l and r localized solvabilities of max separable interval linear equations and its applications in this paper we propose new types of solutions an l localized solution and an r localized solution to interval systems of max separable linear equations a x b where a a a and b b b we provide some small application examples to emphasize the importance of our proposed solutions we give necessary and sufficient conditions for checking l localized and r localized solvabilities and apply them to achieve the closed forms of the sets of all l localized solutions and r localized solutions respectively

resolution of a system of fuzzy polynomial equations using the grobner basis the occurrence of imprecision in the real world is inevitable due to some unexpected situations the imprecision is often involved in any engineering design process the imprecision and uncertainty are often interpreted as fuzziness fuzzy systems have an essential role in the uncertainty modelling which can formulate the uncertainty in the actual environment in this paper a new approach is proposed to solve a system of fuzzy polynomial equations based on the grobner basis in this approach first the h cut of a system of fuzzy polynomial equations is computed and a parametric form for the fuzzy system with respect to the parameter of h is obtained then a grobner basis is computed for the ideal generated by the h cuts of the system with respect to the lexicographical order using faugere s algorithm i e f 4 algorithm the grobner basis of the system has an upper triangular structure therefore the system can be solved using the forward substitution hence all the solutions of the system of fuzzy polynomial equations can easily be obtained finally the proposed approach is compared with the current numerical methods some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods

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

strong solvability of interval max plus systems and applications to optimal control abstract this paper considers the strong solvability of interval max plus systems the solvable entry and solvable interval are introduced to establish the necessary and sufficient condition for strong solvability of interval max plus systems it is proved that such a criterion is equivalent to the criterion proposed by cechlarova and cuninghame green the interval strong solution of interval max plus systems is described and the necessary and sufficient condition for uniquely strong solvability is given the method of testing strong solvability of interval max plus systems is constructive and leads to an effective algorithm to illustrate our results a workshop optimal control problem is addressed

interval systems over idempotent semiring this paper deals with solution of inequality a x b where a x and b are interval matrices with entries dened over idempotent semiring it deals also with the computation of a pair of intervals x y which satises the equation

x simple image eigencones of tropical matrices abstract we investigate max algebraic tropical one sided systems a x b where b is an eigenvector and x lies in an interval x a matrix a is said to have x simple image eigencone associated with an eigenvalue if any eigenvector x associated with and belonging to the interval x is the unique solution of the system a y x in x we characterize matrices with x simple image eigencone geometrically and combinatorially and for some special cases derive criteria that can be efficiently checked in practice

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

on measuring nondeterminism in regular languages abstract it is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language this paper studies situations intermediate between forbidding nondeterminism and allowing it the amount of nondeterminism used by a finite automaton is quantified so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied these intermediate situations are shown always to lie between two extremes 1 there are no savings as the amount of nondeterminism increases incrementally so that savings occur only when the amount of nondeterminism becomes unlimited 2 each increment of nondeterminism results in additional savings the number s of states decreasing approximately as s 1 i until exponential savings have been achieved after about i log s log log s increments

adaptive coded modulation for fading channels we apply coset codes to adaptive modulation in fading channels adaptive modulation is a powerful technique to improve the energy efficiency and increase the data rate over a fading channel coset codes are a natural choice to use with adaptive modulation since the channel coding and modulation designs are separable therefore trellis and lattice codes designed for additive white gaussian noise awgn channels can be superimposed on adaptive modulation for fading channels with the same approximate coding gains we first describe the methodology for combining coset codes with a general class of adaptive modulation techniques we then apply this methodology to a spectrally efficient adaptive m ary quadrature amplitude modulation mqam to obtain trellis coded adaptive mqam we present analytical and simulation results for this design which show an effective coding gain of 3 db relative to uncoded adaptive mqam for a simple four state trellis code and an effective 3 6 db coding gain for an eight state trellis code more complex trellis codes are shown to achieve higher gains we also compare the performance of trellis coded adaptive mqam to that of coded modulation with built in time diversity and fixed rate modulation the adaptive method exhibits a power savings of up to 20 db

numerical solution of fuzzy max min systems abstract for solvable systems of linear equations of the form a x b over the max min algebra b 0 1 max min we propose efficient methods for finding fuzzy determinant and fuzzy solution we formulate a condition for linear systems of equations over b to have not a unique solution by using fuzzy determinant

