dimension of slices through the sierpinski carpet for lebesgue typical the intersection of the sierpinski carpet with a line has if non empty dimension where fix the slope then we shall show on the one hand that this dimension is strictly less than for lebesgue almost every on the other hand for almost every according to the angle projection of the natural measure on this dimension is at least for any we find a connection between the box dimension of this intersection and the local dimension of at

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

approximating markov processes by averaging normally one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition we however take a dual view of markov processes as transformers of bounded measurable functions this is very much in the same spirit as a predicate transformer view which is dual to the state transformer view of transition systems we redevelop the theory of labelled markov processes from this viewpoint in particular we explore approximation theory we obtain three main results i it is possible to define bisimulation on general measure spaces and show that it is an equivalence relation the logical characterization of bisimulation can be done straightforwardly and generally ii a new and flexible approach to approximation based on averaging can be given this vastly generalizes and streamlines the idea of using conditional expectations to compute approximations iii we show that there is a minimal process bisimulation equivalent to a given process and this minimal process is obtained as the limit of the finite approximants

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

formally reasoning about quality in recent years there has been a growing need and interest in formally reasoning about the quality of software and hardware systems as opposed to traditional verification in which one considers the question of whether a system satisfies a given specification or not reasoning about quality addresses the question of how well the system satisfies the specification we distinguish between two approaches to specifying quality the first propositional quality extends the specification formalism with propositional quality operators which prioritize and weight different satisfaction possibilities the second temporal quality refines the eventually operators of the specification formalism with discounting operators whose semantics takes into an account the delay incurred in their satisfaction in this article we introduce two quantitative extensions of linear temporal logic ltl one by propositional quality operators and one by discounting operators in both logics the satisfaction value of a specification is a number in 0 1 which describes the quality of the satisfaction we demonstrate the usefulness of both extensions and study the decidability and complexity of the decision and search problems for them as well as for extensions of ltl that combine both types of operators

approximating markov processes through filtration in this paper we define a probabilistic version of filtration and use it to provide a finite approximation of markov processes in order to measure the approximation we employ probability logic to construct the final markov process and define a metric on the set of markov processes through this logic moreover we show that the set endowed with this metric is a polish space finally we point to some questions connecting approximation to uniformity and approximate bisimilarity as topics for future research

probabilistic propositional temporal logics we present two closely related propositional probabilistic temporal logics based on temporal logics of branching time as introduced by ben ari pnueli and manna acta inform 20 1983 207 226 emerson and halpern proceedings 14th acm sympos theory of comput 1982 pp 169 179 and emerson and clarke sci comput program 2 1982 241 266 the first logic ptlf is interpreted over finite models while the second logic ptlb which is an extension of the first one is interpreted over infinite models with transition probabilities bounded away from 0 the logic ptlf allows us to reason about finite state sequential probabilistic programs and the logic ptlb allows us to reason about finite state concurrent probabilistic programs without any explicit reference to the actual values of their state transition probabilities a generalization of the tableau method yields deterministic single exponential time decision procedures for our logics and complete axiomatizations of them are given several meta results including the absence of a finite model property for ptlb and the connection between satisfiable formulae of ptlb and finite state concurrent probabilistic programs are also discussed

weak bisimulation for probabilistic timed automata we are interested in describing timed systems that exhibit probabilistic behaviour to this purpose we consider a model of probabilistic timed automata and introduce a concept of weak bisimulation for these automata together with an algorithm to decide it the weak bisimulation relation is shown to be preserved when either time or probability is abstracted away as an application we use weak bisimulation for probabilistic timed automata to model and analyze a timing attack on the dining cryptographers protocol

approximating labelled markov processes labelled markov processes are probabilistic versions of labelled transition systems in general the state space of a labelled markov process may be a continuum in this paper we study approximation techniques for continuousstate labelled markov processes we show that the collection of labelled markov processes carries a polish space structure with a countable basis given by finite state markov chains with rational probabilities thus permitting the approximation of quantitative observations e g an integral of a continuous function of a continuous state labelled markov process by the observations on finite state markov chains the primary technical tools that we develop to reach these results are a variant of a finite model theorem for the modal logic used to characterize bisimulation and an isomorphism between the poset of markov processes ordered by simulation with the continuous dcpo proc defined as the solution of the recursive domain equation proc l ppr proc the isomorphism between labelled markov processes and proc can be independently viewed as a full abstraction result relating an operational labelled markov process and a denotational proc model and yields a logic complete for reasoning about simulation for continuous state processes

