kato class measures of symmetric markov processes under heat kernel estimates abstract we establish the coincidence of two classes of kato class measures in the framework of symmetric markov processes admitting upper and lower estimates of heat kernel under mild conditions one class of kato class measures is defined by way of the heat kernel another is defined in terms of the green kernel depending on some exponents related to the heat kernel estimates we also prove that p th integrable functions on balls with radius 1 having a uniformity of its norm with respect to centers are of kato class if p is greater than a constant related to the estimate under the same conditions these are complete extensions of some results for the brownian motion on euclidean space by aizenman and simon our result can be applicable to many examples for instance symmetric relativistic stable processes jump processes on d sets brownian motions on riemannian manifolds diffusions on fractals and so on

density and tails of unimodal convolution semigroups abstract we give sharp bounds for the isotropic unimodal probability convolution semigroups when their levy khintchine exponent has matuszewska indices strictly between 0 and 2

asymptotic behavior of densities of unimodal convolution semigroups we prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on r d under the assumption that its levy khintchine exponent is regularly varying of index between 0 and 2

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

linguistic quantifiers modeled by sugeno integrals since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning this paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures and the truth value of a quantified proposition is evaluated by using sugeno s integral this framework allows us to have some elegant logical properties of linguistic quantifiers we compare carefully our new model of quantification and other approaches to linguistic quantifiers a set of criteria for linguistic quantification was proposed in the previous literature the relationship between these criteria and the results obtained in the present paper is clarified some simple applications of the sugeno s integral semantics of quantifiers are presented

asymptotic behaviour of densities of stable semigroups of measures we prove that densities of the measures in a strictly stable semigroup ht of symmetric measures on a homogeneous group if they exist have the following asymptotic behaviour mathop lim limits x to infty x q alpha cdot h 1 x k bar x where is the characteristic exponent bar x x 1 x andk is the density of the levy measure associated to the semigroup moreover if k bar x 0 a more precise description is given

continuous time modeling of random searches statistical properties and inference mathematical modeling of random searches is of great relevance in the fields of physics chemistry biology or modern ecology a large number of existing studies record the search movement at equidistant time intervals and model such time series data directly with discrete time random walks such as levy flights and correlated random walks given the increasing availability of high resolution observation data statistical inference for search paths based on such high resolution data has recently become one of the major interests and has raised an important issue of robustness of random walk models to the sampling rate as estimation results for the discrete observation data are found to be largely different at different sampling rates even when the underlying movement is supposedly independent of scale to address this issue in this paper we propose to model the continuous time search paths directly with a continuous time stochastic process for which the observer makes statistical inference based on its discrete observation as continuous time counterparts of levy flights we consider two dimensional levy processes and discuss the relevance of those models based upon advantages and limitations in terms of statistical properties and inference among the proposed models the brownian motion is most tractable in various ways while its gaussianity and infinite variation of sample paths do not well describe the reality such drawbacks in statistical properties may be remedied by employing the stable and tempered stable levy motions while those models are less tractable and cause an issue in statistical inference

newton type minimization via the lanczos method this paper discusses the use of the linear conjugate gradient method developed via the lanczos method in the solution of large scale unconstrained minimization problems at each iteration of a newton type method the direction of search is defined as the solution of a quadratic subproblem when the number of variables is very large this subproblem may be solved using the linear conjugate gradient method of hestenes and stiefel we show how the equivalent lanczos characterization of the linear conjugate gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive definite hessian matrices at all points of the region of interest this derivation also makes it possible to compute a negative curvature direction at a stationary point the idea of a truncated newton method is to perform only a limited number of iterations of the quadratic subproblem this effectively gives a search direction that interpolates between the steepest de

