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13
Tropical matrix duality and Green’s D relation
- Journal of the London Mathematical Society
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Rational semimodules over the max-plus semiring and geometric . . .
, 2002
"... We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring. We say that a subsemimodule of the free semimodule S n over a semiring S is rational if it has a generating family that is a rational subset of S n, S n being thought of as a monoid under the ..."
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Cited by 9 (3 self)
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We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring. We say that a subsemimodule of the free semimodule S n over a semiring S is rational if it has a generating family that is a rational subset of S n, S n being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules are stable under the natural algebraic operations (union, product, direct and inverse image, intersection, projection, etc). Rational semimodules are a tool to extend the geometric approach of linear control to discrete event systems. In particular, we show that the reachable and observable spaces of max-plus linear dynamical systems
Multiplicative structure of 2 × 2 tropical matrices
, 907
"... We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup. 1 ..."
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Cited by 4 (4 self)
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We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup. 1
SEMIGROUP IDENTITIES IN THE MONOID OF TWO-BY-TWO TROPICAL MATRICES
, 2009
"... We show that the monoid M2 ( ) of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity ..."
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Cited by 3 (0 self)
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We show that the monoid M2 ( ) of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity
Backward Reachability of Autonomous Max-Plus-Linear Systems ⋆
"... Abstract: This work discusses the backward reachability of autonomous Max-Plus-Linear (MPL) systems, a class of continuous-space discrete-event models that are relevant for applications dealing with synchronization and scheduling. Given an MPL system and a continuous set of final states, we charact ..."
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Abstract: This work discusses the backward reachability of autonomous Max-Plus-Linear (MPL) systems, a class of continuous-space discrete-event models that are relevant for applications dealing with synchronization and scheduling. Given an MPL system and a continuous set of final states, we characterize and compute its "backward reach tube" and "backward reach sets," namely the set of states that can reach the final set within a given event interval or at a fixed event step, respectively. We show that, in both cases, the computation can be done exactly via manipulations of difference-bound matrices. Furthermore, we illustrate the application of the backward reachability computations over safety and transient analysis of MPL systems.
Computational Techniques for Reachability Analysis of Max-Plus-Linear Systems ⋆
"... Abstract This work discusses a computational approach to reachability analysis of Max-Plus-Linear (MPL) systems, a class of discreteevent systems widely used in synchronization and scheduling applications. Given a set of initial states, we characterize and compute its "reach tube," namely ..."
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Abstract This work discusses a computational approach to reachability analysis of Max-Plus-Linear (MPL) systems, a class of discreteevent systems widely used in synchronization and scheduling applications. Given a set of initial states, we characterize and compute its "reach tube," namely the collection of set of reachable states (regarded step-wise as "reach sets"). By an alternative characterization of the MPL dynamics, we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference-bound matrices, and further derive worst-case bounds on the complexity of these operations. The approach is also extended to backward reachability analysis. The concepts and results are elucidated by a running example, and we further illustrate the performance of the approach by a numerical benchmark: the technique comfortably handles twenty-dimensional MPL systems (i.e., with twenty continuous state variables), and as such it outperforms the state-of-the-art alternative approaches in the literature.
Improved Matrix Pair . . .
, 2006
"... We improve undecidability bounds for problems involving two integer matrices. We prove ..."
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We improve undecidability bounds for problems involving two integer matrices. We prove
unknown title
, 2006
"... Algèbres max-plus et mathématiques de la décision/Max-plus algebras and mathematics of decision ..."
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Algèbres max-plus et mathématiques de la décision/Max-plus algebras and mathematics of decision
