strategy iteration is strongly polynomial for 2 player turn based stochastic games with a constant discount factor ye 2011 showed recently that the simplex method with dantzig s pivoting rule as well as howard s policy iteration algorithm solve discounted markov decision processes mdps with a constant discount factor in strongly polynomial time more precisely ye showed that both algorithms terminate after at most o mn1 log n1 iterations where n is the number of states m is the total number of actions in the mdp and 0

the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

invariance concepts in infinite dimensions the concepts of a b and c a invariance are examined for infinite dimensional linear systems examples such as reachability a reachability inobservability and a inobservability subspaces are given these concepts are then used to solve various disturbance decoupling problems

what s decidable about weighted automata abstract weighted automata map input words to values and have numerous applications in computer science a result by krob from the 90s implies that the universality problem is decidable for weighted automata over the tropical semiring with weights in n and is undecidable when the weights are in z we continue the study of the borders of decidability in weighted automata over the tropical semiring we give a complete picture of the decidability and complexity of various decision problems for them including non emptiness universality equality and containment for the undecidability results we provide direct proofs which stay in the terrain of state machines this enables us to tighten the results and apply them to a very simple class of automata in addition we provide a toolbox of algorithms and techniques for weighted automata on top of which we establish the complexity bounds

quantitative supervisory control game for discrete event systems we formulate an optimal supervisory control problem for quantitative non terminating discrete event systems dess modeled by finite weighted automata the control performance of a supervisor is evaluated by the worst case limit average weight of the infinite sequences generated by the supervised des an optimal supervisor is a supervisor that avoids deadlocks and maximizes the control performance we propose a game theoretical design method for an optimal supervisor using a two player turn based mean payoff game automaton as the first player the objective of the supervisor is to maximize the worst case limit average weight of the generated sequences as the second player the des aims to minimize it we show that an optimal supervisor can be computed from an optimal strategy of the first player for this game then we propose an algorithm to compute an f minimally restrictive optimal supervisor which is a finite memory optimal supervisor that enables as many sequences as possible and can be represented by an optimal strategy for a finite version of the two player game

weak dual residuations applied to tropical linear equations an extension to an algorithm of r a cuninghame green and k zimmermann for solving equations with residuated functions is presented this extension relies on the concept of weak residuation and in the so called strong property it is shown that a contextualization of this method to tropical linear equations which will be denoted as primal method in contrast with the dual method another algorithm described in literature generates a non decreasing sequence which converges to the smallest solution in a special semimodule it is also shown the connections of this method with previously published works

a new lagrangian method for time dependent inviscid flow computation this paper presents a new lagrangian approach for the two dimensional 2 d time dependent euler equations it may be considered as a sequel to the authors previous lagrangian approaches for steady supersonic flow computations c y loh and w h hui j comput phys 89 1990 pp 207 240 w h hui and c y loh j comput phys 103 1992 pp 450 464 w h hui and c y loh j comput phys 103 1992 pp 465 471 c y loh and m s liou j comput phys 104 1993 pp 150 161 c y loh and m s liou siam j sci comput 15 1994 pp 1038 1058 c y loh and m s liou j comput phys 113 1994 pp 224 248 the theoretical background and the intrinsic flow coordinates as well as the lagrangian conservation form are introduced based on the concept of material functions or path functions a tvd scheme of the godunov type is chosen to describe the numerical procedure several examples are then given to justify the claimed advantages of the new methodology namely a any contact discontinuities are crisply solved and b grids are automatically and accurately generated following pathlines

fast algorithms for computing isogenies between elliptic curves we survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime we introduce a new algorithm that computes an isogeny of degree l l different from the characteristic in time quasi linear with respect to l this is based in particular on fast algorithms for power series expansion of the weierstrass function and related functions

just in time control of timed event graphs update of reference input presence of uncontrollable input a linear system theory has been developed for the class of discrete event systems subject to synchronization this paper presents the just in time control of such systems when reference input is updated and or in the presence of uncontrollable input s the proposed controls are the solutions to an optimization problem under equality constraint

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

combinatorial structure and randomized subexponential algorithms for infinite games the complexity of solving infinite games including parity mean payoff and simple stochastic is an important open problem in verification automata and complexity theory in this paper we develop an abstract setting for studying and solving such games based on function optimization over certain discrete structures we introduce new classes of recursively local global rlg and partial recursively local global prlg functions and show that strategy evaluation functions for simple stochastic mean payoff and parity games belong to these classes in this setting we suggest randomized subexponential algorithms appropriate for rlg and prlg function optimization we show that the subexponential algorithms for combinatorial linear programming due to kalai and matousek sharir welzl can be adapted for optimizing the rlg and prlg functions

an efficient deterministic test for kloosterman sum zeros we propose a simple deterministic test for deciding whether or not a non zero element a in mathbb f 2 n or mathbb f 3 n is a zero of the corresponding kloosterman sum over these fields and analyse its complexity the test seems to have been overlooked in the literature for binary fields the test has an expected operation count dominated by just two mathbb f 2 n multiplications when n is odd with a slightly higher cost for even extension degrees making its repeated invocation the most efficient method to date to find a non trivial kloosterman sum zero in these fields the analysis depends on the distribution of sylow p subgroups in two corresponding families of elliptic curves which we prove using a theorem due to howe

