blind spectral unmixing based on sparse nonnegative matrix factorization nonnegative matrix factorization nmf is a widely used method for blind spectral unmixing su which aims at obtaining the endmembers and corresponding fractional abundances knowing only the collected mixing spectral data it is noted that the abundance may be sparse i e the endmembers may be with sparse distributions and sparse nmf tends to lead to a unique result so it is intuitive and meaningful to constrain nmf with sparseness for solving su however due to the abundance sum to one constraint in su the traditional sparseness measured by l0 l1 norm is not an effective constraint any more a novel measure termed as s measure of sparseness using higher order norms of the signal vector is proposed in this paper it features the physical significance by using the s measure constraint smc a gradient based sparse nmf algorithm termed as nmf smc is proposed for solving the su problem where the learning rate is adaptively selected and the endmembers and abundances are simultaneously estimated in the proposed nmf smc there is no pure index assumption and no need to know the exact sparseness degree of the abundance in prior yet it does not require the preprocessing of dimension reduction in which some useful information may be lost experiments based on synthetic mixtures and real world images collected by aviris and hydice sensors are performed to evaluate the validity of the proposed method

diagnosability of discrete event systems subject to permanent sensor failures this paper considers a failure diagnosis problem for discrete event systems subject to permanent sensor failures a notion of diagnosability subject to permanent sensor failures is introduced with respect to a certain nondeterministic observation mask for its verification an aggregated mealy automaton with a deterministic and state dependent observation mask is defined it is shown that the diagnosability of the aggregated mealy automaton is equivalent to the diagnosability of the original system subject to permanent sensor failures then a method for verifying the diagnosability of the aggregated mealy automaton is presented moreover the delay bound within which the occurrence of any failure string can be detected subject to permanent sensor failures is computed

model uncertainty in discrete event systems earlier work concerning control of discrete event systems usually assumed that a correct model of the system to be controlled was available a goal of this work is to provide an algorithm for determining the correct model from a set of models the result of the algorithm is a finite language that can be used to test for the correct model or notification that the remaining models cannot be controllably distinguished we use the finite state machine model with controllable and uncontrollable events presented by ramadge and wonham

on a weaker notion of controllability of a language k with respect to a language l a necessary and sufficient condition for a prefix closed language k contained in sigma to be controllable with respect to another prefix closed language l contained in sigma is that k contained in l a weaker notion of controllability where it is not required that k contained in l is considered here if l is the prefix closed language generated by a plant automaton g then essentially there exists a supervisor theta that is complete with respect to g such that l theta mod g k intersection l if and only if k is weakly controllable with respect to l for an arbitrary modeling formalism it is shown that the inclusion problem is reducible to the problem of deciding the weaker notion of controllability therefore removing the requirement that k contained in l from the original definition of controllability does not help the situation from a decidability viewpoint this observation is then used to identify modeling formalisms that are not viable for supervisory control of the untimed behaviors of discrete event dynamic systems

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

robust and nonblocking supervisor for discrete event systems with model uncertainty under partial observation addresses a robust and nonblocking supervisory control problem based on a framework of discrete event systems with model uncertainty under partial observation the uncertainty introduced in the paper is associated with internal and unobservable events occurring in systems the paper presents a systematic method for modeling uncertainty described as a spl delta transition representing the internal and unobservable events in particular at a state with the assigned positive integer value p a system is assumed to experience at most a p step state transition by the internal and unobservable event under the framework for model uncertainty the paper presents necessary and sufficient conditions for the existence of a robust supervisor with the nonblocking property

collaborative sparse regression using spatially correlated supports application to hyperspectral unmixing this paper presents a new bayesian collaborative sparse regression method for linear unmixing of hyperspectral images our contribution is twofold first we propose a new bayesian model for structured sparse regression in which the supports of the sparse abundance vectors are a priori spatially correlated across pixels i e materials are spatially organized rather than randomly distributed at a pixel level this prior information is encoded in the model through a truncated multivariate ising markov random field which also takes into consideration the facts that pixels cannot be empty i e there is at least one material present in each pixel and that different materials may exhibit different degrees of spatial regularity second we propose an advanced markov chain monte carlo algorithm to estimate the posterior probabilities that materials are present or absent in each pixel and conditionally to the maximum marginal a posteriori configuration of the support compute the minimum mean squared error estimates of the abundance vectors a remarkable property of this algorithm is that it self adjusts the values of the parameters of the markov random field thus relieving practitioners from setting regularization parameters by cross validation the performance of the proposed methodology is finally demonstrated through a series of experiments with synthetic and real data and comparisons with other algorithms from the literature

an input output construction of finite state rho mu approximations for control design we consider discrete time plants that interact with their controllers via fixed discrete alphabets for this class of systems and in the absence of exogenous inputs we propose a general methodical approach for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs we explicitly derive conditions under which the proposed construct used in conjunction with a particular generalized structure satisfies desirable properties of approximations thereby leading to nominal deterministic finite state machine models that can be used in certified by design controller synthesis we also show that the cardinality of the min imal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet finally we show that the pro posed construct satisfies a relevant semi completeness property

supervisory control of hybrid systems within a behavioural framework abstract this contribution addresses the synthesis of supervisory control for hybrid systems with discrete external signals such systems are in general neither l complete nor can they be represented by finite state machines we find an l complete approximation abstraction l for represent it by a finite state machine and investigate the control problem for the approximation if a solution exists we synthesize the maximally permissive supervisor for l we show that it also solves the control problem for the hybrid system if no solution exists approximation accuracy can be increased by computing a k complete abstraction k k l this paper is entirely set within the framework on willems behavioural systems theory

optimal nonblocking directed control of discrete event systems for the control of discrete event systems the notion of directed control refines that of supervisory control a directed controller is one that selects at most one controllable event to be enabled at any state without disabling any uncontrollable event which is in fact how a discrete event control is implemented in contrast a supervisory controller computes a maximal allowable set of controllable events at each state leaving undecided exactly which controllable event should be enabled in previous works we developed a framework for the computation of optimal directed controllers and a polynomial synthesis algorithm for acyclic plants in this paper we present a novel synthesis approach for general plants i e plants with or without cycles thus providing a complete solution to the optimal directed control problem the complexity of the approach remains polynomial in the size of plant

