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17
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 252 (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 ev ..."
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Cited by 76 (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...
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 33 (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.
The Minkowski Theorem for Maxplus Convex Sets
, 2006
"... We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones a ..."
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Cited by 22 (11 self)
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We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones and closed unbounded maxplus convex sets. In particular, we show that a closed maxplus convex set can be decomposed as a maxplus sum of its recession cone and of the maxplus convex hull of its extreme points.
Algebraic System Analysis of Timed Petri Nets
, 1997
"... We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (inputoutput) properti ..."
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Cited by 15 (6 self)
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We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (inputoutput) properties, algebraic representations, asymptotic regime. A particular attention is paid to the subclass of stable systems (with asymptotic linear growth). 1 Introduction The fact that a subclass of Discrete Event Systems equations write linearly in the (min,+) or in the (max,+) semiring is now almost classical [9, 2]. The (min,+) linearity allows the presence of synchronization and saturation features but unfortunately prohibits the modeling of many interesting phenomena such as "birth" and "death" processes (multiplication of tokens) and concurrency. The purpose of this paper is to show that after some simplifications, these additional features can be represented by polynomial recurrences in the ...
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.
The number of extreme points of tropical polyhedra
, 2009
"... The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in ..."
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Cited by 8 (5 self)
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The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on the sign pattern, from which we deduce explicit bounds for the number of extreme points, showing in particular that the upper bound is asymptotically tight as the dimension tends to infinity, keeping the number of constraints fixed. When transposed to the classical case, the previous constructions yield some lattice path generalizations of Gale’s evenness criterion.
Deciding unambiguity and sequentiality from a finitely ambiguous maxplus automaton
 THEORET. COMPUT. SCI
, 2004
"... Finite automata with weights in the maxplus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous maxplus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illu ..."
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Cited by 5 (2 self)
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Finite automata with weights in the maxplus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous maxplus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illustrate the hierarchy of maxplus series with respect to ambiguity.