected to establish the categorical foundation of probabilistic partial metric spaces observing that the quantale of distance distributions w r t an arbitrary continuous t norm is non divisible we precisely characterize diagonals between distance distributions and prove that one step functions are the only distance distributions on which the set of diagonals coincides with the generated down set

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

encoding deductive argumentation in quantified boolean formulae there are a number of frameworks for modelling argumentation in logic they incorporate a formal representation of individual arguments and techniques for comparing conflicting arguments a common assumption for logic based argumentation is that an argument is a pair where f is minimal subset of the knowledge base such that f is consistent and f entails the claim a different logics provide different definitions for consistency and entailment and hence give us different options for argumentation classical propositional logic is an appealing option for argumentation but the computational viability of generating an argument is an issue to better explore this issue we use quantified boolean formulae to characterise an approach to argumentation based on classical logic