Skip to Main content Skip to Navigation
Reports

Reachability problems for products of matrices in semirings

Abstract : 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 >= 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 U {-infinity},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 U {+infinity},min,+).
Document type :
Reports
Complete list of metadatas

https://hal.inria.fr/inria-00071635
Contributor : Rapport de Recherche Inria <>
Submitted on : Tuesday, May 23, 2006 - 6:22:23 PM
Last modification on : Friday, May 25, 2018 - 12:02:03 PM
Long-term archiving on: : Sunday, April 4, 2010 - 10:30:22 PM

Identifiers

  • HAL Id : inria-00071635, version 1

Collections

Citation

Stéphane Gaubert, Ricardo David Katz. Reachability problems for products of matrices in semirings. [Research Report] RR-4944, INRIA. 2003. ⟨inria-00071635⟩

Share

Metrics

Record views

320

Files downloads

316