regularity theory for general stable operators abstract we establish sharp regularity estimates for solutions to l u f in r n l being the generator of any stable and symmetric levy process such nonlocal operators l depend on a finite measure on s n 1 called the spectral measure first we study the interior regularity of solutions to l u f in b 1 we prove that if f is c then u belong to c 2 s whenever 2 s is not an integer in case f l we show that the solution u is c 2 s when s 1 2 and c 2 s for all 0 when s 1 2 then we study the boundary regularity of solutions to l u f in u 0 in r n in c 1 1 domains we show that solutions u satisfy u d s c s for all 0 where d is the distance to finally we show that our results are sharp by constructing two counterexamples

a maximum entropy approach to nonmonotonic reasoning an approach to nonmonotonic reasoning that combines the principle of infinitesimal probabilities with that of maximum entropy thus extending the inferential power of the probabilistic interpretation of defaults is proposed a precise formalization of the consequences entailed by a conditional knowledge base is provided the computational machinery necessary for drawing these consequences is developed and the behavior of the maximum entropy approach is compared to related work in default reasoning the resulting formalism offers a compromise between two extremes the cautious approach based on the conditional interpretations of defaults and the bold approach based on minimizing abnormalities

asymptotic estimates of multi dimensional stable densities and their applications the relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed in particular the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of pruitt and taylor in 1969 are solved the proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere or discrete with the support being a finite set or a mixture of such cases are obtained those results are applied to the moment of the last exit time from a ball and the spitzer type limit theorem involving capacity for a multi dimensional transient stable process

estimates of transition densities and their derivatives for jump levy processes abstract we give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding levy measure and the levy khinchin exponent we obtain also estimates of derivatives of densities

default reasoning from conditional knowledge bases complexity and tractable cases abstract conditional knowledge bases have been proposed as belief bases that include defeasible rules also called defaults of the form which informally read as generally if then such rules may have exceptions which can be handled in different ways a number of entailment semantics for conditional knowledge bases have been proposed in the literature however while the semantic properties and interrelationships of these formalisms are quite well understood about their computational properties only partial results are known so far in this paper we fill these gaps and first draw a precise picture of the complexity of default reasoning from conditional knowledge bases given a conditional knowledge base kb and a default does kb entail we classify the complexity of this problem for a number of well known approaches including goldszmidt et al s maximum entropy approach and geffner s conditional entailment where we consider the general propositional case as well as natural syntactic restrictions in particular to horn and literal horn conditional knowledge bases as we show the more sophisticated semantics for conditional knowledge bases are plagued with intractability in all these fragments we thus explore cases in which these semantics are tractable and find that most of them enjoy this property on feedback free horn conditional knowledge bases which constitute a new meaningful class of conditional knowledge bases furthermore we generalize previous tractability results from horn to q horn conditional knowledge bases which allow for a limited use of disjunction our results complement and extend previous results and contribute in refining the tractability intractability frontier of default reasoning from conditional knowledge bases they provide useful insight for developing efficient implementations

semantic characterization of rational closure from propositional logic to description logics abstract in this paper we provide a semantic reconstruction of rational closure we first consider rational closure as defined by lehman and magidor 33 for propositional logic and we provide a semantic characterization based on a minimal models mechanism on rational models then we extend the whole formalism and semantics to description logics by focusing our attention to the standard alc we first naturally adapt to description logics lehman and magidor s propositional rational closure starting from an extension of alc with a typicality operator t that selects the most typical instances of a concept c hence t c stands for typical c then for the description logics we define a minimal model semantics for the logic alc and we show that it provides a semantic characterization for the rational closure of a knowledge base we consider both the rational closure of the tbox and the rational closure of the abox

uncertainty measures decomposability and admissibility abstract uncertainty measures are defined in a general manner as morphism between certain posets which permits a unified treatment of different measures for classical and for fuzzy events admissible measures are discussed in a decision theoretical notation making use of quasilinear entropies connections and differences to decomposable measures are illustrated