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

undirected edge geography abstract the game of edge geography is played by two players who alternately move a token on a graph from one vertex to an adjacent vertex erasing the edge in between the player who first has no legal move is the loser we show that the decision problem of determining whether a position in this game is a win for the first player is pspace complete further the problem remains pspace complete when restricted to planar graphs with maximum degree 3 however if the underlying graph is bipartite we provide 1 a linear algebraic characterization of the p and n positions yielding 2 a polynomial time algorithm for deciding whether any given position is p or n and also 3 a polynomial time algorithm to find winning moves

looking at mean payoff and total payoff through windows we consider two player games played on weighted directed graphs with mean payoff and total payoff objectives two classical quantitative objectives while for single dimensional games the complexity and memory bounds for both objectives coincide we show that in contrast to multi dimensional mean payoff games that are known to be conp complete multi dimensional total payoff games are undecidable we introduce conservative approximations of these objectives where the payoff is considered over a local finite window sliding along a play instead of the whole play for single dimension we show that i if the window size is polynomial deciding the winner takes polynomial time and ii the existence of a bounded window can be decided in np conp and is at least as hard as solving mean payoff games for multiple dimensions we show that i the problem with fixed window size is exptime complete and ii there is no primitive recursive algorithm to decide the existence of a bounded window

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

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

optimal control synthesis of timed event graphs with interval model specifications the purpose of this technical note is the optimal control synthesis of a timed event graph when the state and control trajectories should follow the specifications defined by an interval model the problem is reformulated in the fixed point form and the spectral theory gives the conditions of existence of a solution

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

a unified coordinate system for solving the three dimensional euler equations it is well known that the use of eulerian coordinates for shock capturing methods results in badly smeared slip lines and that lagrangian coordinates while capable of producing sharp slip line resolution may result in severe grid deformation causing inaccuracy and even breakdown of computation a unified coordinate system is introduced in which the flow variables are considered to be functions of time and of some permanent identification of pseudo particles which move with velocity hq q being the velocity of fluid particles it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 for two dimensional inviscid flow the free function h is chosen so as to preserve the grid angles this results in a coordinate system which avoids excessive numerical diffusion across slip lines in the eulerian coordinates and avoids severe grid deformation in the lagrangian coordinates yet it retains sharp resolution of slip lines especially for steady flow furthermore the two dimensional unsteady euler equations of gasdynamics in the unified coordinates are found to be hyperbolic for all values of h except when h 1 i e lagrangian in the latter case the euler equations are only weakly hyperbolic lacking one eigenvector although all eigenvalues are real the consequences of this deficiency of the lagrangian coordinates are pointed out in connection with numerical computation

recursive markov decision processes and recursive stochastic games we introduce recursive markov decision processes rmdps and recursive simple stochastic games rssgs which are classes of finitely presented countable state mdps and zero sum turn based perfect information stochastic games they extend standard finite state mdps and stochastic games with a recursion feature we study the decidability and computational complexity of these games under termination objectives for the two players one player s goal is to maximize the probability of termination at a given exit while the other player s goal is to minimize this probability in the quantitative termination problems given an rmdp or rssg and probability p we wish to decide whether the value of such a termination game is at least p or at most p in the qualitative termination problem we wish to decide whether the value is 1 the important 1 exit subclasses of these models 1 rmdps and 1 rssgs correspond in a precise sense to controlled and game versions of classic stochastic models including multitype branching processes and stochastic context free grammars where the objective of the players is to maximize or minimize the probability of termination extinction we provide a number of upper and lower bounds for qualitative and quantitative termination problems for rmdps and rssgs we show both problems are undecidable for multi exit rmdps but are decidable for 1 rmdps and 1 rssgs specifically the quantitative termination problem is decidable in pspace for both 1 rmdps and 1 rssgs and is at least as hard as the square root sum problem a well known open problem in numerical computation we show that the qualitative termination problem for 1 rmdps i e a controlled version of branching processes can be solved in polynomial time both for maximizing and minimizing 1 rmdps the qualitative problem for 1 rssgs is in np conp and is at least as hard as the quantitative termination problem for condon s finite state simple stochastic games whose complexity remains a well known open problem finally we show that even for 1 rmdps more general qualitative and quantitative model checking problems with respect to linear time temporal properties are undecidable even for a fixed property