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

metrics for labelled markov processes the notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions thus a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent this instability is due to the quantitative nature of probabilistic processes in a situation where the process behavior has a quantitative aspect there should be a more robust approach to process equivalence this paper studies a metric between labelled markov processes this metric has the property that processes are at zero distance if and only if they are bisimilar the metric is inspired by earlier work on logics for characterizing bisimulation and is related in spirit to the hutchinson metric

bisimulation for probabilistic transition systems a coalgebraic approach the notion of bisimulation as proposed by larsen and skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of aczel and mendier in terms of a set functor this coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving borel probability measures under reasonable conditions generalized probabilistic bisimilarity can be characterized categorically application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation

the category theoretic solution of recursive metric space equations it is well known that one can use an adaptation of the inverse limit construction to solve recursive equations in the category of complete ultrametric spaces we show that this construction generalizes to a large class of categories with metric space structure on each set of morphisms the exact nature of the objects is less important in particular the construction immediately applies to categories where the objects are ultrametric spaces with extra structure and where the morphisms preserve this extra structure the generalization is inspired by classical domain theoretic work by smyth and plotkin for many of the categories we consider there is a natural subcategory in which each set of morphisms is required to be a compact metric space our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category as another application we present a construction that relates solutions of generalized domain equations in the sense of smyth and plotkin to solutions of equations in our class of categories our primary motivation for solving generalized recursive metric space equations comes from recent and ongoing work on kripke style models in which the sets of worlds must be recursively defined we show a series of examples motivated by this line of work

polynomial time algorithms for computing distances of fuzzy transition systems behaviour distances to measure the resemblance of two states in a nondeterministic fuzzy transition system have been proposed recently in literature such a distance defined as a pseudo ultrametric over the state space of the model provides a quantitative analogue of bisimilarity in this paper we focus on the problem of computing these distances we first extend the definition of the pseudo ultrametric by introducing discount such that the discounting factor being equal to 1 captures the original definition we then provide polynomial time algorithms to calculate the behavioural distances in both the non discounted and the discounted setting the algorithm is strongly polynomial in the former case

differential privacy in probabilistic systems abstract ever since proposed by dwork differential privacy has been a hot topic in academia however few attempts have been made on reasoning about differential privacy at a system level in this paper we propose a formal framework to verify differential privacy in probabilistic systems with a metric on the states of a system we formalize differential privacy by the ratio of the probabilities in the distributions after the same labeled transitions of relevant states we explain how traditional differential privacy can be embedded in our framework and raise an infimum metric the least distance between two states while not violating differential privacy it is proven that the infimum metric is also a metric instance of differential privacy itself furthermore we propose a two level logic a privacy variant of the familiar hennessy milner logic to characterize differential privacy in our framework our results have close relations to probabilistic bisimilarity as well

differentiable points of sierpinski like sponges abstract for the sierpinski like sponge in r n we obtain the hausdorff dimension of the set of differentiable points such that there exists an n 1 rectifiable subsurface which has a tangent hyperplane at the point using the techniques such as multi dynamical systems geometric types of slices and restricted frobenius problem furthermore we can use the dimension of the set of differentiable points as a bilipschitz invariant to classify sierpinski like sponges in terms of bilipschitz equivalence

reactive generative and stratified models of probabilistic processes abstract we introduce three models of probabilistic processes namely reactive generative and stratified these models are investigated within the context of pccs an extension of milner s sccs in which each summand of a process summation expression is guarded by a probability and the sum of these probabilities is 1 for each model we present a structural operational semantics of pccs and a notion of bisimulation equivalence which we prove to be a congruence we also show that the models form a hierarchy the reactive model is derivable from the generative model by abstraction from the relative probabilities of different actions and the generative model is derivable from the stratified model by abstraction from the purely probabilistic branching structure moreover the classical nonprobabilistic model is derivable from each of these models by abstraction from all probabilities