on the solvability of interval max min matrix equations abstract max min algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and minimum respectively the notation a x c b where a b and c are given interval matrices represents an interval max min matrix equation the paper deals with the solvability of interval matrix equations in max min algebra we define three types of solvability of interval max min matrix equations namely the strongly universal universal and weakly universal solvability we provide the equivalent conditions for each type of solvability that can be verified in polynomial times

optimal adaptive modulation and coding with switching costs we present an optimal adaptive modulation and coding policy that minimizes the transmission latency and modulation coding switching cost across a finite state markovian fading channel we formulate the optimal tradeoff between transmission latency and modulation coding switching cost as a discounted infinite horizon markov decision problem mdp by exploiting special structures of the formulated mdp and under certain sufficient conditions we show that optimal modulation and coding selection policies are monotone in the state variables these monotone optimal policies are computationally inexpensive to implement and are scalable in terms of channel and switching cost parameters numerical results confirm the monotonicity and threshold based structure of the optimal modulation and coding scheme mcs selection policies under the proposed sufficient conditions

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

a note on unsolvable systems of max min fuzzy equations abstract for unsolvable systems of linear equations of the form a x b over the max min fuzzy algebra r 0 1 max min we propose an efficient method for finding a chebychev best approximation of the matrix a r m n in the set s b d r m n d x b is solvable

unique solvability of max min fuzzy equations and strong regularity of matrices over fuzzy algebra a matrix is said to have strongly linearly independent columns or in the case of square matrices to be strongly regular if for some vector b the system a x b has a unique solution we formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property moreover an algorithm for testing these properties is reviewed

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

descriptional complexity of unambiguous input driven pushdown automata it is known that a nondeterministic input driven pushdown automaton idpda a k a visibly pushdown automaton a k a nested word automaton with n states can be transformed to an equivalent deterministic automaton with 2 n 2 states b von braunmuhl and r verbeek 1983 8 and that this size is necessary in the worst case r alur and p madhusudan 2009 4 this paper demonstrates that the same worst case 2 n 2 size blow up occurs when converting a nondeterministic idpda to an unambiguous one and an unambiguous idpda to a deterministic one in addition the methods developed in this paper are used to demonstrate that the descriptional complexity of complementation for nondeterministic idpdas is 2 n 2 and that the descriptional complexity of homomorphisms for deterministic idpdas is 2 n 2

some properties of recognizable l subsets abstract we study some properties of recognizable z subsets of a and its subfamilies the simple z subsets the limited z subsets the recognizable m subsets the simple m subsets and the m subsets which are nondeterministic complexities of finite automata at first we study some necessary conditions for membership in each one of these families and after we investigate the closure properties of these families under several operations we also study the relations existing among some subfamilies of recognizable m subsets and the families h p p 0 obtained by simon

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

ae solutions to interval linear systems over max plus algebra abstract this paper introduces a concept of ae solutions to interval max plus linear systems a rather general concept which includes many known concepts of solutions to interval systems weak solutions strong solutions tolerance solutions and control solutions as its special cases we state full characterizations of ae solutions for the interval max plus systems including both linear inequalities and linear equations moreover a kind of dependency characterized by double appearance of some sub matrix of the constraint matrix is discussed and we prove that which can be relaxed with no change in ae solvability

an algorithm for solving two sided interval system of max plus linear equations we study six types of solutions weak strong tolerance control left localized and right localized solutions to a two sided interval system of max plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions these conditions are in the form of two sided systems of max plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems therefore we could work on the interval inclusion linear systems instead of directly solving these six types of solutions themselves an optimization problem with the two sided interval system of max plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems

tolerance types of interval eigenvectors in max plus algebra interval eigenvectors of interval matrices corresponding to steady states of systems with inexact data are investigated tolerable strongly tolerable and weakly tolerable interval eigenvectors are described polynomial solutions to the recognition versions of all three types of tolerability are presented max plus algebra plays a key role in the study of discrete event systems in connection with optimization problems such as scheduling or project management in which the objective function depends on the operations maximum and plus the steady states of such systems correspond to the eigenvectors of max plus matrices therefore the investigation of the properties of these eigenvectors is important for applications matrices and vectors with interval coefficients in a max plus algebra are studied in this paper and polynomial algorithms are presented for the recognition problem with three types of tolerance interval eigenvector