fuzzy integrals and fuzzy measures with their values in complete lattices abstract real valued fuzzy measures and their associated fuzzy integrals were proposed by sugeno in this paper we consider fuzzy measures valued in complete lattices and their associated upper and lower fuzzy integrals moreover we define a class of functionals called fuzzy integrals in the setting of the complete lattices which are both brouwerian and dually brouwerian the upper and lower fuzzy integrals are fuzzy integrals while in that of the completely distributive complete lattices every fuzzy integral is both lower and upper fuzzy integral with respect to a fuzzy measure

from statistical knowledge bases to degrees of belief an intelligent agent will often be uncertain about various properties of its environment and when acting in that environment it will frequently need to quantify its uncertainty for example if the agent wishes to employ the expected utility paradigm of decision theory to guide its actions it will need to assign degrees of belief subjective probabilities to various assertions of course these degrees of belief should not be arbitrary but rather should be based on the information available to the agent this paper describes one approach for inducing degrees of belief from very rich knowledge bases that can include information about particular individuals statistical correlations physical laws and default rules we call our approach the random worlds method the method is based on the principle of indifference it treats all of the worlds the agent considers possible as being equally likely it is able to integrate qualitative default reasoning with quantitative probabilistic reasoning by providing a language in which both types of information can be easily expressed our results show that a number of desiderata that arise in direct inference reasoning from statistical information to conclusions about individuals and default reasoning follow directly from the semantics of random worlds for example random worlds captures important patterns of reasoning such as specificity inheritance indifference to irrelevant information and default assumptions of independence furthermore the expressive power of the language used and the intuitive semantics of random worlds allow the method to deal with problems that are beyond the scope of many other nondeductive reasoning systems

conditional logics of normality a modal approach abstract several conditional theories of default reasoning have recently been proposed for the representation of statements about normal states of affairs or prototypical properties the natural semantics of these systems and their ability to reason about default rules make these approaches quite appealing we present a family of modal logics in which we define a conditional connective for statements of normality and examine its properties we also demonstrate that two of the most important conditional approaches are equivalent to fragments of our conditional logics of normality and to standard modal logics the approach we take is general enough to allow the expression of a number of different forms of defeasible reasoning and can be used to illustrate the relationship between these types of reasoning e g belief revision subjunctive and autoepistemic reasoning and our default logics this relationship is explored in a companion paper

nonmonotonic reasoning preferential models and cumulative logics abstract many systems that exhibit nonmonotonic behavior have been described and studied already in the literature the general notion of nonmonotonic reasoning though has almost always been described only negatively by the property it does not enjoy i e monotonicity we study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties we concentrate on a number of families of nonmonotonic consequence relations defined in the style of gentzen 13 both proof theoretic and semantic points of view are developed in parallel the former point of view was pioneered by gabbay 10 while the latter has been advocated by shoham 38 five such families are defined and characterized by representation theorems relating the two points of view one of the families of interest that of preferential relations turns out to have been studied by adams 2 the preferential models proposed here are a much stronger tool than adams probabilistic semantics the basic language used in this paper is that of propositional logic the extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper

on tikhonov regularisation abstract we introduce bayesian estimation into the tikhonov regularisation process for recovering functions of high bandwidth from noisy band limited observations the method allows for the incorporation of featural constraints and requires the global minimisation of a functional derived from damped pseudospectral approximation

a classical measure of evidence for general null hypotheses abstract in science the most widespread statistical quantities are perhaps p values a typical advice is to reject the null hypothesis h 0 if the corresponding p value is sufficiently small usually smaller than 0 05 many criticisms regarding p values have arisen in the scientific literature the main issue is that in general optimal p values based on likelihood ratio statistics are not measures of evidence over the parameter space here we propose an objective measure of evidence for very general null hypotheses that satisfies logical requirements i e operations on the subsets of that are not met by p values e g it is a possibility measure we study the proposed measure in the light of the abstract belief calculus formalism and we conclude that it can be used to establish objective states of belief on the subsets of based on its properties we strongly recommend this measure as an additional summary of significance tests at the end of the paper we give a short listing of possible open problems