probabilistic supervisory control of probabilistic discrete event systems this paper considers supervisory control of probabilistic discrete event systems pdess pdess are modeled as generators of probabilistic languages the supervisory control problem considered is to find if possible a supervisor under whose control the behavior of a plant is identical to a given probabilistic specification the probabilistic supervisors we employ are a generalization of the deterministic ones previously employed in the literature at any state the supervisor enables disables events with certain probabilities necessary and sufficient conditions for the existence of such a supervisor and an algorithm for its computation are presented

spectral statistics of bernoulli matrix ensembles a random walk approach i we investigate the eigenvalue statistics of random bernoulli matrices where the matrix elements are chosen independently from a binary set with equal probability this is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues an approach which is similar to dyson s brownian motion model but with important modifications in particular we show our process is described by a fokker planck equation up to an error margin which vanishes in the limit of large matrix dimension the stationary solution of which corresponds to the joint probability density function of certain well known fixed trace gaussian ensembles

on the continuity of the hausdorff dimension of the univoque set abstract in a recent paper komornik et al 2017 19 proved a long conjectured formula for the hausdorff dimension of the set u q of numbers having a unique expansion in the non integer base q and showed that this hausdorff dimension is continuous in q unfortunately their proof contained a gap which appears difficult to fix this article gives a completely different proof of these results using a more direct combinatorial approach

trace formulae and spectral statistics for discrete laplacians on regular graphs i following the derivation of the trace formulas in the first paper in this series we establish here a connection between the spectral statistics of random regular graphs and the predictions of random matrix theory rmt this follows from the known poisson distribution of cycle counts in regular graphs in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely the result is analogous to the so called diagonal approximation in quantum chaos we also show that by assuming that the spectral correlations are given by rmt to all orders we can compute the leading deviations from the poisson distribution for cycle counts we provide numerical evidence which supports this conjecture

random intersections of thick cantor sets let cl c2 be cantor sets embedded in the real line and let ci t2 be their respective thicknesses if rljr2 1 then it is well known that the difference set ci c2 is a disjoint union of closed intervals b williams showed that for some t c int cl c2 it may be that c n c2 t is as small as a single point however the author previously showed that generically the other extreme is true c n c2 t contains a cantor set for all t in a generic subset of cl c2 this paper shows that small intersections of thick cantor sets are also rare in the sense of lebesgue measure if t1lt2 1 then ci n c2 t contains a cantor set for almost all t in ci c2 if cl c2 are cantor sets embedded in the real line then their difference set is ci c2 x yjxcc1c andy 0c2 the difference set has another more dynamical definition as c0 0c2 t ic i 2 c2t 0 where c2 t x t i x c c0 is the translation of c2 by the amount t there are two reasons to say that the second definition is dynamical first it gives a dynamic way of visualizing the difference set if we think of c0 as being fixed in the real line and think of c2 as sliding across co with unit speed then c0 c2 can be thought of as giving those times when the moving copy of c2 intersects c1 second it has become a tool for studying dynamical systems one cantor set sliding over another one comes up in various studies of homoclinic phenomena such as infinitely many sinks ni antimonotonicity kky and q explosions pt1 for an elementary explanation of this see gh pp 331 342 or r pp 110 115 this has led to a number of problems and results of the following form given conditions on the sizes of c0 and c2 what can be said of the sizes of either c 0 c2 or c1n 02 1c t for t e co c2 a wide variety of notions of size have been used such as cardinality topology measure hausdorff dimension limit capacity and thickness see for example hky kp mo pt2 ps s and w in this paper we will be concerned with the thickness of c0 and c2 and our conclusion will be about the topology of c l n c2 t for almost every t e c l c2 it is not hard to show that the difference set of two cantor sets cl c2 is always a compact perfect set so the simplest structure that we can expect c0 c2 to have is the disjoint union of closed intervals there is a condition we can put on c0 and c2 that will guarantee this if ti f2 are the thicknesses of c0 c2 and received by the editors october 14 1997 1991 mathematics subject classification primary 28a80 secondary 58f99