communication complexity method for measuring nondeterminism in finite automata while deterministic finite automata seem to be well understood surprisingly many important problems concerning nondeterministic finite automata nfa s remain open one such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata in this paper the concept of communication complexity is applied in order to achieve progress in this problem area the main results are as follows 1 deterministic communication complexity provides lower bounds on the size of nfa s with bounded unambiguity applying this fact the proofs of several results about nfa s with limited ambiguity can be simplified and presented in a uniform way 2 there is a family of languages konk2 with an exponential size gap between nfa s with polynomial leaf number ambiguity and nfa s with ambiguity k this partially provides an answer to the open problem posed by b ravikumar and o ibarra 1989 siam j comput 18 1263 1282 and h leung 1998 siam j comput 27 1073 1082

x robustness of interval circulant matrices in fuzzy algebra abstract fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b min a b a matrix is x robust if its eigenspace is achieved starting at each vector from a given interval vector x the possible and universal x robustness of interval matrices over fuzzy algebra are defined polynomial algorithms for checking the possible and universal x robustness of interval circulant matrices are described

interval max plus matrix equations abstract this paper deals with the solvability of interval matrix equations in max plus algebra max plus algebra is the algebraic structure in which classical addition and multiplication are replaced by and where a b max a b and a b a b the notation a x c b represents an interval max plus matrix equation where a b and c are given interval matrices we define four types of solvability of interval max plus matrix equations i e the tolerance weak tolerance left weak tolerance and right weak tolerance solvability we derive the necessary and sufficient conditions for checking each of them whereby all can be verified in polynomial time

optimal control of markov decision processes with linear temporal logic constraints in this paper we develop a method to automatically generate a control policy for a dynamical system modeled as a markov decision process mdp the control specification is given as a linear temporal logic ltl formula over a set of propositions defined on the states of the mdp motivated by robotic applications requiring persistent tasks such as environmental monitoring and data gathering we synthesize a control policy that minimizes the expected cost between satisfying instances of a particular proposition over all policies that maximize the probability of satisfying the given ltl specification our approach is based on the definition of a novel optimization problem that extends the existing average cost per stage problem we propose a sufficient condition for a policy to be optimal and develop a dynamic programming algorithm that synthesizes a policy that is optimal for a set of ltl specifications

interval strong solutions of interval systems of max plus linear equations abstract this paper considers the existence and uniqueness of interval strong solutions of interval systems of max plus linear equations a necessary and sufficient condition for the existence of interval strong solutions is presented the proof of the existence of interval strong solutions is constructive and results in a formula for computing such solutions a necessary and sufficient condition for the uniqueness of interval strong solutions is established by testing the unique solvability of a finite number of subsystems rather than all subsystems on this basis a polynomial algorithm is developed to verify the uniqueness of interval strong solutions

contributions to max min convex geometry i segments abstract we give some contributions to the theory of max min convex geometry that is convex geometry in the semimodule r max min n over the max min semiring r max min r we introduce elementary segments that generalize from n 2 the horizontal vertical or oblique segments contained in the main bisector of r max min 2 we show that every segment in r max min n is a concatenation of a finite number of elementary subsegments at most 2 n 1 respectively at most 2 n 2 in the case of comparable respectively incomparable endpoints x y in this first part we study max min segments and in the subsequent second part submitted we study max min semispaces and some of their relations to max min convex sets

closed form of l localized solution set of max plus interval linear system and its application on optimization problem a closed form of the l localized solution set of an interval linear system with max plus algebra is a finite union of interval vectors this paper analyzes and proves some properties to reduce number of the union sets to be a smaller number of the union of interval vectors that are not subsets of each other this result helps simplify the feasible set of an optimization problem with max plus linear equation constraints

solution of non linear fuzzy systems by decomposition of incremental fuzzy numbers this paper presents a proposal for decomposing large uncertainties associated with fuzzy numbers which can be useful for solving non linear fuzzy problems with crisp coefficients the proposed procedure is based on a property of incremental fuzzy numbers and improves the quality of solution of the linearization process the methodology is illustrated by some numeric studies

interval systems of max separable linear equations an interval system of equations is weakly solvable if at least one of its subsystems is solvable and it is strongly solvable if all its subsystems are solvable we give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max plus and max min algebras

the robustness of interval matrices in max plus algebra abstract robust interval matrices in max plus algebra are studied and equivalent conditions for interval matrices to be robust are presented the necessary and sufficient conditions for robustness of interval matrices are introduced in addition efficient algorithms for verifying the robustness of interval circulant and interval zero one matrices are described

residuation in fuzzy algebra and some applications abstract the system 0 1 max min has a number of well known applications many of which involve the inversion of linear relations we show the advantage of residuation theory for this process and apply it to the analysis of the problem of chebychev approximation