on feedback invariance properties for systems over a principal ideal domain simple necessary and sufficient conditions for the solvability of many control problems for linear systems over a field are based on the equivalence between the a b invariance property and the a bf invariance or feedback invariance property for systems over a ring this equivalence is no longer true and many results of the geometric control theory cannot be extended we present new algorithmically checkable characterizations of the a bf invariance property for systems defined over a principal ideal domain and a new solvability condition for the disturbance rejection problem

scheduling with and or precedence constraints in many scheduling applications it is required that the processing of some job be postponed until some other job which can be chosen from a pregiven set of alternatives has been completed the traditional concept of precedence constraints fails to model such restrictions therefore the concept has been generalized to so called and or precedence constraints which can cope with this kind of requirement in the context of traditional precedence constraints feasibility transitivity and the computation of earliest start times for jobs are fundamental well studied problems the purpose of this paper is to provide efficient algorithms for these tasks for the more general model of and or precedence constraints we show that feasibility as well as many questions related to transitivity can be solved by applying essentially the same linear time algorithm in order to compute earliest start times we propose two polynomial time algorithms to cope with different classes of time distances between jobs

cyclical games with prohibitions we consider a certain combinatorial game on a digraph for two cases of the price function for one case the game in question extends the cyclical game studied in ehrenfeucht and mycielski 1979 and gurvitch karzanov and khachiyan 1988 which in its turn is a generalization of the well known problem of finding a minimum mean cycle in an edge weighted digraph we prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies

a high order streamline godunov scheme for steady supersonic flow computation abstract a space marching godunov method using control volumes consisted of streamlines and the coordinate lines in the marching direction is described for the computation of two dimensional and axisymmetric steady supersonic flows these streamline meshes are solution generated they follow the fluid particles closely and also simplify the implementation of the godunov method a second order extension to the godunov scheme using essentially nonoscillatory interpolation is also presented computations of various supersonic and hypersonic flow problems indicate that the method accurately represents smooth flow and crisply resolves all flow discontinuities such as shocks and slip lines even at high mach numbers

mean payoff games with partial observation mean payoff games are important quantitative models for open reactive systems they have been widely studied as games of perfect information in this paper we investigate the algorithmic properties of several subclasses of mean payoff games where the players have asymmetric information about the state of the game these games are in general undecidable and not determined according to the classical definition we show that such games are determined under a more general notion of winning strategy we also consider mean payoff games where the winner can be determined by the winner of a finite cycle forming game this yields several decidable classes of mean payoff games of asymmetric information that require only finite memory strategies including a generalization of perfect information games where positional strategies are sufficient we give an exponential time algorithm for determining the winner of the latter

memoryless determinacy of parity and mean payoff games a simple proof we give a simple direct and constructive proof of memoryless determinacy for parity and mean payoff games first we prove by induction that the finite duration versions of these games played until some vertex is repeated are determined and both players have memoryless winning strategies in contrast to the proof of ehrenfeucht and mycielski internat j game theory 8 1979 109 113 our proof does not refer to the infinite duration versions second we show that memoryless determinacy straightforwardly generalizes to infinite duration versions of parity and mean payoff games

determining the 2 sylow subgroup of an elliptic curve over a finite field in this paper we describe an algorithm that outputs the order and the structure including generators of the 2 sylow subgroup of an elliptic curve over a finite field to do this we do not assume any knowledge of the group order the results that lead to the design of this algorithm are of inductive type then a right choice of points allows us to reach the end within a linear number of successive halvings the algorithm works with abscissas so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field efficient methods for this computation determine the efficiency of our algorithm

optimistic optimization for model predictive control of max plus linear systems model predictive control for max plus linear discrete event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables which may be hard to solve efficiently an alternative approach is to transform the given problem into a mixed integer linear programming problem however the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon the focus of this paper is on making optimistic optimization suited to solve the given problem optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization a key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum we prove that optimistic optimization can be applied for the given problem by developing a dedicated semi metric and by proving it satisfies the necessary requirements for optimistic optimization moreover we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon hence using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large

approximating the minimum cycle mean we consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight our contributions are twofold 1 first we show that the algorithmic question is reducible to the problem of a logarithmic number of min plus matrix multiplications of nxn matrices where n is the number of vertices of the graph 2 second when the weights are nonnegative we present the first 1 e approximation algorithm for the problem and the running time of our algorithm is o n wlog 3 nw e e where o n w is the time required for the classicnxn matrix multiplication and w is the maximum value of the weights with an additional o log nw e factor in space a cycle with approximately optimal weight can be computed within the same time bound

a subexponential randomized algorithm for the simple stochastic game problem we describe a randomized algorithm for the simple stochastic game problem that requires 2o n expected operations for games with n vertices this is the first subexponential time algorithm for this problem