theory of extended fuzzy discrete event systems for handling ranges of knowledge uncertainties and subjectivity in 2001 we originated a theory of fuzzy discrete event systems fdess that generalized the conventional crisp discrete event systems dess vagueness and imprecision concerning states and event transitions of dess were represented by membership grades and computed via fuzzy logic our application of the fdes theory to computerized human immunodeficiency virus acquired immune deficiency syndrome treatment regimen selection although preliminarily successful suggests that a more comprehensive fdes theory is needed to address two general issues critically important not only to biomedical applications but also to real world problems in other industries first domain experts should have means other than point estimates and type 1 fuzzy sets mandated in the current framework to describe uncertainties subjectivity and imprecision in their complex knowledge and experience second when a group of experts with distinct opinions is involved they should not be forced to reach consensus for the sake of system development this is because collective consensus may not be achievable which is often the case in medicine where individual expertspsila opinions should be equally respected since the underlying ground truth is unknown most of the time the theory of extended fdes presented in this paper addresses both the problems and contains the fdes theory as a special case experts are now allowed to use interval numbers and type 1 and type 2 fuzzy sets to intuitively and quantitatively express their diverse knowledge and experience which will then be processed by the new theory to form fuzzy state vectors and fuzzy event transition matrices accordingly we have established mathematical operations that cover the computations of fuzzy states fuzzy event transitions and parallel composition numerical examples are provided

characterizing intransitive noninterference for 3 domain security policies with observability this note introduces a new algorithmic approach to the problem of checking the property of intransitive noninterference ini using discrete event systems dess tools and concepts ini property is widely used in formal verification of security problems in computer systems and protocols the approach consists of two phases first a new property called ip observability observability based on a purge function is introduced to capture ini we prove that a system satisfies ini if and only if it is ip observable second a relation between ip observability and p observability observability as used in des is established by transforming the automaton modeling a system protocol into an automaton where p observability and hence ip observability can be determined this allows us to check ini by checking p observability which can be done efficiently our approach can be used for all systems protocols with three domains or levels which is sufficient for most noninterference problems for cryptographic protocols and systems

doomsday equilibria for omega regular games abstract two player games on graphs provide the theoretical framework for many important problems such as reactive synthesis while the traditional study of two player zero sum games has been extended to multi player games with several notions of equilibria they are decidable only for perfect information games whereas several applications require imperfect information in this paper we propose a new notion of equilibria called doomsday equilibria which is a strategy profile where all players satisfy their own objective and if any coalition of players deviates and violates even one of the players objective then the objective of every player is violated we present algorithms and complexity results for deciding the existence of doomsday equilibria for various classes of regular objectives both for imperfect information games and for perfect information games we provide optimal complexity bounds for imperfect information games and in most cases for perfect information games

nonlinear fokker planck equations whose stationary solutions make entropy like functionals stationary abstract the present study extends the correspondence principle of martinez et al that establishes a link between nonlinear fokker planck equations nlfpes and the variational principle approach of the theory of canonical ensembles by virtue of the extended correspondence principle we reobtain results of kaniadakis and quarati for bose and fermi systems and find relations similar to those derived by plastino and plastino and recently by borland concerning the entropy of nonextensive thermostatistics introduced in physics by tsallis moreover we propose nlfpes related to the renyi entropy and to the entropy proposed by sharma and mittal the latter comprises tsallis entropy and the renyi entropy

similarity based supervisory control of discrete event systems due to the appearance of uncontrollable events in discrete event systems one may wish to replace the behavior leading to the uncontrollability of pre specified language by some quite similar one to capture this similarity we introduce metric to traditional supervisory control theory and generalize the concept of original controllability to spl lambda controllability where spl lambda indicates the similarity degree of two languages a necessary and sufficient condition for a language to be spl lambda controllable is provided we then examine some properties of spl lambda controllable languages and present an approach to optimizing a realization

on petri net models of infinite state supervisors a class of supervisory control problems that require infinite state supervisors is considered and petri nets with inhibitor arcs pns are introduced to model the supervisors this pn based approach to supervisory control is compared to automata based approaches the primary advantage of a pn based supervisory controller is that it provides a finite representation of an infinite state supervisor for verification implementation and testing reasons a finite pn based representation of an infinite state supervisor is preferred over an automata based supervisor it is shown that this modeling advantage is accompanied by a decision disadvantage in that in general the controllability of a language that can be generated by the closed loop system is undecidable

supervisory control of nondeterministic systems with driven events via prioritized synchronization and trajectory models the supervisory control of nondeterministic discrete event dynamical systems dedss with driven events in the setting of prioritized synchronization and trajectory models introduced by heymann are studied prioritized synchronization captures the notions of controllable uncontrollable and driven events in a natural way and the authors use it for constructing supervisory controllers the trajectory model is used for characterizing the behavior of nondeterministic dedss since it is a sufficiently detailed model in contrast to the less detailed language or failures models and serves as a language congruence with respect to the operation of prioritized synchronization results concerning controllability and observability in this general setting are obtained

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

statistical mechanics based on renyi entropy abstract in this work we show that it is possible to obtain a generalized statistical mechanics thermostatistics based on renyi entropy to be maximized with adequate constraints the equilibrium probability distribution thus obtained has a very interesting property indeed it reminds us the statistical distribution proposed by tsallis known to conveniently describe a variety of phenomena in nonextensive systems moreover some examples are worked out in order to illustrate the main features of the herein introduced formalism

robust supervisory control of a class of timed discrete event systems under partial observation this paper studies robust supervisory control of timed discrete event systems proposed by brandin and wonham given a set of possible models which includes the exact model of the plant the objective is to synthesize a robust supervisor such that it achieves legal behavior for all possible models we show that controllability for each possible model and observability for a suitably defined aggregate model are necessary and sufficient conditions for the existence of a solution to the robust supervisory control problem moreover when there does not exist a solution a maximally permissive robust supervisor is synthesized under the assumption that all controllable events are observable