inverse problem in fuzzy relational equations abstract the paper deals with the problem of handling fuzzy relational equations of the form x r y in the case of an empty solution set with respect to x x r y discussing the essence of the structure of the fuzzy relation r with respect to provided y algorithms have been proposed which enable us to modify the original equation i e the fuzzy set y in a way leading to its genuine solution or at least an approximate one it is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation a grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set y which forms a cornerstone of the proposed procedures numerical examples are provided some aspects related to the area of diagnostic models in which these equations are studied are also treated

on the relation between ambiguity and nondeterminism in finite automata abstract nondeterminism in a finite automaton is measured dynamically by counting the number of guesses that the automaton has to make in order to recognize an input string when the amount of nondeterminism is small bounded or large linear in the input length nothing can be concluded about the amount of ambiguity in the automaton but when the amount of nondeterminism is intermediate between these extremes the degree of ambiguity must be infinite

the robustness of interval fuzzy matrices abstract robust interval matrices over max min algebra fuzzy matrices are studied and equivalent conditions for interval fuzzy matrices to be robust are presented polynomial algorithms for checking the necessary and sufficient conditions for robustness of interval fuzzy matrices are introduced in addition more efficient algorithms for verifying the robustness of interval circulant matrices are described

formal verification of stochastic max plus linear systems this work investigates the computation of finite abstractions of stochastic max plus linear smpl systems and their formal verification against general bounded time linear temporal specifications smpl systems are probabilistic extensions of discrete event mpl systems which are widely employed for modeling engineering systems dealing with practical timing and synchronization issues departing from the standard existing approaches for the analysis of smpl systems we newly propose to construct formal finite abstractions of a given smpl system the smpl system is first re formulated as a discrete time markov process then abstracted as a finite state markov chain mc the derivation of precise guarantees on the level of the introduced formal approximation allows us to probabilistically model check the obtained mc against bounded time linear temporal specifications which are of rather general applicability and to reliably export the obtained results over the original smpl system the approach is practically implemented on a dedicated software and is elucidated and run over numerical examples

resolving infeasibility in extremal algebras abstract two extremal algebras b based on a linearly ordered set b are considered in the maxmin algebra max min and in the maxgroup algebra max and is a group operation if a system a x b of linear equations over an extremal algebra is insolvable then any subset of equations such that its omitting leads to a solvable subsystem is called a relieving set we show that the problem of finding the minimum cardinality relieving set is np complete in the maxmin algebra already for bivalent systems while it is polynomially solvable for bivalent systems in maxgroup algebra and also np complete for trivalent systems

probabilistic reachability and safety for controlled discrete time stochastic hybrid systems in this work probabilistic reachability over a finite horizon is investigated for a class of discrete time stochastic hybrid systems with control inputs a suitable embedding of the reachability problem in a stochastic control framework reveals that it is amenable to two complementary interpretations leading to dual algorithms for reachability computations in particular the set of initial conditions providing a certain probabilistic guarantee that the system will keep evolving within a desired safe region of the state space is characterized in terms of a value function and maximally safe markov policies are determined via dynamic programming these results are of interest not only for safety analysis and design but also for solving those regulation and stabilization problems that can be reinterpreted as safety problems the temperature regulation problem presented in the paper as a case study is one such case

interval max plus systems of linear equations abstract in this paper we shall deal with solvability of interval systems of linear equations in max plus algebra max plus algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b a b the notation a x b represents an interval system of linear equations where a a a b b b are given interval matrix and interval vector respectively and a solution is from a given interval vector x x x we can define several types of solvability of interval systems in this paper we define six types of solvability of interval max plus systems of linear equations and give necessary and sufficient conditions for them

a coupling approach to estimating the lyapunov exponent of stochastic max plus linear systems this paper addresses the problem of approximately computing the lyapunov exponent of stochastic max plus linear systems our approach allows for an efficient simulation of bounds for the lyapunov expo nent we provide sufficient conditions for the convergence of the bounds in particular a perfect sampling scheme for the lyapunov exponent is established we illustrate the effectiveness of our bounds with an application to real life railway systems

iteration algorithm for solving ax b in max min algebra abstract for solving the linear system a x b we introduce some iteration algorithms where a aij and b bi a given matrix and a column vector with elements from a brouwerian lattice and is a max min composite operation