complexity of path forming games abstract for a number of two player games where players alternately choose the next vertex of a simple or an elementary path in a graph we consider the problem to determine whether for a given game instance there is a winning strategy for the first player we show several of these problems to be pspace complete in some special cases we obtain polynomial time algorithms based on graph rewriting or an intricate form of dynamic programming e g we show generalized geography and some other pspace complete problems to be linear time solvable on graphs with constant bounded treewidth

a note on elliptic curves over finite fields let e be an elliptic curve over a finite field k and let e k be the group of k rational points on e we evaluate all the possible groups e k where e runs through all the elliptic curves over a given fixed finite field k

the complexity of stochastic games abstract we consider the complexity of stochastic games simple games of chance played by two players we show that the problem of deciding which player has the greatest chance of winning the game is in the class np co np

qualitative analysis of concurrent mean payoff games we consider concurrent games played by two players on a finite state graph where in every round the players simultaneously choose a move and the current state along with the joint moves determine the successor state we study the most fundamental objective for concurrent games namely mean payoff or limit average objective where a reward is associated to each transition and the goal of player 1 is to maximize the long run average of the rewards and the objective of player 2 is strictly the opposite i e the games are zero sum the path constraint for player 1 could be qualitative i e the mean payoff is the maximal reward or arbitrarily close to it or quantitative i e a given threshold between the minimal and maximal reward we consider the computation of the almost sure resp positive winning sets where player 1 can ensure that the path constraint is satisfied with probability 1 resp positive probability almost sure winning with qualitative constraint exactly corresponds to the question of whether there exists a strategy to ensure that the payoff is the maximal reward of the game our main results for qualitative path constraints are as follows 1 we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost sure resp positive winning against all player 2 strategies or player 2 has a spoiling strategy to falsify almost sure resp positive winning against all player 1 strategies 2 we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost sure and positive winning for both players and 3 we present quadratic time algorithms to compute the almost sure and the positive winning sets matching the best known bound of the algorithms for much simpler problems such as reachability objectives for quantitative constraints we show that a polynomial time solution for the almost sure or the positive winning set would imply a solution to a long standing open problem of solving the value problem of turn based deterministic mean payoff games that is not known to be solvable in polynomial time

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

an extended lagrangian method a unique formulation of describing fluid motion is presented the method referred to as extended lagrangian method is interesting from both theoretical and numerical points of view the formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the eulerian description the present method and the arbitrary lagrangian eulerian ale method have a similarity in spirit eliminating the cross streamline numerical diffusion for this purpose we suggest a simple grid constraint condition and utilize an accurate discretization procedure this grid constraint is only applied to the transverse cell face parallel to the local stream velocity and hence our method for the steady state problems naturally reduces to the streamline curvature method without explicitly solving the steady streamline coordinate equations formulated a priori unlike the lagrangian method proposed by loh and hui which is valid only for steady supersonic flows the present method is general and capable of treating subsonic flows and supersonic flows as well as unsteady flows simply by invoking in the same code an appropriate grid constraint suggested in this paper the approach is found to be robust and stable it automatically adapts to flow features without resorting to clustering thereby maintaining rather uniform grid spacing throughout and large time step moreover the method is shown to resolve multi dimensional discontinuities with a high level of accuracy similar to that found in one dimensional problems

max plus a b invariant spaces and control of timed discrete event systems the concept of a b invariant subspace or controlled invariant of a linear dynamical system is extended to linear systems over the max plus semiring although this extension presents several difficulties which are similar to those encountered in the same kind of extension to linear dynamical systems over rings it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings sufficient conditions are given for computing the maximal a b invariant subspace contained in a given space and the existence of linear state feedbacks is discussed an application to the study of transportation networks which evolve according to a timetable is considered

a solution to the problem of a b invariance for series the aim of this article is to study a standard problem of control theory the a b invariance problem which amounts to computing a maximal element x subject to conditions of the form ax 6 x b and x 6 k we give a solution to the problem in the framework of formal series over particular complete idempotent semirings over nite idempotent semirings we show that under the assumption that b and k are recognizable series the maximal solution exists and is also recognizable we obtain a similar result for the in nite tropical semiring with additional hypothesis that the series a is a language but the notion of recognizable series has to be extended to the weaker notion of pseudo recognizable series c 2002 elsevier science b v all rights reserved

concurrent reachability games we consider concurrent two player games with reachability objectives in such games at each round player 1 and player 2 independently and simultaneously choose moves and the two choices determine the next state of the game the objective of player 1 is to reach a set of target states the objective of player 2 is to prevent this these are zero sum games and the reachability objective is one of the most basic objectives determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification there are three types of winning states according to the degree of certainty with which player 1 can reach the target from type 1 states player 1 has a deterministic strategy to always reach the target from type 2 states player 1 has a randomized strategy to reach the target with probability 1 from type 3 states player 1 has for every real e 0 a randomized strategy to reach the target with probability greater than email protected we show that for finite state spaces all three sets of winning states can be computed in polynomial time type 1 states in linear time and type 2 and type 3 states in quadratic time the algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies

interval analysis and dioid application to robust controller design for timed event graphs this paper deals with feedback controller synthesis for timed event graphs where the number of initial tokens and time delays are only known to belong to intervals we discuss here the existence and the computation of a robust controller set for uncertain systems that can be described by parametric models the unknown parameters of which are assumed to vary between known bounds each controller is computed in order to guarantee that the closed loop system behavior is greater than the lower bound of a reference model set and is lower than the upper bound of this set the synthesis presented here is mainly based on dioid interval analysis and residuation theory

an algorithm to describe the solution set of any tropical linear system a x b x abstract an algorithm to give an explicit description of all the solutions to any tropical linear system a x b x is presented the given system is converted into a finite rather small number p of pairs s t of classical linear systems a system s of equations and a system t of inequalities the notion introduced here that makes p small is called compatibility the particular feature of both s and t is that each item equation or inequality is bivariate i e it involves exactly two variables one variable with coefficient 1 and the other one with 1 s is solved by gaussian elimination we explain how to solve t by a method similar to gaussian elimination to achieve this we introduce the notion of sub special matrix the procedure applied to t is therefore called sub specialization

a new lagrangian method for steady supersonic flow computation i godunov scheme abstract this paper studies the problem of steady two dimensional supersonic flow of an inviscid compressible fluid using the new lagrangian formulation of hui and van roessel in which the stream function and the lagrangian time are used as independent variables a shock capturing method is developed by applying the first order godunov scheme to the conservation form equations of this formulation the method is fast and robust furthermore extensive comparisons with exact solutions and with the second order godunov scheme of glaz and wardlaw based on the eulerian formulation show that the first order lagrangian method generally attains the same level of accuracy as the second order eulerian method and is even better in resolving slip line discontinuities

feedback control for a class of discrete event systems with critical time this paper presents an algebraic approach for control laws synthesis of timed event graphs subjected to strict temporal constraints this class of discrete event systems is deterministic in the sense that its behaviour only depends on the initial marking and on the control that is applied this behaviour can be modelled by a linear equations system in min plus algebra the temporal constraint is represented by an inequality that is also linear in the min plus algebra then a method for the synthesis of control laws ensuring the respect of constraints is described we give explicit formulas characterising a control law which ensures the validity of the temporal constraints it is a causal feedback control involving delays the method is illustrated on an example

observer design for max linear systems this technical note deals with the state estimation for max plus linear systems this estimation is carried out following the ideas of the observer method for classical linear systems the system matrices are assumed to be known and the observation of the input and of the output is used to compute the estimated state the observer design is based on the residuation theory which is suitable to deal with linear mapping inversion in idempotent semiring

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

the equation a x b y over max for the two sided homogeneous linear equation system a x b y over max with no infinite rows or columns in a or b an algorithm is presented which converges to a finite solution from any finite starting point whenever a finite solution exists if the finite elements of a b are all integers convergence is in a finite number of steps for which a precise bound can be calculated if moreover one of a b has only finite elements the algorithm is thus pseudopolynomial in complexity

quantitative fair simulation games simulation is an attractive alternative for language inclusion for automata as it is an under approximation of language inclusion but usually has much lower complexity for non deterministic automata while language inclusion is pspace complete simulation can be computed in polynomial time simulation has also been extended in two orthogonal directions namely 1 fair simulation for simulation over specified set of infinite runs and 2 quantitative simulation for simulation between weighted automata again while fair trace inclusion is pspace complete fair simulation can be computed in polynomial time for weighted automata the quantitative language inclusion problem is undecidable for mean payoff automata and the decidability is open for discounted sum automata whereas the quantitative simulation reduce to mean payoff games and discounted sum games which admit pseudo polynomial time algorithms in this work we study quantitative simulation for weighted automata with buchi acceptance conditions i e we generalize fair simulation from non weighted automata to weighted automata we show that imposing buchi acceptance conditions on weighted automata changes many fundamental properties of the simulation games for example whereas for mean payoff and discounted sum games the players do not need memory to play optimally we show in contrast that for simulation games with buchi acceptance conditions i for mean payoff objectives optimal strategies for both players require infinite memory in general and ii for discounted sum objectives optimal strategies need not exist for both players while the simulation games with buchi acceptance conditions are more complicated e g due to infinite memory requirements for mean payoff objectives as compared to their counterpart without buchi acceptance conditions we still present pseudo polynomial time algorithms to solve simulation games with buchi acceptance conditions for both weighted mean payoff and weighted discounted sum automata