estimating the granularity coefficient of a potts markov random field within a markov chain monte carlo algorithm this paper addresses the problem of estimating the potts parameter jointly with the unknown parameters of a bayesian model within a markov chain monte carlo mcmc algorithm standard mcmc methods cannot be applied to this problem because performing inference on requires computing the intractable normalizing constant of the potts model in the proposed mcmc method the estimation of is conducted using a likelihood free metropolis hastings algorithm experimental results obtained for synthetic data show that estimating jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of on the other hand choosing an incorrect value of can degrade estimation performance significantly to illustrate the interest of this method the proposed algorithm is successfully applied to real bidimensional sar and tridimensional ultrasound images

signed real measure of regular languages for discrete event supervisory control this paper reviews expands and clarifies the underlying concepts of a signed real measure of regular languages which has been used as a novel tool for synthesis of discrete event supervisory control systems the language measure is constructed upon the principles of automata theory and real analysis it allows total ordering of a set of partially ordered sublanguages of a regular language for quantitative evaluation of the supervised behaviour of deterministic finite state automata dfsa under different supervisors in the setting of the language measure a supervisor s performance is superior if the supervised plant is more likely to terminate at a good marked state and or less likely to terminate at a bad marked state the computational complexity of the language measure algorithm is polynomial in the number of dfsa states

verification of initial state opacity in security applications of discrete event systems in this paper we formulate and analyze methodologies for verifying the notion of initial state opacity in discrete event systems that are modeled as non deterministic finite automata with partial observation on their transitions a system is initial state opaque if the membership of its true initial state to a set of secret states remains opaque i e uncertain to an intruder who observes system activity through some projection map initial state opacity can be used to characterize security requirements in a variety of applications including tracking problems in sensor networks in order to model and analyze the intruder capabilities regarding initial state opacity we first address the initial state estimation problem in a non deterministic finite automaton via the construction of an initial state estimator we analyze the properties and complexity of the initial state estimator and show how the complexity of the verification method can be greatly reduced in the special case when the set of secret states is invariant we also establish that the verification of initial state opacity is a pspace complete problem

control of discrete event systems modeled as hierarchical state machines discrete event systems dess are systems in which state changes take place in response to events that occur discretely asynchronously and often nondeterministically a class of dess modeled as hierarchical state machines hsms a special case of the statecharts formalism introduced recently is considered an efficient algorithm is provided for solving reachability problems in the hsm framework it utilizes the hierarchical structure of hsms this efficient solution is used extensively in control applications where controllers achieving a desired behavior are synthesized online

the computational complexity of decentralized discrete event control problems computational complexity results are obtained for decentralized discrete event control problems these results generalize the earlier work of tsitsiklis 1989 who showed that for a special class of centralized supervisory control problems under partial observation there is an algorithm for determining in polynomial time whether or not a solution exists the negative complexity results associated with tsitsiklis work also carry over to the decentralized case so that solution existence for the more general class is not decidable in polynomial time nor does there exist a polynomial time algorithm for producing supervisor solutions when such solutions exist

decentralized prognosis of failures in discrete event systems we study the prognosis of failures i e their prediction prior to their occurrence in discrete event systems in a decentralized setting where multiple prognosers use their local observations to issue local prognosis decisions we define the notion of correctness of a decentralized set of prognosers in terms of no missed detections each failure is prognosed prior to its occurrence and no false alarms an incorrect prognostic decision is never issued and introduce the notion of coprognosability as an existence condition when specialized to the centralized case i e the case of a single prognoser this condition turns out to be weaker than the one introduced by genc and lafortune in 2006 since a uniform bound on the number of steps within which a failure will occur is not required for comparison we also introduce the stronger notion of uniformly bounded coprognosability and identify the subclass of decentralized prognosers for which it serves as an existence condition we show that the two notions coincide when the underlying system and its nonfailure specification possess finite state representations and present a verification algorithm whose complexity is polynomial in the sizes of the system being prognosed and its nonfailure specification and is exponential in the number of the local prognosers we also introduce the notion of reaction bound for coprognosis as the earliest time beyond a prognostic decision when a failure can occur and present an algorithm for computing it the complexity of this algorithm is identical to that of the verification algorithm an algorithm with complexity linear in the size of the specification and the number of local prognosers is also presented for an online prognosis of failures we show that the notions of coprognosability and its uniformly bounded version are in general incomparable with the notion of codiagnosability that guarantees a uniformly bounded delay detection of a failure by a local diagnoser when the system cannot execute an unbounded sequence of unobservable events uniformly bounded coprognosability implies codiagnosability whereas coprognosability and codiagnosability remain incomparable

nonlinear spectral unmixing of hyperspectral images using gaussian processes this paper presents an unsupervised algorithm for nonlinear unmixing of hyperspectral images the proposed model assumes that the pixel reflectances result from a nonlinear function of the abundance vectors associated with the pure spectral components we assume that the spectral signatures of the pure components and the nonlinear function are unknown the first step of the proposed method estimates the abundance vectors for all the image pixels using a bayesian approach an a gaussian process latent variable model for the nonlinear function relating the abundance vectors to the observations the endmembers are subsequently estimated using gaussian process regression the performance of the unmixing strategy is first evaluated on synthetic data the proposed method provides accurate abundance and endmember estimations when compared to other linear and nonlinear unmixing strategies an interesting property is its robustness to the absence of pure pixels in the image the analysis of a real hyperspectral image shows results that are in good agreement with state of the art unmixing strategies and with a recent classification method

robust optimal control of regular languages this paper presents an algorithm for robust optimal control of regular languages under specified uncertainty bounds on the event cost parameters of the language measure that has been recently reported in literature the performance index for the proposed robust optimal policy is obtained by combining the measure of the supervised plant language with uncertainty the performance of a controller is represented by the language measure of the supervised plant and is minimized over the given range of event cost uncertainties synthesis of the robust optimal supervisory control policy requires at most n iterations where n is the number of states of the deterministic finite state automaton dfsa model generated from the regular language of the unsupervised plant behavior the computational complexity of the control synthesis method is polynomial in n

