Abstract
Two types of simulations and four types of bisimulations for weighted finite automata over the complete max-plus semiring we define as solutions of particular systems of matrix inequations. We provide a procedure that either decides that there is a simulation or bisimulation of a given type between two automata, and outputs the greatest one, or decides that no simulation or bisimulation of that type exists. The procedure is iterative and does not have to end in a finite number of steps. Certain conditions under which this procedure must terminate in a finite number of steps are described in a slightly more general context in Stamenković et al. (Discrete Event Dynamic Systems, 32:1–25, 2022). We also propose a modification of this procedure which, in case there is no simulation or bisimulation of a given type between two max-plus automata, detects this in finitely many steps and faster than the original procedure. In the same case, that modification also finds a natural number such that containment or equivalence is valid for all input words of length less than that number. For max-plus automata with non-negative weights, we point out the differences that occur when the above mentioned procedure is applied over the complete max-plus semiring, and when it is applied over its non-negative part with minus infinity added.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability statement
Not applicable.
Notes
Here we use that notion in the sense of non-negativity in the field of real numbers, i.e., an element \(x\in \mathbb {R}\) is non-negative if \(x\geqslant 0\). This should not be confused with the notion of positivity in the semiring \(\mathbb {R}_{\max }\), where the zero is \(-\infty \), and thus, no negative elements at all.
References
Baccelli F, Cohen G, Olsder G, Quadrat J (1992) Synchronization and linearity. Wiley, Chichester
Bezem M, Nieuwenhuis R, Rodríguez-Carbonell E (2008) Exponential behaviour of the Butkovic–Zimmermann algorithm for solving two-sided linear systems in max-algebra. Discret Appl Math 156:3506–3509
Boukra R, Lahaye S, Boimond J-L (2012) New representations for (max,+) automata with applications to the performance evaluation of discrete event systems. IFAC PapersOnLine 45:29:116–121
Butkovič P, Zimmermann K (2006) A strongly polynomial algorithm for solving two-sided linear systems in max-algebra. Discret Appl Math 154:437–446
Ćirić M, Ignjatović J, Damljanović N, Bašić M (2012a) Bisimulations for fuzzy automata. Fuzzy Sets Syst 186:100–139
Ćirić M, Ignjatović J, Jančić I, Damljanović N (2012b) Computation of the greatest simulations and bisimulations between fuzzy automata. Fuzzy Sets Syst 208:22–42
Ćirić M, Ignjatović J, Stanimirović P (2022) Bisimulations for weighted finite automata over semirings. Research Square, 19 Dec 2022. https://doi.org/10.21203/rs.3.rs-2386298/v1
Colcombet T, Daviaud L, Zuleger F (2014) Size-change abstraction and max-plus automata. In: Csuhaj-Varjú E, Dietzfelbinger M, Ésik Z (eds), Mathematical foundations of computer science – MFCS 2014. Lecture Notes in Computer Science, vol 8634, pp 208–219
Damljanović N, Ćirić M, Ignjatović J (2014) Bisimulations for weighted automata over an additively idempotent semiring. Theoret Comput Sci 534:86–100
Daviaud L, Guillon P, Merlet G (2017) Comparison of max-plus automata and joint spectral radius of tropical matrices. In: Larsen KG, Bodlaender HL, Raskin J-F (eds) 42nd International symposium on mathematical foundations of computer science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), vol 83, pp 19:1–19:14
Droste M, Kuich W, Vogler H (eds) (2009) Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer-Verlag
Gaubert S (1995) Performance evaluation of (max,+) automata. IEEE Trans Autom Control 40(12):2014–2025
Gaubert S, Mairesse J (1999) Modeling and analysis of timed petri nets using heaps of pieces. IEEE Transaction on Automatic Control 44(4):683–698
Golan J (1999) Semirings and their Applications. Springer, Dordrecht
Komenda J, Lahaye S, Boimond J-L (2009) Supervisory control of (max,+) automata: a behavioral approach. Discrete Event Dynamic Systems 19(4):525–549
Komenda J, Lahaye S, Boimond J-L, van den Boom T (2017) Max-Plus algebra and discrete event systems. IFAC PapersOnLine 50–1:1784–1790
Komenda J, Lahaye S, Boimond J-L (2018a) (Max,+)-automata with partial observations. IFAC PapersOnLine 51–7:192–197
Komenda J, Lahaye S, Boimond J-L, van den Boom T (2018b) Max-plus algebra in the history of discrete event systems. Annu Rev Control 45:240–249
Lahaye S, Komenda J, Boimond J-L (2014) Modeling of timed Petri nets using deterministic (max,+) automata. IFAC Proceedings Volumes 47–2:471–476
Lahaye S, Komenda J, Boimond J-L (2015) Compositions of (max,+) automata. Discrete Event Dyn Syst 25:323–344
Lahaye S, Lai A, Komenda J, Boimond J-L (2020) A contribution to the determinization of max-plus automata. Discrete Event Dynamic Systems 30:155–174
Nguyen LA, Micić I, Stanimirović S (2023) Depth-bounded fuzzy simulations and bisimulations between fuzzy automata. Fuzzy Sets Syst 473:108729. https://doi.org/10.1016/j.fss.2023.108729
Stamenković A, Ćirić M, Djurdjanović D (2022) Weakly linear systems for matrices over the max-plus quantale. Discrete Event Dynamic Systems 32:1–25
Stanković M, Ćirić M, Ignjatović J (2022) Hennessy-Milner type theorems for fuzzy multimodal logics over Heyting algebras. Journal of Multiple-Valued Logic and Soft Computing 39(2–4):341–379
Stanković I, Ćirić M, Ignjatović J (2023a) Bisimulations for weighted networks with weights in a quantale. Filomat 37:11:3335–3355
Stanković M, Ćirić M, Ignjatović J (2023b) Simulations and bisimulations for fuzzy multimodal logics over Heyting algebras. Filomat 37:3:711–743
Urabe N, Hasuo I (2017) Quantitative simulations by matrices. Inf Comput 252:110–137
Acknowledgements
The authors wish to thank the reviewers for helpful comments and suggestions that significantly improved the quality of the article.
Funding
This research was supported by the Science Fund of the Republic of Serbia, Grant no 7750185, Quantitative Automata Models: Fundamental Problems and Applications - QUAM. All authors are also supported by the Ministry of Education, Science, and Technological Development, Republic of Serbia, grant no. 451-03-68/2022-14/200124.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. The first draft of the manuscript was written by Miroslav Ćirić, and all authors commented on previous versions of the manuscript and contributed improvements. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest statement
The authors have no relevant financial or non-financial interests to disclose.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ćirić, M., Micić, I., Matejić, J. et al. Simulations and bisimulations for max-plus automata. Discrete Event Dyn Syst (2024). https://doi.org/10.1007/s10626-024-00395-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10626-024-00395-1