weak nonmonotonic probabilistic logics we present an approach where probabilistic logic is combined with default reasoning from conditional knowledge bases in kraus et al s system p pearl s system z and lehmann s lexicographic entailment the resulting probabilistic generalizations of default reasoning from conditional knowledge bases allow for handling in a uniform framework strict logical knowledge default logical knowledge as well as purely probabilistic knowledge interestingly probabilistic entailment in system p coincides with probabilistic entailment under g coherence from imprecise probability assessments we then analyze the semantic and nonmonotonic properties of the new formalisms it turns out that they all are proper generalizations of their classical counterparts and have similar properties as them in particular they all satisfy the rationality postulates of system p and some conditioning property moreover probabilistic entailment in system z and probabilistic lexicographic entailment both satisfy the property of rational monotonicity and some irrelevance property while probabilistic entailment in system p does not we also analyze the relationships between the new formalisms here probabilistic entailment in system p is weaker than probabilistic entailment in system z which in turn is weaker than probabilistic lexicographic entailment moreover they all are weaker than entailment in probabilistic logic where default sentences are interpreted as strict sentences under natural conditions probabilistic entailment in system z and lexicographic entailment even coincide with such entailment in probabilistic logic while probabilistic entailment in system p does not finally we also present algorithms for reasoning under probabilistic entailment in system z and probabilistic lexicographic entailment and we give a precise picture of its complexity

what does a conditional knowledge base entail abstract this paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation a conditional knowledge base consisting of a set of conditional assertions of the type if then represents the explicit defeasible knowledge an agent has about the way the world generally behaves we look for a plausible definition of the set of all conditional assertions entailed by a conditional knowledge base in a previous paper kraus and the authors defined and studied preferential consequence relations they noticed that not all preferential relations could be considered as reasonable inference procedures this paper studies a more restricted class of consequence relations rational relations it is argued that any reasonable nonmonotonic inference procedure should define a rational relation it is shown that the rational relations are exactly those that may be represented by a ranked preferential model or by a nonstandard probabilistic model the rational closure of a conditional knowledge base is defined and shown to provide an attractive answer to the question of the title global properties of this closure operation are proved it is a cumulative operation it is also computationally tractable this paper assumes the underlying language is propositional

saturation nonmonotonic reasoning and the closed world assumption abstract to deal with the problem of implicit negative information in databases it is necessary to use a nonmonotonic form of reasoning the form of reasoning described in this paper uses an ordering relation on interpretations of the type i is inferior to j if every elementary fact true in i is true in j in simple cases this rule may be defined by a set p of formulas implies a formula q if every minimal model of p satisfies q in more general cases this definition leads to paradox if p is consistent and has no minimal model p must not imply q and its negation we have 1 1 carefully formulated the definition of a special type of implication called sub implication to avoid this type of paradox 2 2 proved that if p is a set of clauses then p has a minimal model and more generally that every model of a set of clauses is minored by one of its minimal models 3 3 developed in this last case subject to certain conditions such as the use of a particular set of clauses called groundable clauses or g clauses a decidable algorithm to compute whether p implies q q being practically any formula

a modal logic for subjective default reasoning abstract in this paper we introduce dml default modal logic dml is a logic endowed with a two place modal connective that has the intended meaning of if then normally on top of providing a well defined tool for analyzing common default reasoning dml allows nesting of the default operator we present a semantic framework in which many of the known default proof systems can be naturally characterized and prove soundness and completeness theorems for several such proof systems our semantics is a neighborhood modal semantics and it allows for subjective defaults that is defaults may vary among different worlds within the same model the semantics has an appealing intuitive interpretation and may be viewed as a set theoretic generalization of the probabilistic interpretations of default reasoning we show that our semantics is most general in the sense that any modal semantics that is sound for some basic axioms for default reasoning is a special case of our semantics such a generality result may serve to provide a semantical analysis of the relative strength of different proof systems and to show the nonexistence of semantics with certain properties