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

Multiplicative structure of 2x2 tropical matrices

We study the algebraic structure of the semigroup of all $2 \times 2$ 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.Comment: 21 pages, 5 figure

Rational semimodules over the max-plus semiring and geometric approach of discrete event systems

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. 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). We show that the reachable and observable spaces of max-plus linear dynamical systems are rational, and give various examples.Comment: 24 pages, 9 postscript figures; example in section 4.3 expande

On undecidability bounds for matrix decision problems

In this paper we consider several reachability problems such as vector reachability, membership in matrix semigroups and reachability problems in piecewise linear maps. Since all of these questions are undecidable in general, we work on lowering the bounds for undecidability. In particular, we show an elementary proof of undecidability of the reachability problem for a set of 5 two-dimensional affine transformations. Then, using a modified version of a standard technique, we also prove that the vector reachability problem is undecidable for two (rational) matrices in dimension 11. The above result can be used to show that the system of piecewise linear functions of dimension 12 with only two intervals has an undecidable set-to-point reachability problem. We also show that the “zero in the upper right corner” problem is undecidable for two integral matrices of dimension 18 lowering the bound from 23

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 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

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

The paper is devoted to several infinite-state Attacker–Defender games with reachability objectives. We prove the undecidability of checking for the existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games. To prove these results, we consider a model of weighted automata operating on infinite words and prove that the universality problem is undecidable for this new class of weighted automata. We show that the universality problem is undecidable by using a non-standard encoding of the infinite Post correspondence problem