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Tropical Abstractions of Max-Plus Linear Systems

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Formal Modeling and Analysis of Timed Systems (FORMATS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11022))

Abstract

This paper describes the development of finite abstractions of Max-Plus-Linear (MPL) systems using tropical operations. The idea of tropical abstraction is inspired by the fact that an MPL system is a discrete-event model updating its state with operations in the tropical algebra. The abstract model is a finite-state transition system: we show that the abstract states can be generated by operations on the tropical algebra, and that the generation of transitions can be established by tropical multiplications of matrices. The complexity of the algorithms based on tropical algebra is discussed and their performance is tested on a numerical benchmark against an existing alternative abstraction approach.

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Notes

  1. 1.

    A permutation \(\alpha \) is called maximal if \(\bigotimes _{i=1}^n A(i,\alpha (i))=\text {per}(A)\), where \(\text {per}(A)\) is the permanent of A [6, 17].

  2. 2.

    \(\hat{R}\) is the collection of non-empty \(R_g\). We use small letter \(\hat{r}_i\) for sake of simplicity.

References

  1. Adzkiya, D.: Finite abstractions of max-plus-linear systems: theory and algorithms. Ph.D. thesis, Delft University of Technology (2014)

    Google Scholar 

  2. Adzkiya, D., Abate, A.: VeriSiMPL: verification via biSimulations of MPL models. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 274–277. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40196-1_22

    Chapter  Google Scholar 

  3. Adzkiya, D., De Schutter, B., Abate, A.: Finite abstractions of max-plus-linear systems. IEEE Trans. Autom. Control 58(12), 3039–3053 (2013)

    Article  MathSciNet  Google Scholar 

  4. Adzkiya, D., De Schutter, B., Abate, A.: Computational techniques for reachability analysis of max-plus-linear systems. Automatica 53, 293–302 (2015)

    Article  MathSciNet  Google Scholar 

  5. Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, Hoboken (1992)

    MATH  Google Scholar 

  6. Butkovič, P.: Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl. 367, 313–335 (2003)

    Article  MathSciNet  Google Scholar 

  7. Dill, D.L.: Timing assumptions and verification of finite-state concurrent systems. In: Sifakis, J. (ed.) CAV 1989. LNCS, vol. 407, pp. 197–212. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52148-8_17

    Chapter  Google Scholar 

  8. Floyd, R.W.: Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962)

    Article  Google Scholar 

  9. Heemels, W., De Schutter, B., Bemporad, A.: Equivalence of hybrid dynamical models. Automatica 37(7), 1085–1091 (2001)

    Article  Google Scholar 

  10. Heidergott, B., Olsder, G.J., Van der Woude, J.: Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications. Princeton University Press, Princeton (2014)

    Google Scholar 

  11. Itenberg, I., Mikhalkin, G., Shustin, E.I.: Tropical Algebraic Geometry, vol. 35. Springer, Basel (2009). https://doi.org/10.1007/978-3-0346-0048-4

    Book  MATH  Google Scholar 

  12. Lu, Q., Madsen, M., Milata, M., Ravn, S., Fahrenberg, U., Larsen, K.G.: Reachability analysis for timed automata using max-plus algebra. J. Logic Algebraic Program. 81(3), 298–313 (2012)

    Article  MathSciNet  Google Scholar 

  13. Mufid, M.S., Adzkiya, D., Abate, A.: Tropical abstractions of max-plus-linear systems (2018). arXiv:1806.04604

  14. Péron, M., Halbwachs, N.: An abstract domain extending difference-bound matrices with disequality constraints. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 268–282. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-69738-1_20

    Chapter  MATH  Google Scholar 

  15. Pin, J.-E.: Tropical semirings. Idempotency, pp. 50–69 (1998)

    Google Scholar 

  16. Richards, A.: University of Oxford Advanced Research Computing (2015). Zenodo. https://doi.org/10.5281/zenodo.22558

  17. Sergeev, S.: Max-plus definite matrix closures and their eigenspaces. Linear Algebra Appl. 421(2–3), 182–201 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is supported by Indonesia Endowment Fund for Education (LPDP), while the third acknowledges the support of the Alan Turing Institute, London, UK.

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Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Muhammad Syifa’ul Mufid & Alessandro Abate

  2. Department of Mathematics, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia

    Dieky Adzkiya

Authors
  1. Muhammad Syifa’ul Mufid
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  2. Dieky Adzkiya
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  3. Alessandro Abate
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Corresponding author

Correspondence to Muhammad Syifa’ul Mufid .

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Editors and Affiliations

  1. Institute of Software, Chinese Academy of Sciences, Beijing, China

    David N. Jansen

  2. Kansas State University, Manhattan, Kansas, USA

    Pavithra Prabhakar

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Mufid, M.S., Adzkiya, D., Abate, A. (2018). Tropical Abstractions of Max-Plus Linear Systems. In: Jansen, D., Prabhakar, P. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2018. Lecture Notes in Computer Science(), vol 11022. Springer, Cham. https://doi.org/10.1007/978-3-030-00151-3_16

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  • DOI: https://doi.org/10.1007/978-3-030-00151-3_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00150-6

  • Online ISBN: 978-3-030-00151-3

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