self similar measures and intersections of cantor sets it is natural to expect that the arithmetic sum of two cantor sets should have positive lebesgue measure if the sum of their dimensions exceeds 1 but there are many known counterexamples e g when both sets are the middle ca cantor set and ca e 1 1 we show that for any compact set k and for a e ag e 0 1 the arithmetic sum of k and the middle ae cantor set does indeed have positive lebesgue measure when the sum of their hausdorff dimensions exceeds 1 in this case we also determine the essential su premum as the translation parameter t varies of the dimension of the intersection of k t with the middle ca cantor set we also establish a new property of the infinite bernoutlli convolutions lj the distributions of random series e 0 a where the signs are chosen independently with probabilities p 1 p let 1 ql q2 2 for p 2 near 2 and for a e a in some nonempty interval iv is absolutely continuous 2 1 and its density is in lil but not in lq2 we also answer a question of kahane concerning the fourier transform of vl 2

bisimulation and cocongruence for probabilistic systems we introduce a new notion of bisimulation called event bisimulation on labelled markov processes and compare it with the now standard notion of probabilistic bisimulation originally due to larsen and skou event bisimulation uses a sub algebra as the basic carrier of information rather than an equivalence relation the resulting notion is thus based on measurable subsets rather than on points hence the name event bisimulation applies smoothly for general measure spaces bisimulation on the other hand is known only to work satisfactorily for analytic spaces we prove the logical characterization theorem for event bisimulation without having to invoke any of the subtle aspects of analytic spaces that feature prominently in the corresponding proof for ordinary bisimulation these complexities only arise when we show that on analytic spaces the two concepts coincide we show that the concept of event bisimulation arises naturally from taking the cocongruence point of view for probabilistic systems we show that the theory can be given a pleasing categorical treatment in line with general coalgebraic principles as an easy application of these ideas we develop a notion of almost sure bisimulation the theory comes almost for free once we modify giry s monad appropriately

intersecting random translates of invariant cantor sets given two cantor setsx andy in 0 1 invariant under the mapx b x mod 1 the hausdorff dimension of x t y is constant almost everywhere whenx y are defined by admissible digits in baseb and more generally by sofic systems we compute this dimension in terms of the largest lyapunov exponent of a random product of matrices the results are extended to higher dimensions and multiple intersections

the asymptotic distribution of a single eigenvalue gap of a wigner matrix we show that the distribution of a suitable rescaling of a single eigenvalue gap lambda i 1 m n lambda i m n of a random wigner matrix ensemble in the bulk is asymptotically given by the gaudin mehta distribution if the wigner ensemble obeys a finite moment condition and matches moments with the gue ensemble to fourth order this is new even in the gue case as prior results establishing the gaudin mehta law required either an averaging in the eigenvalue index parameter i or fixing the energy level u instead of the eigenvalue index the extension from the gue case to the wigner case is a routine application of the four moment theorem the main difficulty is to establish the approximate independence of the eigenvalue counting function n infty x tilde m n where tilde m n is a suitably rescaled version of m n with the event that there is no spectrum in an interval x x s in the case of a gue matrix this will be done through some general considerations regarding determinantal processes given by a projection kernel

the complexity of probabilistic verification we determine the complexity of testing whether a finite state sequential or concurrent probabilistic program satisfies its specification expressed in linear time temporal logic for sequential programs we present an algorithm that runs in time linear in the program and exponential in the specification and also show that the problem is in pspace matching the known lower bound for concurrent programs we show that the problem can be solved in time polynomial in the program and doubly exponential in the specification and prove that it is complete for double exponential time we also address these questions for specifications described by automata or formulas in extended temporal logic

composition and behaviors of probabilistic i o automata abstract we augment the i o automaton model of lynch and tuttle with probability as a step toward the ultimate goal of obtaining a useful tool for specifying and reasoning about asynchronous probabilistic systems our new model called probabilistic i o automata preserves the fundamental properties of the i o automaton model such as the asymmetric treatment of input and output and the pleasant notion of asynchronous composition for certain classes of probabilistic i o automata we show that probabilistic behavior maps which are an abstract representation of i o automaton behavior in terms of a certain expectation operator are compositional and fully abstract with respect to a natural notion of probabilistic testing