semi supervised linear spectral unmixing using a hierarchical bayesian model for hyperspectral imagery this paper proposes a hierarchical bayesian model that can be used for semi supervised hyperspectral image unmixing the model assumes that the pixel reflectances result from linear combinations of pure component spectra contaminated by an additive gaussian noise the abundance parameters appearing in this model satisfy positivity and additivity constraints these constraints are naturally expressed in a bayesian context by using appropriate abundance prior distributions the posterior distributions of the unknown model parameters are then derived a gibbs sampler allows one to draw samples distributed according to the posteriors of interest and to estimate the unknown abundances an extension of the algorithm is finally studied for mixtures with unknown numbers of spectral components belonging to a know library the performance of the different unmixing strategies is evaluated via simulations conducted on synthetic and real data

relative observability and coobservability of timed discrete event systems we study supervisory control of timed discrete event systems tdes under partial observation and propose new observability concepts effective for supervisor synthesis first we consider monolithic centralized supervisory control and introduce timed relative observability and timed relative weak observability the former concept extends our previous work to the timed case while the latter exploits choices of forcible events to preempt the clock event tick we prove that timed relative respectively weak observability is stronger than timed respectively weak observability weaker than normality and closed under set union hence there exists the supremal relatively respectively weakly observable sublanguage of a given language

generalized detectability for discrete event systems in our previous work we investigated detectability of discrete event systems which is defined as the ability to determine the current and subsequent states of a system based on observation for different applications we defined four types of detectabilities weak detectability strong detectability weak periodic detectability and strong periodic detectability in this paper we extend our results in three aspects 1 we extend detectability from deterministic systems to nondeterministic systems such a generalization is necessary because there are many systems that need to be modeled as nondeterministic discrete event systems 2 we develop polynomial algorithms to check strong detectability the previous algorithms are based on observer whose construction is of exponential complexity while the new algorithms are based on a new automaton called detector 3 we extend detectability to d detectability while detectability requires determining the exact state of a system d detectability relaxes this requirement by asking only to distinguish certain pairs of states with these extensions the theory on detectability of discrete event systems becomes more applicable in solving many practical problems

limited lookahead policies in supervisory control of discrete event systems a supervisory control scheme which is based on limited lookahead control policies is proposed using an online scheme where after the occurrence of an event the next control action is determined on the basis of an n step ahead projection of the behavior of the process represented as an n level tree this procedure then repeats after the execution of the next event different attitudes can be adopted to calculate a control action for the given n level tree in order to resolve uncertainties about the process behavior beyond n steps two such attitudes termed conservative and optimistic are studied results are presented pertaining to monotonicity and convergence properties of the optimistic and conservative n step policies in terms of n and comparison of these policies with the optimal offline solution closed form expression for the language generated by the n step conservative policy for prefix closed legal languages and lower bounds for n that guarantee that the online scheme performs as well as an offline scheme with complete information

renyi entropy and complexity measure for skew gaussian distributions and related families abstract in this paper we provide the renyi entropy and complexity measure for a novel flexible class of skew gaussian distributions and their related families as a characteristic form of the skew gaussian shannon entropy we give closed expressions considering a more general class of closed skew gaussian distributions and the weighted moments estimation method in addition closed expressions of renyi entropy are presented for extended skew gaussian and truncated skew gaussian distributions finally additional inequalities for skew gaussian and extended skew gaussian renyi and shannon entropies are reported

residual component analysis of hyperspectral images application to joint nonlinear unmixing and nonlinearity detection this paper presents a nonlinear mixing model for joint hyperspectral image unmixing and nonlinearity detection the proposed model assumes that the pixel reflectances are linear combinations of known pure spectral components corrupted by an additional nonlinear term affecting the end members and contaminated by an additive gaussian noise a markov random field is considered for nonlinearity detection based on the spatial structure of the nonlinear terms the observed image is segmented into regions where nonlinear terms if present share similar statistical properties a bayesian algorithm is proposed to estimate the parameters involved in the model yielding a joint nonlinear unmixing and nonlinearity detection algorithm the performance of the proposed strategy is first evaluated on synthetic data simulations conducted with real data show the accuracy of the proposed unmixing and nonlinearity detection strategy for the analysis of hyperspectral images

output stabilizability of discrete event dynamic systems the authors investigate the problem of designing stabilizing feedback compensators for discrete event dynamic systems deds modeled as finite state automata in which some transition events are controllable and some events are observed the problem of output stabilization is defined as the construction of a compensator such that all state trajectories in the closed loop system go through a given set e infinitely often the authors also define a stronger notion of output stabilizability which requires that the state not only pass through e infinitely often but that the set of instants when the state is in e and one knows it is in e is also infinite necessary and sufficient conditions are presented for both notions the authors also introduce and characterize a notion of resiliency that corresponds to the system being able to recover from observation errors in addition they provide some general bounds for the algorithms considered and discuss several conditions under which far smaller bounds can be achieved

a note on the properties of the supremal controllable sublanguage in pushdown systems consider an event alphabet sigma the supervisory control theory of ramadge and wonham asks the question given a plant model g with language lm g sube sigma and another language k sube lm g is there a supervisor phi such that lm phi g k ramadge and wonham showed that a necessary condition for this to be true is the so called controllability of k with respect to lm g they showed that when g is a finite state automaton and k is a regular language also generated by a finite state automaton then there is a regular supremal controllable sublanguage supc k sube k that is effectively constructable from generators of k and g in this paper we show 1 there is an algorithm to compute the supremal controllable sublanguage of a prefix closed k accepted by a deterministic pushdown automaton dpda when the plant language is also prefix closed and accepted by a finite state automaton and 2 in this case we show that this supremal controllable sublanguage is also accepted by a dpda

control of networked discrete event systems dealing with communication delays and losses in this paper we investigate supervisory control of networked discrete event systems such systems are now widely used in practice the key feature of a networked discrete event system is communication delays and losses in observation and control without addressing communication delays and losses in networked systems no control will be effective we propose to use some special observation mappings to deal with observation delays and losses based on these mappings we define network observability we also consider control delays and losses in networked discrete event systems and define network controllability we show that these definitions capture precisely the requirements on effective supervisory control in networked discrete event systems in the sense that a networked supervisor exists if and only if both network controllability and network observability are satisfied

on supervisory control of sequential behaviors the authors address the supervisory synthesis problem of controlling the sequential behaviors of discrete event dynamical systems dedss under complete and partial information through the use of synchronous composition of the plants and the supervisors the authors present the notion of complete languages discuss some of its algebraic properties and show its close relation to omega languages the authors prove that the supremal closed complete and controllable sublanguage of a given language exists and present an algorithm to compute it they present a closed form expression for the supremal omega controllable sublanguage of a given omega language in terms of the supremal closed complete and controllable sublanguage this closed form expression suggests that certain operations on a given omega language can alternatively be achieved by performing certain other similar operations on its prefix which is a finite language and then taking the limit to obtain the desired omega language a necessary and sufficient condition for the existence of a supervisor in case of partial observation is presented in terms of omega observability notion of omega normality is also introduced and a closed form expression for the supremal omega normal sublanguage in terms of the supremal closed complete and normal sublanguage is presented

a language measure for partially observed discrete event systems recent literature has introduced and validated a signed real measure of regular languages for quantitative analysis and synthesis of discrete event supervisory des control systems where all events are assumed to be observable this paper presents a modification of the language measure for supervisory control under partial observation and shows how to generalize the analysis when some of the events may not be observable at the supervisory level in the context of des control synthesis the language measure of partially observable discrete event processes is expressed in a closed form which is structurally similar to that of completely observable discrete event processes examples are provided to elucidate the concept of des control under partial observation

bayesian estimation of linear mixtures using the normal compositional model application to hyperspectral imagery this paper studies a new bayesian unmixing algorithm for hyperspectral images each pixel of the image is modeled as a linear combination of so called endmembers these endmembers are supposed to be random in order to model uncertainties regarding their knowledge more precisely we model endmembers as gaussian vectors whose means have been determined using an endmember extraction algorithm such as the famous n finder n findr or vertex component analysis vca algorithms this paper proposes to estimate the mixture coefficients referred to as abundances using a bayesian algorithm suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas conjugate priors are chosen for the remaining parameters a hybrid gibbs sampler is then constructed to generate abundance and variance samples distributed according to the joint posterior of the abundances and noise variances the performance of the proposed methodology is evaluated by comparison with other unmixing algorithms on synthetic and real images

controlled markov chains with safety upper bound in this note we introduce and study the notion of safety control of stochastic discrete event systems dess modeled as controlled markov chains for nonstochastic dess modeled by state machines or automata safety is specified as a set of forbidden states or equivalently by a binary valued vector that imposes an upper bound on the set of states permitted to be visited we generalize this notion of safety to the setting of stochastic dess by specifying it as an unit interval valued vector that imposes an upper bound on the state probability distribution vector under the assumption of complete state observation we identify 1 the set of all state feedback controllers that satisfy the safety requirement for any given safe initial state probability distribution and 2 the set of all safe initial state probability distributions for a given state feedback controller

on controllability and normality of discrete event dynamical systems abstract this paper studies the supervisor synthesis problem of discrete event dynamical systems deds s through the use of synchronous composition of the plant and the supervisor and discusses some of the simplifications achieved by using the synchronous composition to model supervisory control an alternate definition of controllability is presented this definition of controllability is then used to derive an algorithm that is computationally more efficient than previously existing ones for the construction of the supremal controllable sublanguage the algorithm is also shown to be optimal the observability and normality issues arising due to the partial observation of the system dynamics under an arbitrary mask are then investigated closed form representations of the supremal normal and supremal closd and normal sublanguages are derived in the more general setting of arbitrary mask and nonclosed languages thus extending earlier results of the authors and others

a framework for state attraction of discrete event systems under partial observation state attraction for discrete event systems des addresses the problem of reaching a desired subset of the plant state space after a bounded number of event occurrences the problem of state attraction arises for example in fault tolerant supervisory control or in the control of reconfigurable manufacturing systems and is also applicable to systems biological problems such as the control of gene regulatory networks state attraction is investigated with the assumption of full event observation in the existing literature this paper extends the concept of state attraction to the case of partial observation the notion of weak attraction under partial observation wapo is introduced and necessary and sufficient conditions for the existence of a supervisor under partial observation that achieves wapo are derived furthermore a solution algorithm is proposed that finds such supervisor whenever it exists it is shown that such supervisor can always be realized as a subautomaton of the observer automaton of the des plant an application example from systems biology illustrates the obtained results

approximately bisimilar symbolic models for nonlinear control systems control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs although these models are very powerful when dealing with physical phenomena they are less suited to describe software and hardware interfacing with the physical world for this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics where each symbol corresponds to an aggregate of states in the continuous model since these symbolic models are of the same nature of the models used in computer science to describe software and hardware they provide a unified language to study problems of control in which software and hardware interact with the physical world furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes in this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent bisimilar to a symbolic model the approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired furthermore if the state space of the control system is bounded the obtained symbolic model is finite for digital control systems and under the stronger assumption of incremental input to state stability symbolic models can be constructed through a suitable quantization of the inputs

reciprocity relations and generalized classic entropic quantifiers that lack trace form abstract in this effort we show that the legendre reciprocity relations thermodynamics essential formal feature are respected by general classic entropic functionals even if they are not of trace form nature in contrast with shannon s or tsallis cases further with reference to the maximum entropy maxent variational process we encounter important cases relevant to physical applications currently discussed in the research literature in which the associated reciprocity relations exhibit anomalies we show that these anomalies can be cured by carefully discriminating between apparently equivalent entropic forms

