Reachability problems for products of matrices in semirings

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 $r\geq 2$ 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 max-plus semiring $(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are ``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.Comment: 21 page

Heap Models, Composition and Control

International audienc

A multiform time approach to real-time system modeling: Application to an automotive system

The original publication is available at ieee.org ({http://dx.doi.org/10.1109/SIES.2007.4297340)International audienceIn the context of an effort to answer the OMG RFP for Modeling and Analysis of Real-Time Embedded systems (MARTE), we are defining extensions to the simple time model of UML2. After a brief review of some time-related UML profiles, we focus on the specificity of our approach: the ability to take account of multiform time-a concept inherited from reactive system modeling. Using an example from the automotive industry, we illustrate the use of our profile to represent, to constraint and to analyze behaviors depending on multiform time