duality and interval analysis over idempotent semirings abstract in this paper semirings with an idempotent addition are considered these algebraic structures are endowed with a partial order this allows to consider residuated maps to solve systems of inequalities a x b see 3 the purpose of this paper is to consider a dual product denoted and the dual residuation of matrices in order to solve the following inequality a x x b x sufficient conditions ensuring the existence of a non linear projector in the solution set are proposed the results are extended to semirings of intervals such as they were introduced in 25

energy parity games energy parity games are infinite two player turn based games played on weighted graphs the objective of the game combines a qualitative parity condition with the quantitative requirement that the sum of the weights i e the level of energy in the game must remain positive beside their own interest in the design and synthesis of resource constrained omega regular specifications energy parity games provide one of the simplest model of games with combined qualitative and quantitative objectives our main results are as follows a exponential memory is sufficient and may be necessary for winning strategies in energy parity games b the problem of deciding the winner in energy parity games can be solved in np conp and c we give an algorithm to solve energy parity by reduction to energy games we also show that the problem of deciding the winner in energy parity games is logspace equivalent to the problem of deciding the winner in mean payoff parity games which can thus be solved in np conp as a consequence we also obtain a conceptually simple algorithm to solve mean payoff parity games

hyperbolicity and optimal coordinates for the three dimensional supersonic euler equations the two dimensional 2 d steady euler equations for supersonic flow in cartesian coordinates x y is hyperbolic except on the singular line u a where a is the speed of sound and u is the x component of velocity however the cauchy problem for x x circ with initial data prescribed at x x circ is well posed only when u a for the 2 d case the optimal coordinates have been found w h hui and d l chu comput fluid dynamics 4 1996 pp 403 426 to be the orthogonal system consisting of streamlines and their orthogonal lines this gives the most robust and accurate computation for supersonic flow it is commonly thought e g c y loh and m s liou j comput phys 113 1994 pp 224 248 that the system of three dimensional 3 d steady euler equations for supersonic flow is hyperbolic the present paper shows that it is hyperbolic only when the velocity in the marching direction is supersonic in which case it also simultaneously guarantees the well posedness of the cau

observer based controllers for max plus linear systems max plus algebra is a suitable algebraic setting to model discrete event systems involving synchronization and delay phenomena which are often found in transportation networks communications systems and manufacturing systems one way of controlling this kind of systems consists in choosing the dates of input events in order to achieve the desired performances e g to obtain output events in order to respect given dates this kind of control is optimal according to a just in time criterion if the input event dates are delayed as much as possible while ensuring the output events to occur before a desired reference date this paper presents the observer based controller for max plus linear systems where only estimations of system states are available for the controller as in the classical sense this is a state feedback control problem which is solved in two steps first an observer computes an estimation of the state by using the input and the output measurements then this estimated state is used to compute the state feedback control action as a main result it is shown that the optimal solution of this observer based control problem leads to a greater control input than the one obtained with the output feedback strategy a high throughput screening example in drug discovery illustrates this main result by showing that the scheduling obtained from the observer based controller is better than the scheduling obtained from the output feedback controller

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

on the steady state control of timed event graphs with firing date constraints two algorithms for solving a specific class of steady state control problems for timed event graphs are presented in the first asymptotic convergence to the desired set is guaranteed the second algorithm which builds on from the recent developments in the spectral theory of min max functions guarantees lyapunov stability since the distance between the actual state and the desired set never increases simulation results show the efficiency of the proposed approach in a problem of moderate complexity

computing the l power torsion of an elliptic curve over a finite field the algorithm we develop outputs the order and the structure including generators of the l sylow subgroup of the group of rational points of an elliptic curve defined over a finite field to do this we do not assume any knowledge of the group order we are able to choose points in such a way that a linear number of successive l divisions leads to generators of the subgroup under consideration after the computation of a couple of polynomials each division step relies on finding rational roots of polynomials of degree l we specify in complete detail the case l 3 when the complexity of each trisection is given by the computation of cubic roots in finite fields

mean payoff games and propositional proofs we associate a cnf formula to every instance of the mean payoff game problem in such a way that if the value of the game is non negative the formula is satisfiable and if the value of the game is negative the formula has a polynomial size refutation in s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 frege i e resolution is solvable in polynomial time our result is close to optimal up to the computational complexity of solving mean payoff games the proof of the main result requires building low depth formulas that compute the bits of the sum of a constant number of integers in binary notation and low complexity proofs of the required arithmetic properties