polynomial synthesis of supervisor for partially observed discrete event systems by allowing nondeterminism in control we study the supervisory control of discrete event systems dess under partial observation using nondeterministic supervisors we formally define a nondeterministic control policy and also a control observation compatible nondeterministic state machine and prove their equivalence the control action of a nondeterministic supervisor is chosen online nondeterministically from among a set of choices determined offline also the control action can be changed online nondeterministically prior to any new observation in accordance with choices determined offline the online choices once made can be used to affect the set of control action choices in future we show that when control is exercised using a nondeterministic supervisor the specification language is required to satisfy a weaker notion of observability which we define in terms of recognizability and achievability achievability serves as necessary and sufficient condition for the existence of a nondeterministic supervisor and it is weaker than controllability and observability combined when all events are controllable achievability reduces to recognizability we show that both existence and synthesis of nondeterministic supervisors can be determined polynomially for deterministic supervisors only existence can be determined polynomially both achievability and recognizability are preserved under union and also under intersection when restricted over prefix closed languages using the intersection closure property we derive a necessary and sufficient condition for the range control problem for the prefix closed case unlike the deterministic supervisory setting where the complexity of existence is exponential our existence condition is polynomially verifiable and also a supervisor can be polynomially synthesized

on deadlock free modular supervisory control of discrete event systems in this paper the problem of synthesizing deadlock free modular supervisors for discrete event systems is discussed by introducing the d invariant relation between a pair of supervisors it is shown that when the control objective is described in terms of intersection of two languages a necessary and sufficient condition for the modular supervisor to be deadlock free is that this pair of subsupervisors satisfies a d invariant relation a procedure for synthesizing the deadlock free modular supervisor is presented some issues concerning deadlock free modular supervisory control are also discussed

decentralized observability of discrete event systems with synchronizations abstract this paper deals with the problem of decentralized observability of discrete event systems we consider a set of sites each capable of observing a subset of the total event set when a synchronization occurs each site transmits its own observation to a coordinator that decides if the word observed belongs to a reference language k or not two different properties are studied uniform q observability and q sync observability it is proved that both properties are decidable for regular languages finally under the assumption that languages k and l are regular and all the events are observable by at least one site we propose a procedure to determine the instants at which synchronization should occur to detect the occurrence of any word not in k as soon as it occurs the advantage of the proposed approach is that most of the burdensome computations can be moved off line

failure diagnosis of discrete event systems with linear time temporal logic specifications the paper studies failure diagnosis of discrete event systems dess with linear time temporal logic ltl specifications the ltl formulas are used for specifying failures in the system the ltl based specifications make the specification specifying process easier and more user friendly than the formal language automata based specifications and they can capture the failures representing the violation of both liveness and safety properties whereas the prior formal language automaton based specifications can capture the failures representing the violation of only the safety properties such as the occurrence of a faulty event or the arrival at a failed state prediagnosability and diagnosability of dess in the temporal logic setting are defined the problem of testing prediagnosability and diagnosability is reduced to the problem of model checking an algorithm for the test of prediagnosability and diagnosability and the synthesis of a diagnoser is obtained the complexity of the algorithm is exponential in the length of each specification ltl formula and polynomial in the number of system states and the number of specifications the requirement of nonexistence of unobservable cycles in the system which is needed for the diagnosis algorithms in prior methods to work is relaxed finally a simple example is given for illustration

optimal supervisory control of discrete event systems cyclicity and interleaving of tasks a substantial number of tasks in production systems are executed in a repetitive cyclic fashion specifically production systems run different production cycles of different products as well as different instances of the same production cycle in this paper we consider the optimal control and interleaving of such production cycles in a supervisory control framework for discrete event systems dess that is different from other approaches our work is based on a behavioral specification of each production cycle first we adapt an optimal control approach for dess in order to optimize the operation of individual production cycles second we employ the interleaving composition to design a supervisor that enables the simultaneous execution of different production cycles combining both results we can further determine the maximum number of production cycles that can be executed simultaneously on a given production system

relative entropies in conformal field theory relative entropy is a measure of distinguishability for quantum states and it plays a central role in quantum information theory the family of renyi entropies generalizes to renyi relative entropies that include as special cases most entropy measures used in quantum information theory we construct a euclidean path integral approach to renyi relative entropies in conformal field theory then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick in contrast to the entanglement entropy the relative entropy is free of ultraviolet divergences and is obtained as a limit of certain correlation functions the relative entropy of two states provides an upper bound on their trace distance

robust nonblocking supervisory control of discrete event systems under partial observation in this paper we extend previous results on robust nonblocking supervisory control of discrete event systems with full observation to the case of control under partial observation where only a subset of events are observable and can be monitored the exact model of the plant is not known but is assumed to be among a finite set of possible models for each plant model a legal marked behavior is assumed given we manage to characterize the entire set of solutions of the robust control problem and obtain a set of necessary and sufficient conditions for the existence of a solution for the robust control problem

on closed loop liveness of discrete event systems under maximally permissive control a class of controlled discrete event systems modeled as controlled marked graphs cmgs a special case of controlled petri nets is considered liveness of controlled systems under maximally permissive feedback control policies is investigated in the cmg context closed loop liveness implies that from any reachable marking state any transition can be enabled to fire the concept of synchronic distances in petri nets is used to prove sufficient conditions under which the maximally permissive control results in a live closed loop system

robust prognosability for a set of partially observed discrete event systems abstract in this paper we consider a robust failure prognosis problem for partially observed discrete event systems given a set of possible models each of which has its own nonfailure specification we consider a single prognoser such that for all possible models it predicts any failure prior to its occurrence we call such a prognoser a robust prognoser we introduce a notion of robust prognosability and show that it serves as a necessary and sufficient condition for the existence of a robust prognoser we then present a method for verifying the robust prognosability condition moreover we discuss online synthesis of a robust prognoser

verification of robust diagnosability for partially observed discrete event systems in this paper we study robust failure diagnosis of discrete event systems given a set of possible models each of which has its own nonfailure specification we consider the existence of a single diagnoser such that for all possible models it detects any occurrence of a failure within a uniformly bounded number of steps we call such a diagnoser a robust diagnoser we introduce a notion of robust diagnosability and prove that it serves as a necessary and sufficient condition for the existence of a robust diagnoser we then present an algorithm for verifying the robust diagnosability condition

