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

presburger liveness verification of discrete timed automata using an automata theoretic approach we investigate the decidability of liveness properties called presburger liveness properties for timed automata when presburger formulas on configurations are allowed while the general problem of checking a temporal logic such as tptl augmented with presburger clock constraints is undecidable we show that there are various classes of presburger liveness properties which are decidable for discrete timed automata for instance it is decidable given a discrete timed automaton a and a presburger property p whether there exists an path of a where p holds infinitely often we also show that other classes of presburger liveness properties are indeed undecidable for discrete timed automata e g whether p holds infinitely often for each path of a these results might give insights into the corresponding problems for timed automata over dense domains and help in the definition of a fragment of linear temporal logic augmented with presburger conditions on configurations which is decidable for model checking timed automata

symbolic model checking for probabilistic timed automata probabilistic timed automata are timed automata extended with discrete probability distributions and can be used to model timed randomised protocols or fault tolerant systems we present symbolic model checking algorithms for probabilistic timed automata to verify both qualitative temporal logic properties corresponding to satisfaction with probability 0 or 1 and quantitative properties corresponding to satisfaction with arbitrary probability the algorithms operate on zones which represent sets of valuations of the probabilistic timed automaton s clocks our method considers only those system behaviours which guarantee the divergence of time with probability 1 the paper presents a symbolic framework for the verification of probabilistic timed automata against the probabilistic timed temporal logic ptctl we also report on a prototype implementation of the algorithms using difference bound matrices and present the results of its application to the csma cd and firewire root contention protocol case studies

symbolic optimal expected time reachability computation and controller synthesis for probabilistic timed automata in this paper we consider the problem of computing the optimal minimum or maximum expected time to reach a target and the synthesis of an optimal controller for a probabilistic timed automaton pta although this problem admits solutions that employ the digital clocks abstraction or statistical model checking symbolic methods based on zones and priced zones fail due to the difficulty of incorporating probabilistic branching in the context of dense time we work in a generalisation of the setting introduced by asarin and maler for the corresponding problem for timed automata where simple and nice functions are introduced to ensure finiteness of the dense time representation we find restrictions sufficient for value iteration to converge to the optimal expected time on the uncountable markov decision process representing the semantics of a pta we formulate bellman operators on the backwards zone graph of a pta and prove that value iteration using these operators equals that computed over the pta s semantics this enables us to extract an optimal controller from value iteration in the standard way

reachability analysis of dynamical systems having piecewise constant derivatives abstract in this paper we consider a class of hybrid systems namely dynamical systems with piecewise constant derivatives pcd systems such systems consist of a partition of the euclidean space into a finite set of polyhedral sets regions within each region the dynamics is defined by a constant vector field hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction with respect to such systems we investigate the reachability question given an effective description of the systems and of two polyhedral subsets p and q of the state space is there a trajectory starting at some x p and reaching some point in q our main results are a decision procedure for two dimensional systems and an undecidability result for three or more dimensions

updatable timed automata we investigate extensions of alur and dill s timed automata based on the possibility to update the clocks in a more elaborate way than simply reset them to zero we call these automata updatable timed automata they form an undecidable class of models in the sense that emptiness checking is not decidable however using an extension of the region graph construction we exhibit interesting decidable subclasses in a surprising way decidability depends on the nature of the clock constraints which are used diagonal free or not whereas these constraints play identical roles in timed automata we thus describe in a quite precise way the thin frontier between decidable and undecidable classes of updatable timed automata we also study the expressive power of updatable timed automata it turns out that any up datable automaton belonging to some decidable subclass can be effectively transformed into an equivalent timed automaton without updates but with silent transitions the transformation suffers from an enormous combinatorics blow up which seems unavoidable therefore updatable timed automata appear to be a concise model for representing and analyzing large classes of timed systems

palindromic automorphisms of free groups abstract let f n be the free group of rank n with free basis x x 1 x n a palindrome is a word in x 1 that reads the same backwards as forwards the palindromic automorphism group a n of f n consists of those automorphisms that map each x i to a palindrome in this paper we investigate linear representations of a n and prove that a 2 is linear we obtain conjugacy classes of involutions in a 2 and investigate residual nilpotency of a n and some of its subgroups let ia n be the group of those automorphisms of f n that act trivially on the abelianisation pi n be the palindromic torelli group of f n and let e a n be the elementary palindromic automorphism group of f n we prove that pi n ia n e a n this result strengthens a recent result of fullarton 2

compact families of piecewise constant functions in lp 0 t b abstract a strong compactness result in the spirit of the lions aubin simon lemma is proven for piecewise constant functions in time u with values in a banach space the main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts u u instead of all time shifts u u h for h 0 as required in simon s compactness theorem this simplifies significantly the application of the rothe method in the existence analysis of parabolic problems

generalized discrete timed automata decidable approximations for safety verification we consider generalized discrete timed automata with general linear relations over clocks and parameterized constants as clock constraints and with parameterized durations we look at three approximation techniques i e the r reset bounded approximation the b bounded approximation and the b r crossing bounded approximation and derive automata theoretic characterizations of the binary reachability under these approximations the characterizations allow us to show that the safety analysis problem is decidable for generalized discrete timed automata with unit durations and for deterministic generalized discrete timed automata with parameterized durations an example specification written in astral is used to run a number of experiments using one of the approximation techniques

pushdown timed automata a binary reachability characterization and safety verification abstract we consider pushdown timed automata ptas that are timed automata with dense clocks augmented with a pushdown stack a configuration of a pta includes a state dense clock values and a stack word by using the pattern technique we give a decidable characterization of the binary reachability i e the set of all pairs of configurations such that one can reach the other of a pta since a timed automaton can be treated as a pta without the pushdown stack we can show that the binary reachability of a timed automaton is definable in the additive theory of reals and integers the results can be used to verify a class of properties containing linear relations over both dense variables and unbounded discrete variables the properties previously could not be verified using the classic region technique nor expressed by timed temporal logics for timed automata and ctl for pushdown systems the results are also extended to other generalizations of timed automata

a causal semantics for time petri nets the objective of this work is to give time petri nets a partial order semantics akin to the nonsequential processes of untimed net systems to this end a time process of a time petri net is defined as a traditionally constructed causal process with a valid timing a timing is a labelling that attaches occurrence times to the events of the process that must satisfy specific validness criteria the main result of the paper is the bijective correspondence between firing schedules the classical interleaving semantics of time petri nets and linearizations of time processes the result shows that time processes correctly represent the behavior of the system using the definition of validness an efficient algorithm for checking validness of given timings is derived also a sufficient condition is given for when the invalidity of timings for a process can be inferred from its initial subprocess to compute e g the maximum time separation between two events in a time process an alternative characterization of validness is developed this definition is used to derive an algorithm for constructing the set of all valid timings for a process the set of valid timings is presented as sets of alternative linear constraints which can be used in optimization problems it is shown that the existence of a valid timing for a given process can be decided in np time

analysis of a drift diffusion model with velocity saturation for spin polarized transport in semiconductors abstract a system of drift diffusion equations with electric field under dirichlet boundary conditions is analyzed the system of strongly coupled parabolic equations for particle density and spin density vector describes the spin polarized semi classical electron transport in ferromagnetic semiconductors the presence of a nonconstant and nonsmooth magnetization vector solution of the landau lifshitz equation causes the diffusion matrix to be dependent on space and time and to have in general poor regularity properties thus making the analysis challenging to partially overcome the analytical difficulties the velocity saturation hypothesis is made which results in a bounded drift velocity the global in time existence and uniqueness of weak solutions is shown by means of a semi discretization in time which yields an elliptic semilinear problem and a quadratic entropy inequality which allow for the limit of vanishing time step size the convergence of the weak solutions to the steady state under some restrictions on the parameters and data is shown finally the higher regularity of solutions for a smooth magnetization in two space dimensions is shown through a diagonalization argument which allows to get rid of the cross diffusion terms in the fluid equations and the iterative application of gagliardo nirenberg inequalities and a generalized version of aubin lemma

blowup in reaction diffusion systems with dissipation of mass we prove possible blowup in finite time of the solutions to reaction diffusion systems which preserve nonnegativity and for which the total mass of the components is nonincreasing in time two natural properties in applications this is done by presenting explicit counterexamples constructed with the help of formal computation software several partial results of global existence had been obtained previously in the literature our counterexamples explain a posteriori why extra conditions were needed negative results of independent interest are also provided as a by product for linear parabolic equations in nondivergence form and with discontinuous coefficients and for nonlinear hamilton jacobi evolution equations

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

on the expressiveness of tptl and mtl abstract tptl and mtl are two classical timed extensions of ltl in this paper we prove the 20 year old conjecture that tptl is strictly more expressive than mtl but we show that surprisingly the tptl formula proposed by alur and henzinger for witnessing this conjecture it can be expressed in mtl more generally we show that tptl formulae using only modality f can be translated into mtl

when are timed automata weakly timed bisimilar to time petri nets in this paper we compare timed automata ta and time petri nets tpn with respect to weak timed bisimilarity it is already known that the class of bounded tpns is strictly included in the class of ta it is thus natural to try and identify the subclass ta w t b of ta equivalent to some tpn for the weak timed bisimulation relation we give a characterization of this subclass and we show that the membership problem and the reachability problem for ta w t b are pspace complete furthermore we show that for a ta in ta w t b with integer constants an equivalent tpn can be built with integer bounds but with a size exponential w r t the original model surprisingly using rational bounds yields a tpn whose size is linear

event clock automata a determinizable class of timed automata we introduce event recording automata an event recording automaton is a timed automaton that contains for every event a a clock that records the time of the last occurrence of a the class of event recording automata is on one hand expressive enough to model finite timed transition systems and on the other hand determinizable and closed under all boolean operations as a result the language inclusion problem is decidable for event recording automata we present a translation from timed transition systems to event recording automata which leads to an algorithm for checking if two timed transition systems have the same set of timed behaviors we also consider event predicting automata which contain clocks that predict the time of the next occurrence of an event the class of event clock automata which contain both event recording and event predicting clocks is a suitable specification language for real time properties we provide an algorithm for checking if a timed automaton meets a specification that is given as an event clock automaton

global existence analysis for degenerate energy transport models for semiconductors abstract a class of energy transport equations without electric field under mixed dirichlet neumann boundary conditions is analyzed the system of degenerate and strongly coupled parabolic equations for the particle density and temperature arises in semiconductor device theory the global in time existence of weak nonnegative solutions is shown the proof consists of a variable transformation and a semi discretization in time such that the discretized system becomes elliptic and semilinear positive approximate solutions are obtained by stampacchia truncation arguments and a new cut off test function nonlogarithmic entropy inequalities yield gradient estimates which allow for the limit of vanishing time step sizes exploiting the entropy inequality the long time convergence of the weak solutions to the constant steady state is proved because of the lack of appropriate convex sobolev inequalities to estimate the entropy dissipation only an algebraic decay rate is obtained numerical experiments indicate that the decay rate is typically exponential

discrete time control for rectangular hybrid automata rectangular hybrid automata model digital control programs of analog plant environments we study rectangular hybrid automata where the plant state evolves continuously in real numbered time and the controller samples the plant state and changes the control state discretely only at the integer points in time we prove that rectangular hybrid automata have finite bisimilarity quotients when all control transitions happen at integer times even if the constraints on the derivatives of the variables vary between control states this is in contrast with the conventional model where control transitions may happen at any real time and already the reachability problem is undecidable based on the finite bisimilarity quotients we give an exponential algorithm for the symbolic sampling controller synthesis of rectangular automata we show our algorithm to be optimal by proving the problem to be exptime hard we also show that rectangular automata form a maximal class of systems for which the sampling controller synthesis problem can be solved algorithmically

palindromic width of wreath products abstract a palindrome is a word which reads the same left to right as right to left we show that the wreath product g z n of any finitely generated group g with z n has finite palindromic width this generalizes the main result from 16 we also show that c a has finite palindromic width if c has finite commutator width and a is a finitely generated infinite abelian group further we prove that if h is a non abelian group with finite palindromic width and g any finitely generated group then every element of the subgroup g h can be expressed as a product of uniformly boundedly many palindromes from this we obtain that p h has finite palindromic width if p is a perfect group and further that g f has finite palindromic width for any finite non abelian group f

the semiconductor system with temperature effect in this paper we study the semiconductor system with temperature effect the electric conductivity in the system is assumed to be temperature dependent by taking the joule heating produced by the current flow into consideration we obtain a strongly coupled elliptic parabolic system the existence of a weak solution is proved by using a compactness argument regularity of the weak solution is also investigated

palindromic primitives and palindromic bases in the free group of rank two abstract the present paper records more details of the relationship between primitive elements and palindromes in f 2 the free group of rank two we characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length we identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length we show that under obvious conditions on exponent sums pairs of palindromic primitives form palindromic bases for f 2 further we note that each cyclically reduced primitive element is either a palindrome or the concatenation of two palindromes

palindromic width of free nilpotent groups abstract in this paper we consider the palindromic width of free nilpotent groups in particular we prove that the palindromic width of a finitely generated free nilpotent group is finite we also prove that the palindromic width of a free abelian by nilpotent group is finite

reachability and control synthesis for piecewise affine hybrid systems on simplices in this paper we consider the synthesis of control laws for piecewise affine hybrid systems on simplices the construction is based on the solution to the control to facet problem at the continuous level and on dynamic programming at the discrete level the construction is given as an explicit algorithm using only linear algebra and reach set computations for automata no numerical integration is required the method is conservative in that it may fail to find a control law where one exists but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise affine hybrid systems is undecidable

degenerate semiconductor device equations with temperature effect abstract in this paper we establish an existence assertion for the following elliptic system div a v n a v n 0 in div a v p a v p 0 in div a v p n f in div a v v a v 2 in coupled with suitable boundary conditions this problem arises from the study of semiconductor devices with temperature effect and without recombination we only assume that a is continuous and positive this gives rise to the possibility that the system may be degenerate and or singular we show that if a s does not go to zero too fast as s there exists a bounded weak solution and therefore neither degeneracy nor singularity really occur this also immediately implies some additional regularity properties for the weak solution

enumerating palindromes and primitives in rank two free groups abstract let f be a free group of rank two an element of f is primitive if it along with another group element generates the group if f a b then a word w a b in a and b is a palindrome if it reads the same forwards and backwards it is known that in a rank two free group for any fixed set of two generators a primitive element will be conjugate either to a palindrome or to the product of two palindromes but known iteration schemes for all primitive words give only a representative for the conjugacy class here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes that have already appeared in the scheme we denote these words by e p q where p q is rational number expressed in lowest terms we prove that e p q is a palindrome if pq is even and the unique product of two unique palindromes if pq is odd we prove that the pair x y or x 1 y 1 generates the group if and only if x is conjugate to e p q and y is conjugate to e r s where p s r q 1 this improves a previously known result that held only for pq and rs both even the derivation of the enumeration scheme also gives a new proof of the known results about primitive words

parametric multisingular hybrid petri nets multisingular hybrid petri net mshpn is an extension of hybrid petri nets enriched with the capabilities of hybrid automata to achieve the practical expressive power of multisingular hybrid automata in this paper we define parametric multisingular hybrid petri nets p mshpns as a parametric extension of mshpns we present the parametric reachability analysis techniques and algorithms and prove that the parametric reachability analysis of p mshpns amounts to the analysis of standard mshpns once the parametric state space of a p mshpns model is computed it can either be used for parametric model checking analysis using the existing techniques or be instantiated to obtain non parametric state spaces p mshpns models can be analysed to obtain the set of feasible configurations for the system parameters we also give a method for deriving the set of constraints on the parameters that ensure the correctness of an invariant property and a method for finding the optimum parameter configuration

the expressive power of time petri nets we investigate expressiveness questions for time petri nets tpns and some of their most useful extensions we first introduce generalised time petri nets gtpns as an abstract model that encompasses variants of tpns such as self modifications and read reset and inhibitor arcs we give a syntactical translation from bounded gtpns to timed automata ta that generates isomorphic transition systems we prove that the class of bounded gtpns is strictly less expressive than ta w r t weak timed bisimilarity we prove that bounded gtpns bounded tpns and ta are equally expressive w r t timed language acceptance finally we characterise a syntactical subclass of ta that is equally expressive to bounded gtpns a la merlin w r t weak timed bisimilarity these results provide a unified comparison of the expressiveness of many variants of timed models often used in practice it leads to new important results for tpns among them are 1 safe tpns and bounded tpns are equally expressive e transitions strictly increase the expressive power of tpns self modifying nets as well as read inhibitor and reset arcs do not add expressiveness to bounded tpns