randomness for free we consider two player zero sum games on finite state graphs these games can be classified on the basis of the information of the players and on the mode of interaction between them on the basis of information the classification is as follows a partial observation both players have partial view of the game b one sided complete observation one player has complete observation and c complete observation both players have complete view of the game on the basis of mode of interaction we have the following classification a concurrent players interact simultaneously and b turn based players interact in turn the two sources of randomness in these games are randomness in the transition function and randomness in the strategies in general randomized strategies are more powerful than deterministic strategies and probabilistic transitions give more general classes of games we present a complete characterization for the classes of games where randomness is not helpful in a the transition function probabilistic transitions can be simulated by deterministic transitions and b strategies pure strategies are as powerful as randomized strategies as a consequence of our characterization we obtain new undecidability results for these games

a higher order godunov method for the hyperbolic equations modelling shock dynamics a numerical method of solution is developed for steady state hyperbolic equations in conservation form an upwind direction is identified and a smooth mapping is introduced to align the equations with that direction a flux f u in the upwind direction is regarded as the state variable and advanced numerically on a grid according to a second order extension of godunov method the solution u is determined from the computed flux and a constraint on the mapping is that the jacobian det f u is of one sign the method is applied to the equations modelling shock wave propagation according to whitham theory of geometrical shock dynamics in this context the solution vector u gives the components of the gradient of a function whose level curves determine the shock positions at successive times shock propagation in both uniform and non uniform gases is considered and several example calculations are presented a further application of the numerical method is inviscid supersonic flow as governed by the steady euler equations the steady supersonic flow in a converging diverging duct is computed to illustrate the numerical method for this latter application

contingent planning under uncertainty via stochastic satisfiability we describe a new planning technique that efficiently solves probabilistic propositional contingent planning problems by converting them into instances of stochastic satisfiability ssat and solving these problems instead we make fundamental contributions in two areas the solution of ssat problems and the solution of stochastic planning problems this is the first work extending the planning as satisfiability paradigm to stochastic domains our planner zander can solve arbitrary goal oriented finite horizon partially observable markov decision processes pomdps an empirical study comparing zander to seven other leading planners shows that its performance is competitive on a range of problems

the disturbance decoupling problem for systems over a ring up to now the use of geometric methods in the study of disturbance decoupling problems ddps for systems over a ring has provided only necessary conditions for the existence of solutions in this paper we study such problems considering separately the case in which only static feedback solutions are allowed and the one in which dynamic feedback solutions are admitted in the first case we give a complete geometric characterization of the solvability conditions of such problems for injective systems with coefficients in a commutative ring practical procedures for testing the solvability conditions and for constructing solutions if any exist are given in the case of systems with coefficients in a principal ideal domain pid in the second case we give a complete geometric characterization of the solvability conditions for systems with coefficients in a pid

paper a new approach to the analysis of discrete event dynamic systems we present a new time domain approach to the study of discrete event dynamical systems deds typified by queueing networks and production systems a general state space representation is developed and perturbation analysis is carried out observation of a single sample realization of such a system can be used to predict behavior over other sample realizations when some parameter is perturbed without having to make additional observations conditions under which this is always possible are investigated and explicit results for some special cases are included

new scaling algorithms for the assignment and minimum mean cycle problems in this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems our assignment algorithm is based on applying scaling to a hybrid version of the recentauction algorithm of bertsekas and the successive shortest path algorithm the algorithm proceeds by relaxing the optimality conditions and the amount of relaxation is successively reduced to zero on a network with 2n nodes m arcs and integer arc costs bounded byc the algorithm runs in o sqrt n m log nc time and uses very simple data structures this time bound is comparable to the time taken by gabow and tarjan s scaling algorithm and is better than all other time bounds under thesimilarity assumption i e c o nk for somek we next consider the minimum mean cycle problem themean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains theminimum mean cycle problem is to identify a cycle whose mean cost is minimum we show that by using ideas of the assignment algorithm in an approximate binary search procedure the minimum mean cycle problem can also be solved in o sqrt n m lognc time under the similarity assumption this is the best available time bound to solve the minimum mean cycle problem

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

a new lagrangian method for three dimensional steady supersonic flows in this paper the new lagrangian method introduced by loh and hui is extended for 3d steady supersonic flow computation we present the derivation of the conservation form and the solution of the local riemann solver using the godunov and the high resolution tvd schemes the new approach is accurate and robust capable of handling complicated geometry and interactions between discontinuous waves as shown in the test problems the current lagrangian method retains all the advantages claimed in the 2d method e g crisp resolution of a slip surface contact discontinuity and automatic grid generation in this paper we also suggest a novel 3d riemann problem in which interesting and intricate flow features are present

mean cost cyclical games we study the mean cost cyclical game in a more general setting than that in gurvitch et al 1988 and karzanow and lebedev 1993 the game is played on a directed graph and generalizes the single source shortest path problem the minimum mean cycle problem see karp 1978 and the ergodic extension of matrix games moulin 1976 we prove the existence of a solution in uniform stationary strategies and present an algorithm for finding such optimal strategies in fact our algorithm is an extension of the algorithms due to gurvitch et al 1988 and karzanow and lebedev 1993 which were proved to be finite but exponential in the worst case we prove that all these algorithms are pseudopolynomial