idempotent and tropical mathematics complexity of algorithms and interval analysis abstract a very brief introduction to tropical and idempotent mathematics is presented tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the planck constant tends to zero taking imaginary values in the framework of idempotent mathematics constructions and algorithms are usually simpler than their traditional analogs we examine in particular algorithms of tropical idempotent mathematics generated by a collection of basic semiring or semifield operations and other good operations every algorithm of this type has an interval version the complexity of this interval version coincides with the complexity of the initial algorithm the interval version of an algorithm of this type gives exact interval estimates for the corresponding output data algorithms of linear algebra over idempotent semirings are examined in this case basic algorithms like their interval versions are polynomial this situation is very different from the traditional linear algebra case where basic algorithms are polynomial but the corresponding interval versions are np hard and interval estimates are not exact

discrete event systems with stochastic processing times discrete event dynamic systems are studied in which the underlying algebra is the max algebra and the coefficients in the system referring to processing times in practice are stochastic the processing times and or the transportation times within a network show stochastic fluctuations the restrictions are that the stochastic processing times of the nodes in the network are independent and identically distributed the asymptotic behavior of the system is investigated and the average duration of one cycle of the process is calculated a specific example of the theory is considered the state space is two dimensional and the probability distributions are exponential it is shown that the process approaches a stationary limit as time proceeds the case when the probability distributions are discrete is also treated several examples are given two dimensional systems and more generally finite dimensional systems are considered

mimo transmission control in fading channels a constrained markov decision process formulation with monotone randomized policies this paper addresses the optimal power and rate allocation control in multiple input multiple output mimo wireless systems over markovian fading channels the problem is posed as an infinite horizon average cost constrained markov decision process cmdp with the goal of minimizing the average transmission power subject to delay constraints by using a lagrangian formulation of the cmdp we use the concepts of stochastic dominance submodularity and multimodularity to prove that the optimal randomized policies are monotone three important structural results on the nature of the optimal randomized policies are derived first we show that the action space can be exponentially reduced by decomposing the rate allocation problem into bit loading problem across individual antennas and the total rate allocation based on the current buffer occupancy and channel state second we show that the optimal rate allocation policy is a randomized mixture of two pure policies that are monotonically increasing in the buffer occupancy finally we show that the optimal power allocation is piecewise linear in the delay constraint these three structural results can be exploited to devise efficient online reinforcement learning algorithms for optimal rate allocation

interval eigenproblem in max min algebra the eigenvectors of square matrices in max min algebra correspond to steady states in discrete events system in various application areas such as design of switching circuits medical diagnosis models of organizations and information systems imprecise input data lead to considering interval version of the eigenproblem in which interval eigenvectors of interval matrices in max min algebra are investigated six possible types of an interval eigenvector of an interval matrix are introduced using various combination of quantifiers in the definition the previously known characterizations of the interval eigenvectors were restricted to the increasing eigenvectors see 11 in this chapter the results are extended to the non decreasing eigenvectors and further to all possible interval eigenvectors of a given max min matrix classification types of general interval eigenvectors are studied and characterization of all possible six types is presented

dimension reduction and feedback stabilization for max plus linear systems and applications in vlsi array processors this paper investigates the dimension reduction and feedback stabilization of max plus linear systems and applies them to control and optimize the very large scale integration vlsi array processors we introduce the weakly similar relation between max plus matrices and the pseudoequivalent relation between autonomous max plus linear systems and point out that two systems are pseudoequivalent if and only if their state matrices are weakly similar the reduced system is defined by using the pseudoequivalence whose dimension is determined by the row rank of the original state matrix we focus on obtaining a reduced system which maintains the stability and retains the steady state period an algorithm of polynomial complexity is developed to find such a reduced system the reduced system is then used to design a state feedback controller to stabilize a max plus linear system finally we use the vlsi array processors as an example to demonstrate how the presented methods work in practical applications

finite valued distance automata abstract distance automata are a model of finite state machines which charge to an input word the expense of the cheapest successful computation consuming that word called its distance the distance of such a machine is the maximal distance of an input word recognized by it or is infinite depending on whether or not a maximum exists the valuedness of a distance automaton is introduced by considering this machine to be a transducer with unary output alphabet the distance of a finite valued distance automaton with n states is either infinite or at most 6 n 2 n n 2 1 in the former case its growth is linear in the input length the problem of deciding whether a given finite valued 2 valued respectively distance automaton has infinite distance is pspace complete it is decidable in deterministic double exponential time whether two given finite valued distance automata are equivalent i e every input word has the same distance in both machines there is an inherently infinite valued distance automaton and for each k an inherently k valued distance automaton such that the growth of the distance in all these machines is linear in the input length it is decidable in deterministic polynomial time whether a 2 valued distance automaton given as the disjoint union of two single valued distance automata is inherently 2 valued