Max-plus Semiring Research Articles

Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton

Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illustrate the hierarchy of max-plus series with respect to ambiguity.

Duality and separation theorems in idempotent semimodules

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.

From max-plus algebra to nonexpansive mappings: a nonlinear theory for discrete event systems

Discrete event systems provide a useful abstraction for modelling a wide variety of systems: digital circuits, communication networks, manufacturing plants, etc. Their dynamics—stability, equilibrium states, cyclical behaviour, asymptotic average delays—are of vital importance to system designers. However, in marked contrast to continuous dynamical systems, there has been little systematic mathematical theory that designers can draw upon. In this paper, we survey the development of such a theory, based on the dynamics of maps which are nonexpansive in the ℓ ∞ norm. This has its origins in linear algebra over the max-plus semiring but extends to a nonlinear theory that encompasses a variety of problems arising in other mathematical disciplines. We concentrate on the mathematical aspects and set out several open problems.

Le problème de la réalisation minimale dans le demi-anneau max-plus et le problème de Pisot sont NP-durs

We prove the NP-hardness of two problems. The first is the well-known minimal realization problem in the max-plus semiring. The second problem (Pisot's problem) is the problem of determining if a given integer linear recurrent sequence has a zero coefficient.

Duality of Idempotent Semimodules

We establish general duality results for complete semimodules over idempotent semirings, like the max-plus semiring.

Numerical Computation of Spectral Elements in Max-Plus Algebra

We describe the specialization to max-plus algebra of Howard's policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the max-plus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time.