renyi and tsallis entropies for incomplete or overcomplete systems of events abstract in this article shannon renyi and tsallis entropies are considered for a system of events characterized by an arbitrary probability distribution p that can be incomplete complete or overcomplete after a suitable transformation that leads to the escort probabilities of p these can be written as the canonical probability distribution for a set of pseudo energies hartley information e n ln p n and a dimensionless parameter q that plays the role of thermodynamics several relations between the entropies are presented including the analysis of compound systems the method is illustrated with an example

exact time dependent solutions of the renyi fokker planck equation and the fokker planck equations related to the entropies proposed by sharma and mittal abstract exact time dependent solutions representing generalized one dimensional ornstein uhlenbeck processes are derived from nonlinear fokker planck equations and the corresponding diffusion processes are studied the corresponding systems are related to the renyi entropy and entropies proposed by sharma and mittal in the former case normal diffusion is found with a diffusion rate depending on the order of the renyi entropy in the latter case we find anomalous diffusion processes

fuzzy discrete event systems under fuzzy observability and a test algorithm in order to more effectively cope with the real world problems of vagueness impreciseness and subjectivity fuzzy discrete event systems fdess were proposed recently notably fdess have been applied to biomedical control for human immunodeficiency virus hiv acquired immune deficiency syndrome aids treatment planning and sensory information processing for robotic control qiu independently developed supervisory control theory of fdess we note that the controllability of events in qiu s work is fuzzy but the observability of events is crisp and the observability of events in cao and ying s work is also crisp although the controllability is not completely crisp since the controllable events can be disabled with any degrees motivated by the necessity to consider the situation that the events may be observed or controlled with some membership degrees in this paper we establish the supervisory control theory of fdess with partial observations in which both the observability and controllability of events are fuzzy instead we formalize the notions of fuzzy controllability condition and fuzzy observability condition in addition controllability and observability theorem of fdess is set up in a more generic framework in particular we present a detailed computing flow to verify whether the controllability and observability conditions hold thus this result can decide the existence of supervisors also we use this computing method to check the existence of supervisors in the controllability and observability theorem of classical discrete event systems dess which is a new method and different from classical case a number of examples are elaborated on to illustrate the presented results

on the complexity of synthesizing a minimum weighted supervisor under partial observation abstract with full observation the problem of synthesizing a minimum weighted supervisor has been shown polynomial time solvable with partial observation the problem becomes np hard in this paper we will show that the decision version of the problem becomes np complete when the natural projection is a natural observer this np completeness result is unexpected considering that the logic supervisor synthesis problem under the same assumption becomes polynomial time solvable

supervisory control of partially observable marked graphs this note presents a control synthesis approach for discrete event systems modeled by marked graphs with unobservable transitions it solves forbidden state problems characterized by a set of general mutual exclusion constraints we prove that for any sequence of observable transitions there exist a unique marking from which all other possible current markings can be reached unobservably this salient feature allows us to design efficient control policies based on proper separation of observation and control

control of discrete event systems with minimal switchings this paper studies the problem of controlling discrete event systems as formulated by ramadge and wonham 1987 an optimal d supervisor is introduced which requires a minimal number of switchings i e enabling and disabling controllable events a complete characterization of such controllers as well as a procedure for its construction is given an example to illustrate the result is also included

predictability of event occurrences in partially observed discrete event systems this paper studies the problem of predicting occurrences of a significant event in a partially observed discrete event system the predictability of occurrences of an event in a system is defined in the context of formal languages the predictability of a language is a stronger condition than the diagnosability of the language two necessary and sufficient conditions for predictability of occurrences of an event in systems modeled by regular languages are presented both conditions can be algorithmically tested the first condition employs diagnosers the second condition employs verifiers and results in a polynomial time in the number of states complexity test for verification of predictability when predictability holds diagnosers can be used online to predict the significant event

a tutorial on the extended perturbation analysis abstract in this paper we explain the simple idea of infinitesimal and extended perturbation analysis the motivation of the new technique the difficulties as well as the advantages and contributions

the complexity of mean payoff pushdown games two player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems in this work we consider solving recursive game graphs or pushdown game graphs that model the control flow of sequential programs with recursion while pushdown games have been studied before with qualitative objectives such as reachability and regular objectives in this work we study for the first time such games with the most well studied quantitative objective the mean payoff objective in pushdown games two types of strategies are relevant 1 global strategies which depend on the entire global history and 2 modular strategies which have only local memory and thus do not depend on the context of invocation but rather only on the history of the current invocation of the module our main results are as follows 1 one player pushdown games with mean payoff objectives under global strategies are decidable in polynomial time 2 two player pushdown games with mean payoff objectives under global strategies are undecidable 3 one player pushdown games with mean payoff objectives under modular strategies are np hard 4 two player pushdown games with mean payoff objectives under modular strategies can be solved in np i e both one player and two player pushdown games with mean payoff objectives under modular strategies are np complete we also establish the optimal strategy complexity by showing that global strategies for mean payoff objectives require infinite memory even in one player pushdown games and memoryless modular strategies are sufficient in two player pushdown games finally we also show that all the problems have the same complexity if the stack boundedness condition is added where along with the mean payoff objective the player must also ensure that the stack height is bounded

a maximum entropy approach to unsupervised mixed pixel decomposition due to the wide existence of mixed pixels the derivation of constituent components endmembers and their fractional proportions abundances at the subpixel scale has been given a lot of attention the entire process is often referred to as mixed pixel decomposition or spectral unmixing although various algorithms have been proposed to solve this problem two potential issues still need to be further investigated first assuming the endmembers are known the abundance estimation is commonly performed by employing a least squares error criterion which however makes the estimation sensitive to noise and outliers second the mathematical intractability of the abundance non negative constraint results in computationally expensive numerical approaches in this paper we propose an unsupervised decomposition method based on the classic maximum entropy principle termed the gradient descent maximum entropy gdme aiming at robust and effective estimates we address the importance of the maximum entropy principle for mixed pixel decomposition from a geometric point of view and demonstrate that when the given data present strong noise or when the endmember signatures are close to each other the proposed method has the potential of providing more accurate estimates than the popular least squares methods e g fully constrained least squares we apply the proposed gdme to the subject of unmixing multispectral and hyperspectral data the experimental results obtained from both simulated and real images show the effectiveness of the proposed method

