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An Algorithm for Exact Bounds on the Time Separation of Events in Concurrent Systems
 IEEE Transactions on Computers
, 1993
"... Determining the time separation of events is a fundamental problem in the analysis, synthesis, and optimization of concurrent systems. Applications range from logic optimization of asynchronous digital circuits to evaluation of execution times of programs for realtime systems. We present an efficie ..."
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Cited by 47 (7 self)
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Determining the time separation of events is a fundamental problem in the analysis, synthesis, and optimization of concurrent systems. Applications range from logic optimization of asynchronous digital circuits to evaluation of execution times of programs for realtime systems. We present an efficient algorithm to find exact (tight) bounds on the separation time of events in an arbitrary process graph without conditional behavior. This result is more general than the methods presented in several previously published papers as it handles cyclic graphs and yields the tightest possible bounds on event separations. The algorithm is based on a functional decomposition technique that permits the implicit evaluation of an infinitely unfolded process graph. Examples are presented that demonstrate the utility and efficiency of the solution. The algorithm will form a basis for exploration of timingconstrained synthesis techniques. Index terms: Abstract algebra, asynchronous systems, concurrent ...
Maxplus algebra and system theory: Where we are and where to go now
 Annu. Rev. Control
, 1999
"... Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison ..."
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Cited by 42 (18 self)
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Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to systemtheoretic and controlsynthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the maxplus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre maxplus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre maxplus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.
Duality and separation theorems in idempotent semimodules
 Linear Algebra and its Applications 379 (2004), 395–422. Also arXiv:math.FA/0212294
"... 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 sep ..."
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Cited by 35 (19 self)
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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 halfspaces over the maxplus semiring. 1.
Minimal Realization in the Max Algebra is an Extended Linear Complementarity Problem
 SYSTEMS & CONTROL LETTERS
, 1993
"... In this paper we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an Extended Linear Complementarity Problem (ELCP). We use an algorithm that finds all solutions of an ELCP to find all equivalent minimal state space realizations of a single inpu ..."
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Cited by 29 (26 self)
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In this paper we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an Extended Linear Complementarity Problem (ELCP). We use an algorithm that finds all solutions of an ELCP to find all equivalent minimal state space realizations of a single input single output (SISO) discrete event system. We also give a geometrical description of the set of all minimal realizations of a SISO maxlinear discrete event system.
A Method to Find All Solutions of a System of Multivariate Polynomial Equalities and Inequalities in the Max Algebra
, 1996
"... In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and ine ..."
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Cited by 17 (15 self)
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In this paper we show that finding solutions of a system of multivariate polynomial equalities and inequalities in the max algebra is equivalent to solving an Extended Linear Complementarity Problem. This allows us to find all solutions of such a system of multivariate polynomial equalities and inequalities and provides a geometrical insight in the structure of the solution set. We also demonstrate that this enables us to solve many important problems in the max algebra and the maxminplus algebra such as matrix decompositions, construction of matrices with a given characteristic polynomial, state space transformations and the (minimal) state space realization problem.
Rational Series over Dioids and Discrete Event Systems
 In Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete Event Systems, number 199 in Lect. Notes. in Control and Inf. Sci, Sophia Antipolis
, 1994
"... this paper is obviously too short for such a program, we have chosen to propose an introductive guided tour. A more detailed exposition will be found in our references and in a more complete paper to appear elsewhere. 1 Rational Series in a Single Indeterminate ..."
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Cited by 16 (6 self)
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this paper is obviously too short for such a program, we have chosen to propose an introductive guided tour. A more detailed exposition will be found in our references and in a more complete paper to appear elsewhere. 1 Rational Series in a Single Indeterminate
An introduction to idempotency
 In Idempotency [41
, 1998
"... The word idempotency signifies the study of semirings in which the addition operation is idempotent: a+ a = a. The bestknown example is the maxplus ..."
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Cited by 15 (2 self)
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The word idempotency signifies the study of semirings in which the addition operation is idempotent: a+ a = a. The bestknown example is the maxplus
Minimal (max, +) realization of convex sequences
 SIAM JOURNAL ON CONTROL AND OPTIMIZATION
, 1998
"... We show that the minimal dimension of a linear realization over the (max,+) semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps. The minimaldimensional realization of any convex realizable sequence can thus be found in ..."
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Cited by 15 (4 self)
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We show that the minimal dimension of a linear realization over the (max,+) semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps. The minimaldimensional realization of any convex realizable sequence can thus be found in linear time. The result is based on a bound in terms of minors of the Hankel matrix.
Kernels, Images And Projections In Dioids
 PROCEEDINGS OF WODES’96
, 1996
"... We consider the projection problem for linear spaces and operators over dioids such as the (max, +) semiring. We give existence and uniqueness conditions for the projection onto the image of an operator, parallel to the kernel of another one, together with an explicit formula for the projector. Th ..."
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Cited by 15 (12 self)
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We consider the projection problem for linear spaces and operators over dioids such as the (max, +) semiring. We give existence and uniqueness conditions for the projection onto the image of an operator, parallel to the kernel of another one, together with an explicit formula for the projector. The theory is not limited to linear operators: the result holds more generally for residuated operators over complete dioids. Illustrative examples are provided.