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

approximating and computing behavioural distances in probabilistic transition systems in an earlier paper we presented a pseudometric on the states of a probabilistic transition system yielding a quantitative notion of behavioural equivalence the behavioural pseudometric was defined via the terminal coalgebra of a functor based on a metric on borel probability measures in the present paper we give a polynomial time algorithm based on linear programming to calculate the distances between states up to a prescribed degree of accuracy

logical characterization of branching metrics for nondeterministic probabilistic transition systems abstract in this paper we propose a logical characterization of bi simulation metrics obtained by a probabilistic variant of hennessy milner logic enriched with variables whose semantics is defined following the equational calculus approach our characterization is based on the novel notions of mimicking formulae and syntactical distance on formulae the former ones are the quantitative analogous to characteristic formulae the latter is a 1 bounded pseudometric on formulae measuring their syntactical disparities the characterization is obtained by showing that the bi simulation distance between processes corresponds to the syntactical distance between their mimicking formulae we also discuss the expressive power of mimicking formulae with respect to probabilistic bi simulations we show that two processes are bisimilar if and only if their mimicking formulae are syntactically equivalent moreover we obtain that mimicking formulae of processes coincide with their characteristic formulae for ready simulation and that negation free mimicking formulae coincide with the characteristic formulae for simulation

characterize branching distance in terms of η α bisimilarity in order to describe approximate equivalence between processes the notion of branching distance has been introduced in the framework of quantitative transition systems this paper introduces the notion of h a bisimilarity and explores the relationship between it and the branching distance

testing probabilistic equivalence through reinforcement learning checking if a given system implementation respects its specification is often done by proving that the two are equivalent the equivalence is chosen in particular for its computability and of course for its meaning that is for its adequacy with what is observable from the two systems implementation and specification trace equivalence is easily testable decidable from interaction but often considered too weak in contrast bisimulation is accepted as the canonical equivalence for interaction but it is not testable richer than an equivalence is a form of distance it is zero between equivalent systems and it provides an estimation of their difference if the systems are not equivalent our main contribution is to define such a distance in a context where 1 the two systems to be compared have a stochastic behavior 2 the model of one of them e g the implementation is unknown hence our only knowledge is obtained by interacting with it 3 consequently the target equivalence observed when distance is zero must be testable to overcome the problem that the model is unknown we use a reinforcement learning approach that provides powerful stochastic algorithms that only need to interact with the model our second main contribution is a new family of testable equivalences called k moment the weakest of them 1 moment equivalence is trace equivalence as k grows k moment equivalences become finer all remaining as well as their limit weaker than bisimulation we propose a framework to define and test a bigger class of testable equivalences test observation equivalences toes and we show how they can be made coarser or not by tuning some parameters

recursively defined metric spaces without contraction in this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1 bounded complete metric spaces and nonexpansive functions in contrast to previous approaches we do not assume that the endofunctors are locally contractive and our results do not depend on banach s fixed point theorem our approach is particularly suitable for constructing models of systems that feature quantitative data for instance using the kantorovich metric on probability measures we construct a quantitative model for probabilistic transition systems the metric in our model can reasonably be seen as measuring the behavioural distance between states of the system it depends exclusively on the transition probabilities and not on an arbitrary discount factor

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

bulk eigenvalue statistics for random regular graphs ak was partly supported by swiss national science foundation grant 144662 hty was partly supported by nsf grant dms 1307444 and a simons investigator fellowship

verification of general markov decision processes by approximate similarity relations and policy refinement in this work we introduce new approximate similarity relations that are shown to be key for policy or control synthesis over general markov decision processes the models of interest are discrete time markov decision processes endowed with uncountably infinite state spaces and metric output or observation spaces the new relations underpinned by the use of metrics allow in particular for a useful trade off between deviations over probability distributions on states and distances between model outputs we show that the new probabilistic similarity relations inspired by a notion of simulation developed for finite state models can be effectively employed over general markov decision processes for verification purposes and specifically for control refinement from abstract models

bisimulation metrics and norms for real weighted automata abstract we develop a new bisimulation pseudo metric for weighted finite automata wfa that generalizes boreale s linear bisimulation relation our metrics are induced by seminorms on the state space of wfa our development is based on spectral properties of sets of linear operators in particular the joint spectral radius of the transition matrices of wfa plays a central role we also study continuity properties of the bisimulation pseudometric establish an undecidability result for computing the metric and give a preliminary account of applications to spectral learning of weighted automata

