The presence of a zero in an integer linear recurrent sequence is NP-hard to decide

Linear Algebra and Its Applications

Volumn 351-352, Issue , 2002, Pages 91-98

  Blondel, Vincent D ^a   Portier, Natacha ^b  

^a Catholic University of Louvain   (Belgium)

^b UNIVERSITÉ DE LYON   (France)

Author keywords

Linear recurrent sequence; Max plus rational series; Minimal realization; Pisot's problem

Indexed keywords

LINEAR ALGEBRA;

MINIMAL REALIZATIONS; NP-HARD; PISOT'S PROBLEM; RATIONAL SERIES; RECURRENT SEQUENCES;

DISCRETE EVENT SIMULATION;

EID: 31244436100     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(01)00466-9     Document Type: Article

References (30)

1. F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat Synchronization and Linearity 1992 Wiley New York
2. J. Berstel, and C. Reutenauer Rational Series and their Languages 1988 Springer Berlin
3. V.D. Blondel, S. Gaubert, N. Portier, The set of realizations of a max-plus linear sequence is semi-polyhedral, Preprint
4. V.D. Blondel, and J.N. Tsitsiklis Complexity of stability and controllability of elementary hybrid systems Automatica 35 1988 479 489
5. L. Cerlienco, M. Mignotte, and F. Piras Suites récurrentes linéaires: propriétés algébriques et arithmétiques Enseign. Math. 33 1-2 1987 67 108
6. R.A. Cuninghame-Green, and P. Butkovič Discrete event systems: the strictly convex case Ann. Oper. Res. 57 1995
7. B. DasGupta, and E.D. Sontag A polynomial-time algorithm for an equivalence problem which arises in hybrid systems theory in: Proceedings of the 37th IEEE Conference on Decision and Control 1998 1629 1634
8. B. De Schutter, V.D. Blondel, R. de Vries, and B. De Moor On the boolean minimal realization problem in the max-plus algebra Systems Control Lett. 35 2 1998 69 78
9. B. De Schutter, and B. De Moor Minimal realization in the max algebra is an extended linear complementary problem Systems Control Lett. 25 2 1995 103 111
10. M.R. Garey, and D.S. Johnson Computers and Intractability: A Guide to the Theory of NP-completeness 1979 Freeman New York
11. S. Gaubert, Théorie des systèmes linéaires dans les dioides, Thèse de Doctorat, Ecole des Mines de Paris, 1992
12. S. Gaubert, On rational series in one variable over certain dioids, Technical Report 2162, INRIA, 1994
13. S. Gaubert, and M. Plus Methods and applications of (max,+) linear algebra R. Reischuk, M. Morvan, STACS'97 Lecture Notes in Computer Science, vol. 1200, Lübeck 1997 Springer Berlin
14. S. Gaubert, R.A. Cuninghame-Green, and P. Butkovič Minimal (max,+) realization of convex sequences SIAM J. Control Optim. 36 1 1998 137 147
15. C. Golaszewski, and P.J. Ramadge The complexity of some reachability problems for a system on a finite group Systems Control Lett. 12 1989 431 435
16. J. Gunawardena Idempotency 1998 Publications of the Newton Institute, Cambridge University Press Cambridge
17. M. Kanta, D. Krob, One-letter rational series with multiplicities in the tropical semiring: an algorithmic approach, Prépublication du LIAFA, Université Paris 7, 2 place Jussieu, 75 251 Paris Cédex 05, France, 1997
18. S.C. Kleene, Representation of events in nerve nets and finite automata, in: Automata Studies, Annals of Mathematical Studies, vol. 34, Princeton University Press, Princeton, 1956, pp. 3-41
19. V. Kolokoltsov, and V. Maslov Idempotent Analysis and Applications 1997 Kluwer Academic Publisher Dordrecht, MA
20. W. Kuich, A. Salomaa, Semirings, Automata, Languages, EATCS Monographs on Theoretical Computer Science, vol. 5, Springer, Berlin, 1986
21. S. Lang Algebra third edition 1997 Addison-Wesley Reading, MA
22. V. Maslov, and S. Samborski Idempotent Analysis Advances in Soviet Mathematics vol. 13 1992 AMS Providence, RI
23. G.-J. Olsder, and B. De Schutter The minimal realization problem in the max-plus algebra V. Blondel, E. Sontag, M. Vidyasagar, J. Willems, Open Problems in Mathematical Systems and Control Theory 1999 Springer London
24. K. Ruohonen Zeros of Z-rational functions and DOL-equivalence Theoret. Comput. Sci. 3 1976 282 292
25. A. Salomaa, and M. Soittola Automata-Theoretic Aspects of Formal Power Series 1978 Springer New York
26. M.P. Schützenberger, On the definition of a family of automata, Inform. Control 4 (1961) 245-270
27. L.J. Stockmeyer, and A.R. Meyer Word problems requiring exponential time: preliminary report Fifth Annual ACM Symposium on Theory of Computing 1973 ACM New York
28. O. Toker On the algorithmic unsolvability of some stability problems for discrete event systems Proc. IFAC World Congress 1997 353 358
29. N.K. Vereshchagin On the zeros of linear recurrence sequences Soviet Math. Dokl. 30 2 1984 502 505
30. V.D. Blondel, J.N. Tsitsiklis, A survey of computational complexity results in systems and control, Automatica 36 (9) (2000) 1249-1274