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On Weighted Petri Net Transducers

  • Robert Lorenz
  • Markus Huber
  • Günther Wirsching
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8489)

Abstract

In this paper we present a basic framework for weighted Petri net transducers (PNTs) for the translation of partial languages (consisting of partial words) as a natural generalisation of finite state transducers (FSTs).

Concerning weights, we use the algebraic structure of continuous concurrent semirings which is based on bisemirings and induces a natural order on its elements. Using the operations of this algebra, it is possible to define the weight of sequential parallel partial words in a standard way. We define the weight of a general partial word as the supremum of the weights of all of its sequential parallel extensions. As a fundamental result we show that concurrent semirings are the least restrictive idempotent bisemiring structure such that partial words with fewer dependencies have bigger weights. Moreover, the weight definition turns out to be compositional, i.e. the weight of (sequential or parallel) composed partial words equals the corresponding bisemiring composition of the weights of its components.

To be able to create complex PNTs through composition of simple PNTs, we introduce clean PNTs and the composition operations union, product, closure, parallel product and language composition on clean PNTs, lifting standard composition operations on FSTs. Composed PNTs yield a compositional computation of weights, where in the case of language composition such a compositional computation is possible only in restricted cases. Moreover, we give definitions for equivalent PNTs and show that all composition operations preserve equivalence. We also show that under certain conditions concerning the algebraic weight structure an FST can be represented by an equivalent PNT.

Keywords

Petri Net Petri Net Transducer Weighted Transducer Labelled Partial Order Weighted Labelled Partial Order Partial Language Semiring Bisemiring Concurrent Semiring Cleanness 

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References

  1. 1.
    Azéma, P., Balbo, G. (eds.): ICATPN 1997. LNCS, vol. 1248. Springer, Heidelberg (1997)Google Scholar
  2. 2.
    Best, E., Devillers, R.R., Hall, J.G.: The box calculus: a new causal algebra with multi-label communication. In: Rozenberg [19], pp. 21–69Google Scholar
  3. 3.
    Boudol, G., Castellani, I.: On the semantics of concurrency: Partial orders and transition systems. In: Ehrig, H., Levi, G., Montanari, U. (eds.) CAAP 1987 and TAPSOFT 1987. LNCS, vol. 249, pp. 123–137. Springer, Heidelberg (1987)Google Scholar
  4. 4.
    Chothia, T., Klejin, J.: Q-automata: Modelling the resource usage of concurrent components. Electronic Notes in Theoretical Computer Science 175(175), 153–167 (2007)CrossRefGoogle Scholar
  5. 5.
    Droste, M., Kuich, W.: Semirings and Formal Power Series. In: Droste, et al. (eds.) [6], ch.1, pp. 3–28 (2009)Google Scholar
  6. 6.
    Droste, M., Kuich, W., Vogler, H. (eds.): Handbook of Weighted Automata. Monographs in Theoretical Computer Science. Springer (2009)Google Scholar
  7. 7.
    Esposito, A., Esposito, A.M., Vinciarelli, A., Hoffmann, R., Müller, V.C. (eds.): COST 2102. LNCS, vol. 7403. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Fichtner, I., Kuske, D., Meinecke, I.: Traces, Series-Parallel Posets, and Pictures: A Weighted Study. In: Droste, et al. (eds.) [6], ch. 10, pp. 405–452 (2009)Google Scholar
  9. 9.
    Füllöp, Z., Vogler, H.: Weighted Tree Automata and Tree Transducers. In: Droste, et al. (eds.) [6], ch. 9, pp. 313–404 (2009)Google Scholar
  10. 10.
    Gischer, J.L.: The equational theory of pomsets. Theoretical Computer Science 61, 199–224 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grabowski, J.: On Partial Languages. Fundamenta Informaticae 4(2), 428–498 (1981)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hack, M.: Petri net languages. Technical Report Memo 124, computation structures group, massachusetts institute of technology (1975)Google Scholar
  13. 13.
    Hoare, T., Möller, B., Struth, G., Wehrman, I.: Concurrent Kleene algebra and its foundations. The Journal of Logic and Algebraic Programming 80, 266–296 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kuske, D., Meinecke, I.: Branching automata with costs - a way of reflecting parallelism in costs. Theoretical Computer Science 328, 53–75 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lorenz, R., Huber, M.: Petri net transducers in semantic dialogue modelling. In: Proceedings of “Elektronische Sprachsignalverarbeitung (ESSV)”. Studientexte zur Sprachkommunikation, vol. 64, pp. 286–297 (2012)Google Scholar
  16. 16.
    Lorenz, R., Huber, M.: Realizing the Translation of Utterances into Meanings by Petri Net Transducers. In: Proceedings of “Elektronische Sprachsignalverarbeitung (ESSV)”. Studientexte zur Sprachkommunikation, vol. 65 (2013)Google Scholar
  17. 17.
    Mohri, M.: Weighted Automata Algorithms. In: Droste, et al. (eds.) [6], ch. 6, pp. 213–254 (2009)Google Scholar
  18. 18.
    Pratt, V.: Modelling Concurrency with Partial Orders. Int. Journal of Parallel Programming 15, 33–71 (1986)CrossRefGoogle Scholar
  19. 19.
    Rozenberg, G. (ed.): Advances in Petri Nets 1992, The DEMON Project. Springer (1992)Google Scholar
  20. 20.
    Straßner, D.: Prototypische Implementierung von Petrinetz-Transduktoren mit SNAKES. Bachelor thesis, Augsburg University (2013)Google Scholar
  21. 21.
    van Biljon, W.R.: Extending Petri nets for specifying man-machine dialogues. Int. J. Man-Mach. Stud. 28(4), 437–455 (1988)CrossRefGoogle Scholar
  22. 22.
    van der Aalst, W.M.P.: Verification of workflow nets. In: Azéa, Balbo (eds.) [1], pp. 407–426CrossRefGoogle Scholar
  23. 23.
    Wang, F.-Y., Mittmann, M., Saridis, G.N.: Coordination specification for CIRSSE robotic platform system using Petri net transducers. Journal of Intelligent and Robotic Systems 9, 209–233 (1994)CrossRefGoogle Scholar
  24. 24.
    Wang, F.-Y., Saridis, G.N.: A model for coordination of intelligent machines using Petri nets. In: Proceedings of the IEEE International Symposium on Intelligent Control, pp. 28–33. IEEE Comput. Soc. Press (1989)Google Scholar
  25. 25.
    Wirsching, G., Huber, M., Kölbl, C.: Zur Logik von Bestenlisten in der Dialogmodellierung. In: Proceedings of “Elektronische Sprachsignalverarbeitung (ESSV)”. Studientexte zur Sprachkommunikation, vol. 61, pp. 309–316 (2011)Google Scholar
  26. 26.
    Wirsching, G., Huber, M., Kölbl, C., Lorenz, R., Römer, R.: Semantic dialogue modeling. In: Esposito, et al. (eds.) [7], pp. 104–113CrossRefGoogle Scholar
  27. 27.
    Wolff, M.: Akustische Mustererkennung. Habilitation (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Robert Lorenz
    • 1
  • Markus Huber
    • 1
  • Günther Wirsching
    • 2
  1. 1.Department of Computer ScienceUniversity of AugsburgGermany
  2. 2.Mathematisch-Geographische FakultätCatholic University of EichstättGermany

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