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Comparison of max-plus automata and joint spectral radius of tropical matrices

Daviaud, L., Guillon, P. & Merlet, G. (2017). Comparison of max-plus automata and joint spectral radius of tropical matrices. In: Larsen, K. G., Bodlaender, H. L. & Raskin, J-F. (Eds.), 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), 83. (19:1-19:14). Dagstuhl: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. ISBN 9783959770460


Weighted automata over the max-plus semiring S are closely related to finitely generated semigroups of matrices over S. 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 M and provides as output the joint spectral radius (resp. the ultimate rank) of M. 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.

Publication Type: Conference or Workshop Item (Paper)
Additional Information: ©Mikołaj Bojańczyk, Laure Daviaud, Bruno Guillon, and Vincent Penelle; licensed under Creative Commons License CC-BY
Publisher Keywords: Streaming String Transducers, Origin Semantics, String-to-String Transductions, MSO Definability
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Computer Science
Text - Published Version
Available under License Creative Commons: Attribution 3.0.

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