Comparison of Max-Plus Automata and Joint Spectral Radius of Tropical Matrices

Authors Laure Daviaud, Pierre Guillon, Glenn Merlet



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Laure Daviaud
Pierre Guillon
Glenn Merlet

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Laure Daviaud, Pierre Guillon, and Glenn Merlet. Comparison of Max-Plus Automata and Joint Spectral Radius of Tropical Matrices. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.19

Abstract

Weighted automata over the tropical semiring Zmax are closely related to finitely generated semigroups of matrices over Zmax. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is no algorithm that takes as input a finite set of matrices S and provides as output the joint spectral radius (resp. the ultimate rank) of S. On the other hand, we prove that the joint spectral radius is nevertheless approximable and we exhibit restricted cases in which the joint spectral radius and the ultimate rank are computable. To reach this aim, we study the problem of comparing functions computed by weighted automata over the tropical semiring. This problem is known to be undecidable, and we prove that it remains undecidable in some specific subclasses of automata.
Keywords
  • max-plus automata
  • max-plus matrices
  • weighted automata
  • tropical semiring
  • joint spectral radius
  • ultimate rank

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References

  1. Shaull Almagor, Udi Boker, and Orna Kupferman. What’s decidable about weighted automata? In ATVA 2011, pages 482-491. Springer-Verlag, oct 2011. Google Scholar
  2. Yu. A. Al'pin. Bounds for joint spectral radii of a set of nonnegative matrices. Mathematical Notes, 87(1):12-14, 2010. URL: http://dx.doi.org/10.1134/S0001434610010025.
  3. François Louis Baccelli, Geert Jan Olsder, Jean-Pierre Quadrat, and Guy Cohen. Synchronization and linearity. An algebra for discrete event systems. Chichester: Wiley, 1992. Google Scholar
  4. Vincent D. Blondel, Stéphane Gaubert, and John N. Tsitsiklis. Approximating the spectral radius of sets of matrices in the max-algebra is np-hard. Automatic Control, IEEE Transactions on, 45(9):1762-1765, Sep 2000. URL: http://dx.doi.org/10.1109/9.880644.
  5. Peter Butkovič. Max-linear systems. Theory and algorithms. London: Springer, 2010. URL: http://dx.doi.org/10.1007/978-1-84996-299-5.
  6. Thomas Colcombet. On distance automata and regular cost function. Presented at the Dagstuhl seminar “Advances and Applications of Automata on Words and Trees”, 2010. Google Scholar
  7. Thomas Colcombet and Laure Daviaud. Approximate comparison of distance automata. In Natacha Portier and Thomas Wilke, editors, STACS, volume 20 of LIPIcs, pages 574-585. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.574.
  8. Thomas Colcombet, Laure Daviaud, and Florian Zuleger. Size-change abstraction and max-plus automata. In Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, and Zoltán Ésik, editors, Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part I, volume 8634 of Lecture Notes in Computer Science, pages 208-219. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44522-8_18.
  9. Ludwig Elsner and P. van den Driessche. Bounds for the perron root using max eigenvalues. Linear Algebra and its Applications, 428(8):2000-2005, 2008. URL: http://dx.doi.org/10.1016/j.laa.2007.11.014.
  10. Stéphane Gaubert and Ricardo Katz. Reachability problems for products of matrices in semirings. International Journal of Algebra and Computation, 16(3):603-627, jun 2006. URL: http://arxiv.org/abs/math/0310028, http://arxiv.org/abs/0310028, URL: http://dx.doi.org/10.1142/S021819670600313X.
  11. Stéphane Gaubert and Jean Mairesse. Task resource models and (max,+) automata. In Idempotency (Bristol, 1994), volume 11 of Publ. Newton Inst., pages 133-144. Cambridge Univ. Press, Cambridge, 1998. URL: http://dx.doi.org/10.1017/CBO9780511662508.009.
  12. Stéphane Gaubert. Performance evaluation of (max,+) automata. IEEE Trans. Automat. Control, 40(12):2014-2025, 1995. URL: http://dx.doi.org/10.1109/9.478227.
  13. Stéphane Gaubert. On the Burnside problem for semigroups of matrices in the (max,+) algebra. Semigroup Forum, 52(1):271-294, 1996. URL: http://dx.doi.org/10.1007/BF02574104.
  14. Stéphane Gaubert and Jean Mairesse. Modeling and analysis of timed Petri nets using heaps of pieces. IEEE Trans. Automat. Control, 44(4):683-697, 1999. URL: http://dx.doi.org/10.1109/9.754807.
  15. Pierre Guillon, Zur Izhakian, Jean Mairesse, and Glenn Merlet. The ultimate rank of semi-groups of tropical matrices. Journal of Algebra, 437:222-248, September 2015. URL: http://dx.doi.org/10.1016/j.jalgebra.2015.02.026.
  16. Bernd Heidergott, Geert Jan Oldser, and Jacob van der Woude. Max plus at work. Modeling and analysis of synchronized systems: a course on max-plus algebra and its applications. Princeton, NJ: Princeton University Press, 2006. Google Scholar
  17. James P. Jones. Universal Diophantine equation. J. Symbolic Logic, 47(3):549-571, 1982. URL: http://dx.doi.org/10.2307/2273588.
  18. Raphaël Jungers. The joint spectral radius, volume 385 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 2009. Theory and applications. URL: http://dx.doi.org/10.1007/978-3-540-95980-9.
  19. Jui-Yi Kao, Narad Rampersad, and Jeffrey Shallit. On NFAs where all states are final, initial, or both. Theoretical Computer Science, 410(47):5010-5021, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.07.049.
  20. Daniel Krob. The equality problem for rational series with multiplicities in the tropical semiring is undecidable. In Automata, languages and programming (Vienna, 1992), volume 623 of Lecture Notes in Comput. Sci., pages 101-112. Springer, Berlin, 1992. URL: http://dx.doi.org/10.1007/3-540-55719-9_67.
  21. Sylvain Lombardy and Jean Mairesse. Max-plus automaton. In Handbook of Automata. European Mathematical Society, To appear. Google Scholar
  22. Glenn Merlet. Semigroup of matrices acting on the max-plus projective space. Linear Algebra and its Applications, 432(8):1923-1935, 2010. URL: http://dx.doi.org/10.1016/j.laa.2009.03.029.
  23. Marvin L. Minsky. Recursive unsolvability of Post’s problem of "tag" and other topics in theory of Turing machines. Ann. of Math. (2), 74:437-455, 1961. Google Scholar
  24. Marvin L. Minsky. Computation: finite and infinite machines. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. Prentice-Hall Series in Automatic Computation. Google Scholar
  25. Marcel-Paul Schützenberger. On the definition of a family of automata. Information and Control, 4:245-270, 1961. Google Scholar
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