symbolic model checking for real time systems abstract we describe finite state programs over real numbered time in a guarded command language with real valued clocks or equivalently as finite automata with real valued clocks model checking answers the question which states of a real time program satisfy a branching time specification given in an extension of ctl with clock variables we develop an algorithm that computes this set of states symbolically as a fixpoint of a functional on state predicates without constructing the state space for this purpose we introduce a calculus on computation trees over real numbered time unfortunately many standard program properties such as response for all nonzeno execution sequences during which time diverges cannot be characterized by fixpoints we show that the expressiveness of the timed calculus is incomparable to the expressiveness of timed ctl fortunately this result does not impair the symbolic verification of implementable real time programs those whose safety constraints are machine closed with respect to diverging time and whose fairness constraints are restricted to finite upper bounds on clock values all timed ctl properties of such programs are shown to be computable as finitely approximable fixpoints in a simple decidable theory

dimension of the orbit of marked subspaces given a nilpotent endomorphism we consider the manifold of invariant subspaces having a fixed segre characteristic in linear algebra appl 332 334 2001 569 the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained here we obtain the explicit form of this deformation when the invariant subspace is marked and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces those with open orbit moreover we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace

the natural element method in solid mechanics the application of the natural element method nem 1 2 to boundary value problems in two dimensional small displacement elastostatics is presented the discrete model of the domain consists of a set of distinct nodes n and a polygonal description of the boundary in the natural element method the trial and test functions are constructed using natural neighbour interpolants these interpolants are based on the voronoi tessellation of the set of nodes n the interpolants are smooth c 1 everywhere except at the nodes where they are c 0 in one dimension nem is identical to linear nite elements the nem interpolant is strictly linear between adjacent nodes on the boundary of the convex hull which facilitates imposition of essential boundary conditions a methodology to model material discontinuities and non convex bodies cracks using nem is also described a standard displacement based galerkin procedure is used to obtain the discrete system of linear equations application of nem to various problems in solid mechanics which include the patch test gradient problems bimaterial interface and a static crack problem are presented excellent agreement with exact analytical solutions is obtained which exemplies the accuracy and robustness of nem and suggests its potential application in the context of other classes of problems crack growth plates and large deformations to name a few 1998 john wiley sons ltd

approximate abstractions of stochastic hybrid systems we present a constructive procedure for obtaining a finite approximate abstraction of a discrete time stochastic hybrid system the procedure consists of a partition of the state space of the system and depends on a controllable parameter given proper continuity assumptions on the model the approximation errors introduced by the abstraction procedure are explicitly computed and it is shown that they can be tuned through the parameter of the partition the abstraction is interpreted as a markov set chain we show that the enforcement of certain ergodic properties on the stochastic hybrid model implies the existence of a finite abstraction with finite error in time over the concrete model and allows introducing a finite time algorithm that computes the abstraction

testing preorders for probabilistic processes can be characterized by simulations the domestic dog may be exceptionally well suited for behavioral genetic studies owing to its population history and the striking behavior differences among breeds to explore to what extent and ho

probabilistic bisimilarity as testing equivalence abstract larsen and skou initiated the study of probabilistic bisimilarity and its characterisation in terms of tests later on van breugel et al showed that for labelled markov processes with continuous state spaces probabilistic bisimilarity nicely coincides with a simple notion of testing equivalence their proof employs advanced machinery from topology in the discrete case of finite state reactive probabilistic processes we prove that coincidence result with an elementary and more accessible proof

characteristic invariant subspaces generated by a single vector abstract if f is an endomorphism of a finite dimensional vector space v over a field k then an invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f the characteristic hull of a subset w of v is defined to be the smallest characteristic subspace in v that contains w it is known that characteristic subspaces that are not hyperinvariant can only exist when k 2 in this paper we study subspaces x which are the characteristic hull of a single element in the case where k 2 we derive a necessary and sufficient condition such that x is hyperinvariant

maximal aggregation of polynomial dynamical systems ordinary differential equations odes with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models here we propose an aggregation technique that rests on two notions of equivalence relating ode variables whenever they have the same solution backward criterion or if a self consistent system can be written for describing the evolution of sums of variables in the same equivalence class forward criterion a key feature of our proposal is to encode a polynomial ode system into a finitary structure akin to a formal chemical reaction network this enables the development of a discrete algorithm to efficiently compute the largest equivalence building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements the physical interpretability of the aggregation is shown on polynomial ode systems for biochemical reaction networks gene regulatory networks and evolutionary game theory

a theory of stochastic systems part ii process algebra this paper introduces pronounce spades a stochastic process algebra for discrete event systems that extends traditional process algebra with timed actions whose delay is governed by general a o continuous probability distributions the operational semantics is defined in terms of stochastic automata a model that uses clocks like in timed automata to symbolically represent randomly timed systems cf the accompanying paper p r d argenio j p katoen a theory of stochastic systems part i stochastic automata inf comput 2005 to appear we show that stochastic automata and are equally expressive and prove that the operational semantics of a term up to conversion of clocks is unique modulo symbolic bisimulation open probabilistic and structural bisimulation are proven to be congruences for and are equipped with an equational theory the equational theory is shown to be complete for structural bisimulation and allows to derive an expansion law

the lattice of characteristic subspaces of an endomorphism with jordan chevalley decomposition abstract given an endomorphism a over a finite dimensional vector space having jordan chevalley decomposition the lattices of invariant and hyperinvariant subspaces of a can be obtained from the nilpotent part of this decomposition we extend this result for lattices of characteristic subspaces we also obtain a generalization of shoda s theorem about the characterization of the existence of characteristic non hyperinvariant subspaces

geometric characterization and classification of marked subspaces abstract given an endomorphism of a finite dimensional c vector space we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces w e that is to say the invariant subspaces having some jordan basis which can be extended to a jordan basis of e both conditions are expressed in terms of the double family of subspaces ker f h im f d the starting point is a new criterion for extensibility of jordan bases

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

branching bisimulation congruence for probabilistic systems a notion of branching bisimilarity for the alternating model of probabilistic systems compatible with parallel composition is defined for a congruence result an internal transition immediately followed by a non trivial probability distribution is not considered inert a weaker definition of branching bisimilarity for the same model has been given earlier here we show that our branching bisimulation is the coarsest congruence for parallel composition that is included in the weaker version to support the use of the present equivalence as a reduction technique we also show that probabilistic ctl formulae are preserved by our equivalence and we provide a polynomial time algorithm deciding branching bisimilarity

characteristic subspaces and hyperinvariant frames abstract let f be an endomorphism of a finite dimensional vector space v over a field k an f invariant subspace is called hyperinvariant respectively characteristic if it is invariant under all endomorphisms respectively automorphisms that commute with f we assume k 2 since all characteristic subspaces are hyperinvariant if k 2 the hyperinvariant hull w h of a subspace w of v is defined to be the smallest hyperinvariant subspace of v that contains w the hyperinvariant kernel w h of w is the largest hyperinvariant subspace of v that is contained in w and the pair w h w h is the hyperinvariant frame of w in this paper we study hyperinvariant frames of characteristic non hyperinvariant subspaces w we show that all invariant subspaces in the interval w h w h are characteristic we use this result for the construction of characteristic non hyperinvariant subspaces

additive models of probabilistic processes we propose a new model of probabilistic processes in this model a probability is assigned to the action of a prefix and a probability distribution is assigned to the components of a parallel composition in addition the probability of a transition of a probabilistic summation is evaluated as the sum of the probabilities of the same transition of summands multiplied by the probabilities associated to them in the summation the concepts of strong bisimulation degree and weak bisimulation degree are introduced these notions provide us with continuous spectra of strong bisimilarities weak bisimilarities and observation congruences which equate probabilistic processes with different degrees of belief various equational laws of probabilistic processes with respect to these equivalence relations are presented and substitutivities of these equivalence relations under various combinators are established

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

the theory of interactive generalized semi markov processes in this paper we introduce the calculus of interactive generalized semi markov processes igsmps a stochastic process algebra which can express probabilistic timed delays with general distributions and synchronizable actions with zero duration and where choices may be probabilistic non deterministic and prioritized igsmp is equipped with a structural operational semantics which generates semantic models in the form of generalized semi markov processes gsmps i e probabilistic systems with generally distributed time extended with action transitions representing interaction among system components this is obtained by expressing the concurrent execution of delays through a variant of st semantics which is based on dynamic names the fact that names for delays are generated dynamically by the semantics makes it possible to define a notion of observational congruence for igsmp that abstracts from internal actions with zero duration simply as a combination of standard observational congruence and probabilistic bisimulation we also present a complete axiomatization for observational congruence over igsmp finally we show how to derive a gsmp from a given igsmp specification in order to evaluate the system performance and we present a case study

the characteristic subspace lattice of a linear transformation abstract given a square matrix a m n f the lattices of the hyperinvariant hinv a and characteristic chinv a subspaces coincide whenever f g f 2 if the characteristic polynomial of a splits over f a can be considered nilpotent in this paper we investigate the properties of the lattice chinv j when f g f 2 for a nilpotent matrix j in particular we prove it to be self dual

testing equivalences for processes abstract given a set of processes and a set of tests on these processes we show how to define in a natural way three different equivalences on processes these equivalences are applied to a particular language ccs we give associated complete proof systems and fully abstract models these models have a simple representation in terms of trees

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

quantales finite observations and strong bisimulation abstract it has often been claimed that bisimulation makes distinctions that cannot be observed in practice abramsky and vickers proposed an algebraic framework based on quantales for describing observations on concurrent processes without hidden transitions and used it in order to provide an observational explanation of several process equivalences ranging from hoare trace equivalence to ready simulation we follow their approach and argue that strong bisimulation can be explained in the same way at least in the case of image computable labelled transition systems

measurable stochastics for brane calculus the main aim of this work is to give a stochastic extension of the brane calculus along the lines of recent work by cardelli and mardare 2010 12 in this approach the semantics of a process is a measure of the stochastic distribution of possible derivations to this end we first introduce a compositional finitely branching labelled transition system for brane calculus interestingly the associated strong bisimulation is a congruence then we give a stochastic semantics to brane systems by defining them as markov processes over the measurable space generated by terms up to syntactic congruence and where the measures are indexed by the actions of this new lts finally we provide an sos presentation of this stochastic semantics which is compositional and syntax driven and moreover the induced rate bisimilarity is a congruence

observation equivalence as a testing equivalence a notion of testing is developed for transition systems with divergence the forms of testing include traces refusals copying and global testing both denotational and operational formulations of testing are given the equivalence based on this notion of testing is shown to coincide with observation equivalence

description of characteristic non hyperinvariant subspaces over the field gf 2 given a square matrix a an a invariant subspace is called hyperinvariant respectively characteristic if and only if it is also invariant for all matrices t respectively nonsingular matrices t that commute with a shoda s theorem gives a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces for a nilpotent matrix in gf 2 here we present an explicit construction for all subspaces of this type

process algebra for performance evaluation this paper surveys the theoretical developments in the field of stochastic process algebras process algebras where action occurrences may be subject to a delay that is determined by a random variable a huge class of resource sharing systems like large scale computers client server architectures networks can accurately be described using such stochastic specification formalisms the main emphasis of this paper is the treatment of operational semantics notions of equivalence and sound and complete axiomatisations of these equivalences for different types of markovian process algebras where delays are governed by exponential distributions starting from a simple actionless algebra for describing time homogeneous continuous time markov chains we consider the integration of actions and random delays both as a single entity like in known markovian process algebras like tipp pepa and empa and as separate entities like in the timed process algebras timed csp and tccs in total we consider four related calculi and investigate their relationship to existing markovian process algebras we also briefly indicate how one can profit from the separation of time and actions when incorporating more general non markovian distributions

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

generalized delaunay triangulations of non convex domains akrraet the delaunay triangulation associated with a finite set s of points in the plane is a triangulation of the convex hull of s delaunay triangulations have been used in a number of computational methods for example in lagrangian fluid dynamics in this paper we discuss two natural generalizations of the notion of delaunay triangulation to non convex domains we show that the two generalizations are equivalent and describe a method for the construction of generalized delaunay triangulations

extensions of jordan bases for invariant subspaces of a matrix a characterization is obtained for the matrices a with the property that every some jordan basis of every a invariant subspace can be extended to a jordan basis of a these results are based on a criterion for a jordan basis of an invariant subspace to be extendable to a jordan basis of the whole space the criterion involves two concepts the constancy property and the depth property

computing the cardinality of the lattice of characteristic subspaces abstract we obtain the cardinality of the lattice of characteristic subspaces of a nilpotent jordan matrix when the underlying field is g f 2 the only field where the lattices of characteristic and hyperinvariant subspaces can be different if the characteristic polynomial of the matrix splits in the field the general case can be reduced to the nilpotent jordan case results are complex and highly combinatorial and include the design of an algorithm

probabilistic bisimulations for quantum processes modeling and reasoning about concurrent quantum systems is very important for both distributed quantum computing and quantum protocol verification as a consequence a general framework formally describing communication and concurrency in complex quantum systems is necessary for this purpose we propose a model named qccs it is a natural quantum extension of classical value passing ccs which can deal with input and output of quantum states and unitary transformations and measurements on quantum systems the operational semantics of qccs is given in terms of probabilistic labeled transition system this semantics has many different features compared with the proposals in the available literature in order to describe the input and output of quantum systems which are possibly correlated with other components based on this operational semantics the notions of strong probabilistic bisimulation and weak probabilistic bisimulation between quantum processes are introduced furthermore some properties of these two probabilistic bisimulations such as congruence under various combinators are examined

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

algebraic laws for nondeterminism and concurrency since a nondeterministic and concurrent program may in general communicate repeatedly with its environment its meaning cannot be presented naturally as an input output function as is often done in the denotational approach to semantics in this paper an alternative is put forth first a definition is given of what it is for two programs or program parts to be equivalent for all observers then two program parts are said to be observation congruent if they are in all program contexts equivalent the behavior of a program part that is its meaning is defined to be its observation congruence class the paper demonstrates for a sequence of simple languages expressing finite terminating behaviors that in each case observation congruence can be axiomatized algebraically moreover with the addition of recursion and another simple extension the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties

picturing the lattice of invariant subspaces of a nilpotent complex matrix abstract let q be a nilpotent transformation acting on a finite dimensional complex vector space a method is given by which a diagrammatic representation of lat q the lattice of invariant subspaces of q can be obtained basically the method consists of adding to the hasse diagram of hyperlat q the finite lattice of hyperinvariant subspaces of q in a specified way some applications are given

a lagrangian fractional step method for the incompressible navier stokes equations on a periodic domain abstract in the lagrangian fractional step method introduced in this paper the fluid velocity and pressure are defined on a collection of n fluid markers at each time step these markers are used to generate a voronoi diagram and this diagram is used to construct finite difference operators corresponding to the divergence gradient and laplacian the splitting of the navier stokes equations leads to discrete helmholtz and poisson problems which we solve using a two grid method the nonlinear convection terms are modeled simply by the displacement of the fluid markers we have implemented this method on a periodic domain in the planee we describe an efficient algorithm for the numerical construction of periodic voronoi diagrams and we report on numerical results which indicate that the fractional step method is convergent of first order the overall work per time step is proportional to n log n

calculi for synchrony and asynchrony abstract a calculus for distributed computation is studied based upon four combinators a central idea is an abelian group of actions which models the interfaces between components of a distributed computing agent using a notion of bisimulation congruence relations are defined over computing agents and thence an algebraic theory is derived the calculus models both synchronous and asynchronous computation in particular it is shown that the author s calculus of communicating systems 1980 which is an asynchronous model is derivable from the calculus presented here

imposing essential boundary conditions in the natural element method by means of density scaled shapes a new method is proposed for the exact imposition of essential boundary conditions in the context of the natural element method nem this is a new technique in the field of computational mechanics and can be considered as a meshless method unlike most of these methods the ne shape functions are strictly interpolant and the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations however these shape functions are not strictly linear over non convex boundaries and the approach does not make the test functions vanish over the whole essential boundary region a modification of the initial nem version is considered based on shapes and complexes which are widely used in the field of scientific visualization using shapes in the context of the nem allows the construction of models entirely in terms of nodes and also ensures the linear precision of the interpolant over convex and non convex boundaries results on some benchmark problems are presented after a theoretical description of the method copyright 2000 john wiley sons ltd

a theory of stochastic systems part i stochastic automata this paper presents the theoretical underpinning of a model for symbolically representing probabilistic transition systems an extension of labelled transition systems for the modelling of general discrete as well as continuous or singular probability spaces these transition systems are particularly suited for modelling softly timed systems real time systems in which the time constraints are of random nature for continuous probability spaces these transition systems are infinite by nature stochastic automata represent their behaviour in a finite way this paper presents the model of stochastic automata their semantics in terms of probabilistic transition systems and studies several notions of bisimulation furthermore the relationship of stochastic automata to generalised semi markov processes is established

characteristic and hyperinvariant subspaces over the field gf 2 let f be an endomorphism of a vector space v over a field k an f invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f if k 2 then all characteristic subspaces are hyperinvariant if k 2 then there are endomorphisms f with invariant subspaces that are characteristic but not hyperinvariant in this paper we give a new proof of a theorem of shoda which provides a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces

the max plus algebra of the natural numbers has no finite equational basis this paper shows that the collection of identities which hold in the algebra n of the natural numbers with constant zero and binary operations of sum and maximum is not finitely based moreover it is proven that for every n the equations in at most n variables that hold in n do not form an equational basis as a stepping stone in the proof of these facts several results of independent interest are obtained in particular explicit descriptions of the free algebras in the variety generated by n are offered such descriptions are based upon a geometric characterization of the equations that hold in n which also yields that the equational theory of n is decidable in exponential time