approximating markovian testing equivalence several approaches have been proposed to relax behavioral equivalences for fine grain models including probabilities and time all of them face two problems behind the notion of approximation i e the lack of transitivity and the efficiency of the verification algorithm while the typical equivalence under approximation is bisimulation we present a relaxation of markovian testing equivalence in a process algebraic framework in this coarser setting we show that it is particularly intuitive to manage separately three different dimensions of the approximation execution time event probability and observed behavior by illustrating in each case results concerning the two problems mentioned above finally a unified definition combining the three orthogonal aspects is provided in order to favor trade off analyses

on metrics for probabilistic systems definitions and algorithms in this paper we consider the behavioral pseudometrics for probabilistic systems which are a quantitative analogue of probabilistic bisimilarity in the sense that the distance zero captures the probabilistic bisimilarity the model we are interested in is probabilistic automata which are based on state transition systems and make a clear distinction between probabilistic and nondeterministic choices the pseudometrics are defined as the greatest fixpoint of a monotonic functional on the complete lattice of state metrics a distinguished characteristic of this pseudometric lies in that it does not discount the future which addresses some algorithmic challenges to compute the distance of two states in the model we solve this problem by providing an approximation algorithm up to any desired degree of accuracy e the distance can be approximated to within e in time exponential in the size of the model and logarithmic in email protected one of the key ingredients of our algorithm is to express a pseudometric being a post fixpoint as the elementary sentence over real closed fields which allows us to exploit tarski s decision procedure together with the binary search to approximate the behavioral distance

non expansive e bisimulations for probabilistic processes e bisimulation equivalence has been proposed in the literature as a technique to study the concept of behavioral distance between probabilistic processes in this paper we first consider the generative model of probabilistic processes and introduce two stronger equivalence notions action e bisimulation and global e bisimulation for each of these three equivalence notions we propose an sos transition rule format ensuring the property of non expansiveness non expansiveness means that if the behavioral distance between s i and t i is e i then the behavioral distance between f s 1 s n and f t 1 t n is no more that e 1 e n as expected the stronger the e bisimulation considered the weaker the constraints of the transition rule format then we switch to the reactive model of probabilistic processes and we propose a rule format for e bisimulation and action e bisimulation arguing that global e bisimulation is not needed in such a context

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

approximate bisimulation relations for constrained linear systems in this paper we define the notion of approximate bisimulation relation between two continuous systems while exact bisimulation requires that the observations of two systems are and remain identical approximate bisimulation allows the observations to be different provided the distance between them remains bounded by some parameter called precision approximate bisimulation relations are conveniently defined as level sets of a so called bisimulation function which can be characterized using lyapunov like differential inequalities for a class of constrained linear systems we develop computationally effective characterizations of bisimulation functions that can be interpreted in terms of linear matrix inequalities and optimal values of static games we derive a method to evaluate the precision of the approximate bisimulation relation between a constrained linear system and its projection this method has been implemented in a matlab toolbox matisse an example of use of the toolbox in the context of safety verification is shown

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

intersecting nonhomogeneous cantor sets with their translations abstract a scheme is given to compute the hausdorff dimensions for the intersection of a class of nonhomogeneous cantor sets with their translations

a behavioral distance for fuzzy transition systems in contrast with the existing approaches to exact bisimulation for fuzzy systems we introduce a robust notion of behavioral distance to measure the behavioral similarity of nondeterministic fuzzy transition systems which are a generalization of fuzzy automata this behavioral distance provides a quantitative analogue of bisimilarity and is defined as the greatest fixed point of a suitable monotonic function the behavioral distance has the important property that two systems are at zero distance if and only if they are bisimilar moreover for any given threshold we find that systems with behavioral distances bounded by the threshold are equivalent in addition we show that two system combinators parallel composition and product are nonexpansive with respect to our behavioral distance which makes compositional verification possible the theory developed here is applicable to the quantitative verification approximate reduction and reliability analysis of fuzzy transition systems

metrics for weighted transition systems axiomatization and complexity abstract simulation distances are essentially approximations of simulation which provide a measure of the extent by which behaviors in systems are inequivalent in this paper we consider the general quantitative model of weighted transition systems where transitions are labeled with elements of a finite metric space we study the so called point wise and accumulating simulation distances which provide extensions to the well known boolean notion of simulation on labeled transition systems we introduce weighted process algebras for finite and regular behavior and offer sound and approximate complete inference systems for the proposed simulation distances we also settle the algorithmic complexity of computing the simulation distances

optimal supervisory control of probabilistic discrete event systems probabilistic discrete event systems pdes are modeled as generators of probabilistic languages and the supervisors employed are a probabilistic generalization of deterministic supervisors used in standard supervisory control theory in the case when there exists no probabilistic supervisor such that the behavior of a plant under control exactly matches the probabilistic language given as the requirements specification we want to find a probabilistic control such that the behavior of the plant under control is as close as possible to the desired behavior first as a measure of this proximity a pseudometric on states of generators is defined two algorithms for the calculation of the distance between states in this pseudometric are described then an algorithm to synthesize a probabilistic supervisor that minimizes the distance between generators representing the achievable and required behavior of the plant is presented

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

dimensions of slices through a class of generalized sierpinski sponges abstract in this paper we discuss the intersections of a class of generalized sierpinski sponges in r n with an n 1 dimensional hyperplane of integer orthogonal direction we prove that under a suitable congruence condition the dimension equals the dimension of the generalized sierpinski sponge minus 1

approximations of stochastic hybrid systems this paper develops a notion of approximation for a class of stochastic hybrid systems that includes as special cases both jump linear stochastic systems and linear stochastic hybrid automata our approximation framework is based on the recently developed notion of the so called stochastic simulation functions these lyapunov like functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction for the class of jump linear stochastic systems and linear stochastic hybrid automata we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem this enables us to compute the modeling error incurred by abstracting some of the continuous dynamics or by neglecting the influence of stochastic noise or even the influence of stochastic discrete jumps

characterization and computation of infinite horizon specifications over markov processes this work is devoted to the formal verification of specifications over general discrete time markov processes with an emphasis on infinite horizon properties these properties formulated in a modal logic known as pctl can be expressed through value functions defined over the state space of the process the main goal is to understand how structural features of the model primarily the presence of absorbing sets influence the uniqueness of the solutions of corresponding bellman equations furthermore this contribution shows that the investigation of these structural features leads to new computational techniques to calculate the specifications of interest the emphasis is to derive approximation techniques with associated explicit convergence rates and formal error bounds

value passing ccs with noisy channels value passing ccs a full version of milner s ccs is a process algebra in which actions consist of sending and receiving values through noiseless communication channels the full calculus is a succinct yet expressive language for the specification and verification of reactive systems taking into account the reality of channel noise in reactive systems in this paper we introduce an extension of value passing ccs called value passing ccs with noisy channels vccs n in which noise is described by a probability distribution over the values after presenting the reduction operational semantics and labelled operational semantics of vccs n we develop the theory of behavioural equivalence by introducing barbed equivalence barbed congruence bisimilarity and full bisimilarity in particular we show that barbed equivalence and barbed congruence coincide with bisimilarity and full bisimilarity respectively based upon the labelled operational semantics of vccs n we establish a probabilistic modal logic for expressing system properties and show its connection with the notion of bisimilarity finally we use vccs n to model a communication protocol for ensuring the reliable transmission of data across an error prone channel

hyperbolicity and the creation of homoclinic orbits we consider one parameter families sp t e r of diffeomorphisms on surfaces which display a homoclinic tangency for yt 0 and are hyperbolic for yt 0 i e pf has a hyperbolic nonwandering set the tangency unfolds into transversal homoclinic orbits for yt positive for many of these families we prove that p is also hyperbolic for most small positive values of yt which implies much regularity of the dynamical structure a main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency

the rate distortion dimension of sets and measures data compression of independent samples drawn from a fractal set is considered the asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution this quantity is shown to be identical to renyi s 1959 information dimension for self similar fractal sets this dimension is distribution dependent in sharp contrast with the behavior of absolutely continuous measures the rate distortion dimension of a set is defined as the maximal rate distortion dimension for distributions supported on this set kolmogorov s metric dimension is an upper bound on the rate distortion dimension while the hausdorff dimension is a lower bound examples of sets for which the rate distortion dimension differs from these bounds are provided

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices