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23
Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 250 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
Methods and Applications of (max,+) Linear Algebra
 STACS'97, NUMBER 1200 IN LNCS, LUBECK
, 1997
"... Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; g ..."
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Cited by 73 (26 self)
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Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, HamiltonJacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities) . Despite this apparent profusion, there is a small set of common, nonnaive, basic results and problems, in general not known outside the (max, +) community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of (max, +) results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of...
The duality theorem for minmax functions
 C. R. Acad.Sci.Paris.326, Série I
, 1998
"... Abstract. The set of minmax functions F: R n → R n is the least set containing coordinate substitutions and translations and closed under pointwise max, min, and function composition. The Duality Conjecture asserts that the trajectories of a minmax function, considered as a dynamical system, have ..."
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Cited by 35 (12 self)
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Abstract. The set of minmax functions F: R n → R n is the least set containing coordinate substitutions and translations and closed under pointwise max, min, and function composition. The Duality Conjecture asserts that the trajectories of a minmax function, considered as a dynamical system, have a linear growth rate (cycle time) and shows how this can be calculated through a representation of F as an infimum of maxplus linear functions. We prove the conjecture using an analogue of Howard’s policy improvement scheme, carried out in a lattice ordered group of germs of affine functions at infinity. The methods yield an efficient algorithm for computing cycle times. LE THÉORÈME DE DUALITÉ POUR LES FONCTIONS MINMAX Résumé. L’ensemble des fonctions minmax F: R n → R n est le plus petit ensemble de fonctions qui contient les substitutions de coordonnées et les translations, et qui est stable par les opérations min et max (point par point), ainsi que par composition. La Conjecture de Dualité affirme que les trajectoires d’un système récurrent gouverné par une dynamique minmax ont un taux de croissance linéaire (temps de cycle), qui se calcule à partir d’une représentation de F comme infimum de fonctions maxplus linéaires. Nous montrons cette conjecture en utilisant une itération sur les politiques à la Howard, à valeurs dans un groupe réticulé de germes de fonctions affines à l’infini. On a ainsi un
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.
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.
Linear Projector in the maxplus Algebra
 IEEE MEDITERRANEEN CONFERENCE ON CONTROL, CHYPRE
, 1997
"... In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +) ..."
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Cited by 14 (11 self)
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In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +)semiring, we give necessary and sufficient conditions for an image and a kernel to be direct factors. We characterize the semimodules that admit a direct factor (or equivalently, the semimodules that are the image of a linear projector): their matrices have a ginverse. We give simple effective tests for all these properties, in terms of matrix residuation.
Minplus methods in eigenvalue perturbation theory and generalised LidskiiVishikLjusternik theorem
, 2005
"... Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturba ..."
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Cited by 13 (1 self)
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Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and minplus analogues of eigenvalues. 1.
Resource Optimization and (min,+) Spectral Theory
 IEEE Trans. on Automat. Contr
, 1995
"... We show that certain resource optimization problems relative to Timed Event Graphs reduce to linear programs. The auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the (min,+) algebra. KeywordsResource Optimization, Timed Event Graphs, (max,+) algebra, ..."
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Cited by 9 (2 self)
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We show that certain resource optimization problems relative to Timed Event Graphs reduce to linear programs. The auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the (min,+) algebra. KeywordsResource Optimization, Timed Event Graphs, (max,+) algebra, spectral theory. I. INTRODUCTION Timed Event Graphs (TEGs) are a subclass of timed Petri nets which can be used to model deterministic discrete event dynamic systems subject to saturation and synchronization phenomena: typically, flexible manufacturing systems, multiprocessor systems, transportation networks [5], [1], [3], [2], [16], [17]. The most remarkable result about TEGs [4], [3], [1] is certainly the following: a TEG functioning at maximal speed reaches after a finite time a periodic regime. More precisely, let x denote the counter function of a given transition of the graph. That is, x(t) represents the number of firings of the transition up to time t, usually the number of parts...
Reachability problems for products of matrices in semirings
, 2002
"... Abstract. We consider the following matrix reachability problem: given r square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? We define similarly the vector (resp. scalar) reachability problem, by requiring that the matrix product, actin ..."
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Cited by 7 (1 self)
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Abstract. We consider the following matrix reachability problem: given r square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? We define similarly the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied at left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any r ≥ 2 is equivalent to the specialization to r = 2. As an application of this result and of a theorem of Krob, we show that when r = 2, the vector and matrix reachability problems are undecidable over the maxplus semiring (Z ∪ {−∞},max,+). We also show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are “positive”, like the tropical semiring (N ∪ {+∞}, min, +).