feedback stabilization of some event graph models the authors introduce several notions of stability for event graph models timed or not the stability is similar to the boundedness notion for petri nets the event graph models can be controlled by an output feedback which takes information from some observable transitions and can disable some controllable transitions the controller itself is composed of an event graph in this framework the authors solve the corresponding stabilization problems i e they wonder if such a controller may prevent the explosion of the number of tokens

measuring renyi entanglement entropy in quantum monte carlo simulations we develop a quantum monte carlo procedure in the valence bond basis to measure the renyi entanglement entropy of a many body ground state as the expectation value of a unitary swap operator acting on two copies of the system an improved estimator involving the ratio of swap operators for different subregions enables convergence of the entropy in a simulation time polynomial in the system size we demonstrate convergence of the renyi entropy to exact results for a heisenberg chain finally we calculate the scaling of the renyi entropy in the two dimensional heisenberg model and confirm that the neel ground state obeys the expected area law for systems up to linear size l 32

exploiting symmetry in the synthesis of supervisors for discrete event systems we investigate a method to exploit the symmetry inherent to many discrete event systems in order to reduce the computational complexity of the supervisory control problem scp we characterize symmetry using notions of group theory and derive conditions under which there exists a quotient reduced automaton representation for a language with in general a much smaller state space than the original minimal in number of states automaton representation for the language we then propose an algorithm to synthesize a solution to the scp which is similar to the classical one but performed on reduced automata special attention is given to the particular case of systems whose models contain similar components the approach is illustrated by an example of control of a small production line

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

maximally permissive feedback and modular control synthesis in petri nets with external input places a necessary and sufficient condition is presented for the unique existence of a maximally permissive feedback mpf in a concurrent discrete event system modeled by a petri net the concept of the weak interaction of a predicate is introduced in addition to reviewing fundamental definitions and notations the author defines the control invariance of a predicate and an mpf a modular synthesis problem of mpf is discussed

decentralized control of discrete event systems using prioritized composition with exclusion a necessary and sufficient condition for decentralized control of discrete event systems via prioritized composition with exclusion pcx is provided pcx was extended from prioritized synchronous composition psc in order to model various boolean modes of decision fusion by allowing the exclusivity of participation in system interactions a notion of pcx coobservability is introduced that captures the nature of pcx based decision fusion and a testing algorithm with polynomial complexity in the states of plant and specification is presented the result is extended for multiple supervisors with multiple exclusivity sets based on an extended pcx definition

on optimal attraction in discrete event processes abstract the concept of attraction has been shown to play a basic role in the synthesis of supervisors for stabilizing discrete event processes in this paper we define and investigate optimal attraction in the sense that the cost or distance of the convergent path is minimized with respect to some distance measure we find conditions for the existence of supervisors achieving optimal attraction and provide efficient algorithms for their synthesis

troublesome aspects of the renyi maxent treatment we study in great detail the possible existence of a renyi associated thermodynamics with negative results in particular we uncover a hidden relation in renyi s variational problem maxent this relation connects the two associated lagrange multipliers canonical ensemble with the mean energy u and the renyi parameter as a consequence of such relation we obtain anomalous renyi maxent thermodynamic results

on the notion of uncontrollable marking in supervisory control of petri nets we show that the notion of uncontrollable marking commonly used in the literature on supervisory control theory of petri nets is not sound by means of a counter example we also show how the definition can be corrected and provide an adaptation of a decidability proof for the problem of checking controllability for specifications expressed as deterministic petri net languages

observability and decentralized control of fuzzy discrete event systems fuzzy discrete event systems as a generalization of crisp discrete event systems have been introduced in order that it is possible to effectively represent uncertainty imprecision and vagueness arising from the dynamic of systems a fuzzy discrete event system has been modeled by a fuzzy automaton its behavior is described in terms of the fuzzy language generated by the automaton in this paper we are concerned with the supervisory control problem for fuzzy discrete event systems with partial observation observability normality and co observability of crisp languages are extended to fuzzy languages it is shown that the observability together with controllability of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor when a decentralized solution is desired it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co observable moreover the infimal controllable and observable fuzzy superlanguage and the supremal controllable and normal fuzzy sublanguage are also discussed simple examples are provided to illustrate the theoretical development

minimizing the cardinality of an events set for supervisors of discrete event dynamical systems in this paper we study sufficient observation and work spaces for supervisors of a class of discrete event dynamical systems deds we use finite state automata to model a deds the finite automata generates a formal language defined over the set of events in the deds a supervisor is a feedback system that observes a generated trace of events and dynamically disables or enables a subset of controllable events such that the closed loop system behaves as desired we model the desired behavior of the deds by a sublanguage defined over the set of events a sufficient observation space is a collection of events whose observation by a supervisor is enough to realize a given desired behavior a sufficient work space is a sufficient observation space such that the control action of the supervisor is limited to only the controllable elements of the work space we construct algorithms to evaluate a sufficient work or observation space the reduction of work or observation spaces results in reducing the number of event detectors communication and command channels between the supervisor and the plant

on the existence of finite state supervisors for arbitrary supervisory control problems given two prefix closed languages k l spl sube spl sigma where k spl sube l represents the desired closed loop behavior and l is the open loop behavior there exists a finite state supervisor that enforces k in the closed loop if and only if there is a regular prefix closed language m spl sube spl sigma such that 1 m spl sigma sub u spl cup l spl sube m and 2 m spl cup l k in this paper we show that this is equivalent to 1 the controllability of sup spl lcub p spl sube k spl cup l spl macr spl verbar pr p p spl rcub with respect to spl sigma and 2 the regularity of sup spl lcub p spl sube k spl cup l spl macr spl verbar pr p p spl rcub where l spl macr spl sigma spl minus l and pr spl middot is the set of prefixes of strings in the language argument we use this property to investigate the issue of deciding the existence of a finite state supervisor for different representations we also present some properties of the language sup spl lcub p spl sube k spl cup l spl macr spl verbar pr p p spl rcub along with implications to the synthesis of solutions to the supervisory control problem with the fewest states