a unified coordinates approach to computational fluid dynamics a unified coordinate system is introduced for computational fluid dynamics in which the grid moves with velocity hq q being fluid velocity it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 by suitably choosing the free function h h 1 for 1d flow h chosen to preserve grid angles for 2d flow and to preserve grid skewness for 3d flow the unified coordinate system is shown in a large number of examples involving the euler equations to be superior to both eulerian and lagrangian ones in resolving flow discontinuities shocks and especially slip lines this approach is also successfully extended to shallow water waves and viscous flow

on the dimension of max min convex sets we introduce a notion of dimension of max min convex sets following the approach of tropical convexity we introduce a max min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope we describe the relation between this rank and the notion of strong regularity in max min algebra which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property

an optimal on line algorithm for metrical task system in practice almost all dynamic systems require decisions to be made on line without full knowledge of their future impact on the system a general model for the processing of sequences of tasks is introduced and a general on line decision algorithm is developed it is shown that for an important class of special cases this algorithm is optimal among all on line algorithms specifically a task system s d for processing sequences of tasks consists of a set s of states and a cost matrix d where d i j is the cost of changing from state i to state j we assume that d satisfies the triangle inequality and all diagonal entries are 0 the cost of processing a given task depends on the state of the system a schedule for a sequence t 1 t 2 t k of tasks is a sequence s 1 s 2 s k of states where s i is the state in which t i is processed the cost of a schedule is the sum of all task processing costs and the state transition costs incurred an on line scheduling algorithm is one that chooses s i only knowing t 1 t 2 t i such an algorithm is w competitive if on any input task sequence its cost is within an additive constant of w times the optimal offline schedule cost the competitive ratio w s d is the infimum w for which there is a w competitive on line scheduling algorithm for s d it is shown that w s d 2 s 1 for every task system in which d is symmetric and w s d o s 2 for every task system finally randomized on line scheduling algorithms are introduced it is shown that for the uniform task system in which d i j 1 for all i j the expected competitive ratio w s d o log s

max plus control design for temporal constraints meeting in timed event graphs the aim of the presented work is the control of timed event graph to meet tight temporal constraints the problem of temporal constraints is formulated in terms of control of linear max plus models first the synthesis of a control law that ensures the satisfaction of a single constraint for a single input system is presented then the single input multi constraints problem is tackled and finally the method is extended to the multi inputs multi constraints problem the proposed method is illustrated on the example of a simple production process

on max plus linear dynamical system theory a class of timed discrete event systems can be modeled by using timed event graphs a class of timed petri nets that can have its firing dynamic described by using an algebra called max plus algebra for this kind of systems it may be desirable to enforce some timing constraints in steady state in this paper this problem is called a max plus regulation problem in this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems the obtained controller is a simple linear static state feedback and can be computed using efficient pseudo polynomial algorithms simulation results will illustrate the applicability of the proposed methodology

an integrated control strategy to solve the disturbance decoupling problem for max plus linear systems with applications to a high throughput screening system this paper presents the new investigations on the disturbance decoupling problem ddp for the geometric control of max plus linear systems the classical ddp concept in the geometric control theory means that the controlled outputs will not be changed by any disturbances in practical manufacturing systems solving for the ddp would require further delays on the output parts than the existing delays caused by the system breakdown the new proposed modified disturbance decoupling problem mddp in this paper ensures that the controlled output signals will not be delayed more than the existing delays caused by the disturbances in order to achieve the just in time optimal control furthermore this paper presents the integration of output feedback and open loop control strategies to solve for the mddp as well as for the ddp if these controls can only solve for the mddp but not for the ddp an evaluation principle is established to compare the distance between two output signals generated by controls solving for the mddp and ddp respectively this distance can be interpreted as the number of tokens or firings that are needed in order for the controls to solve for the ddp moreover another alternative approach is finding a new disturbance mapping in order to guarantee the solvability of the ddp by the same optimal control for the mddp the main results of this paper are illustrated by using a timed event graph model of a high throughput screening system in drug discovery

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

a pseudo polynomial algorithm for mean payoff stochastic games with perfect information and few random positions abstract we consider two person zero sum stochastic mean payoff games with perfect information or bwr games given by a digraph g v e with local rewards r e z and three types of positions black v b white v w and random v r forming a partition of v it is a long standing open question whether a polynomial time algorithm for bwr games exists or not even when v r 0 in fact a pseudo polynomial algorithm for bwr games would already imply their polynomial solvability in this paper 1 we show that bwr games with a constant number of random positions can be solved in pseudo polynomial time more precisely in any bwr game with v r o 1 a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in v w v b the maximum absolute local reward and the common denominator of the transition probabilities