Bisimulation Metrics for Weighted Automata

Authors Borja Balle, Pascale Gourdeau, Prakash Panangaden

Thumbnail PDF


  • Filesize: 0.52 MB
  • 14 pages

Document Identifiers

Author Details

Borja Balle
Pascale Gourdeau
Prakash Panangaden

Cite AsGet BibTex

Borja Balle, Pascale Gourdeau, and Prakash Panangaden. Bisimulation Metrics for Weighted Automata. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 103:1-103:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We develop a new bisimulation (pseudo)metric for weighted finite automata (WFA) that generalizes Boreale's linear bisimulation relation. Our metrics are induced by seminorms on the state space of WFA. Our development is based on spectral properties of sets of linear operators. In particular, the joint spectral radius of the transition matrices of WFA plays a central role. We also study continuity properties of the bisimulation pseudometric, establish an undecidability result for computing the metric, and give a preliminary account of applications to spectral learning of weighted automata.
  • weighted automata
  • bisimulation
  • metrics
  • spectral theory
  • learning


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. R. Bailly. Méthodes spectrales pour l'inférence grammaticale probabiliste de langages stochastiques rationnels. PhD thesis, Aix-Marseille Université, 2011. Google Scholar
  2. Borja Balle. Learning Finite-State Machines: Algorithmic and Statistical Aspects. PhD thesis, Universitat Politècnica de Catalunya, 2013. Google Scholar
  3. Borja Balle, Xavier Carreras, Franco M. Luque, and Ariadna Quattoni. Spectral learning of weighted automata: A forward-backward perspective. Machine Learning, 2014. Google Scholar
  4. Borja Balle, Pascale Gourdeau, and Prakash Panangaden. Bisimulation metrics for weighted automata, 2017. URL:
  5. Borja Balle and Mehryar Mohri. Learning weighted automata. In Conference on Algebraic Informatics, 2015. Google Scholar
  6. Borja Balle, Prakash Panangaden, and Doina Precup. A canonical form for weighted automata and applications to approximate minimization. In Proceedings of the Thirtieth Annual ACM-IEEE Symposium on Logic in Computer Science, July 2015. Google Scholar
  7. Nikita E. Barabanov. On the Lyapunov indicator of discrete inclusions, part I, II, and III. Avtomatika i Telemekhanika, 2:40-46, 1988. Google Scholar
  8. Vincent D. Blondel, Yurii Nesterov, and Jacques Theys. On the accuracy of the ellipsoid norm approximation of the joint spectral radius. Linear Algebra and its Applications, 394:91-107, 2005. Google Scholar
  9. Filippo Bonchi, Marcello M. Bonsangue, Helle Hvid Hansen, Prakash Panangaden, Jan Rutten, and Alexandra Silva. Algebra-coalgebra duality in Brzozowski’s minimization algorithm. ACM Transactions on Computational Logic, 2014. Google Scholar
  10. Michele Boreale. Weighted bisimulation in linear algebraic form. In CONCUR 2009 - Concurrency Theory, pages 163-177. Springer, 2009. Google Scholar
  11. Ingrid Daubechies and Jeffrey C. Lagarias. Sets of matrices all infinite products of which converge. Linear algebra and its applications, 161:227-263, 1992. Google Scholar
  12. Ingrid Daubechies and Jeffrey C. Lagarias. Corrigendum/addendum to: Sets of matrices all infinite products of which converge. Linear Algebra and its Applications, 327(1-3):69-83, 2001. Google Scholar
  13. François Denis, Mattias Gybels, and Amaury Habrard. Dimension-free concentration bounds on hankel matrices for spectral learning. Journal of Machine Learning Research, 17(31):1-32, 2016. Google Scholar
  14. J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In Proceedings of CONCUR99, number 1664 in Lecture Notes in Computer Science. Springer-Verlag, 1999. Google Scholar
  15. Josée Desharnais, Vineet Gupta, Radhakrishnan Jagadeesan, and Prakash Panangaden. A metric for labelled Markov processes. Theoretical Computer Science, 318(3):323-354, June 2004. Google Scholar
  16. Manfred Droste, Werner Kuich, and Heiko Vogler, editors. Handbook of weighted automata. EATCS Monographs on Theoretical Computer Science. Springer, 2009. Google Scholar
  17. Norm Ferns, Prakash Panangaden, and Doina Precup. Metrics for finite Markov decision processes. In Proceedings of the 20th conference on Uncertainty in Artificial Intelligence, pages 162-169. AUAI Press, 2004. Google Scholar
  18. Norm Ferns, Prakash Panangaden, and Doina Precup. Metrics for Markov decision processes with infinite state spaces. In Proceedings of the 21st Conference on Uncertainty in Artificial Intelligence, pages 201-208, July 2005. Google Scholar
  19. Norm Ferns, Prakash Panangaden, and Doina Precup. Bisimulation metrics for continuous markov decision processes. SIAM Journal on Computing, 40(6):1662-1714, 2011. Google Scholar
  20. A. Giacalone, C. Jou, and S. Smolka. Algebraic reasoning for probabilistic concurrent systems. In Proceedings of the Working Conference on Programming Concepts and Methods, IFIP TC2, 1990. Google Scholar
  21. Hadrien Glaude and Olivier Pietquin. PAC learning of Probabilistic Automaton based on the Method of Moments. In Proceedings of The 33rd International Conference on Machine Learning, pages 820-829, 2016. Google Scholar
  22. Christopher Heil and Gilbert Strang. Continuity of the joint spectral radius: application to wavelets. In Linear Algebra for Signal Processing, pages 51-61. Springer, 1995. Google Scholar
  23. Daniel Hsu, Sham M Kakade, and Tong Zhang. A spectral algorithm for learning hidden markov models. Journal of Computer and System Sciences, 78(5), 2012. Google Scholar
  24. Manfred Jaeger, Hua Mao, Kim Guldstrand Larsen, and Radu Mardare. Continuity properties of distances for Markov processes. In Proceedings of QEST 2014 Quantitative Evaluation of Systems: 11th International Conference, pages 297-312. Springer International Publishing, 2014. Google Scholar
  25. Raphaël Jungers. The joint spectral radius: theory and applications, volume 385. Springer Science and Business Media, 2009. Google Scholar
  26. Victor Kozyakin. An explicit lipschitz constant for the joint spectral radius. Linear Algebra and its Applications, 433(1):12-18, 2010. Google Scholar
  27. Omid Madani, Steve Hanks, and Anne Condon. On the undecidability of probabilistic planning and related stochastic optimization problems. Artificial Intelligence, 147(1-2):5-34, 2003. Google Scholar
  28. Prakash Panangaden. Labelled Markov Processes. Imperial College Press, 2009. Google Scholar
  29. Gian-Carlo Rota and W. Strang. A note on the joint spectral radius. Indag. Math., 22:379–381, 1960. Google Scholar
  30. Marcel Paul Schützenberger. On the definition of a family of automata. Information and control, 4(2):245-270, 1961. Google Scholar
  31. S. M. Siddiqi, B. Boots, and G. Gordon. Reduced-rank hidden Markov models. In AISTATS, 2010. Google Scholar
  32. Franck van Breugel and James Worrell. Towards quantitative verification of probabilistic systems. In Proceedings of the Twenty-eighth International Colloquium on Automata, Languages and Programming. Springer-Verlag, July 2001. Google Scholar
  33. Fabian Wirth. The generalized spectral radius and extremal norms. Linear Algebra and its Applications, 342(1-3):17-40, 2002. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail