Abstract
Elementary net systems (ENS) are the most fundamental class of Petri nets. Their synthesis problem has important applications in the design of digital hardware and commercial processes. Given a labeled transition system (TS) A, feasibility is the NP-complete decision problem whether A can be synthesized into an ENS. It is known that A is feasible if and only if it has the event state separation property (ESSP) and the state separation property (SSP). Recently, these properties have also been studied individually for their practical implications. A fast ESSP algorithm, for instance, would allow applications to at least validate the language equivalence of A and a synthesized ENS. Being able to efficiently decide SSP, on the other hand, could serve as a quick-fail preprocessing mechanism for synthesis. Although a few tractable subclasses have been found, this paper destroys much of the hope that many practically meaningful input restrictions make feasibility or at least one of ESSP and SSP efficient. We show that all three problems remain NP-complete even if the input is restricted to linear TSs where every event occurs at most three times.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Agostini, A., De Michelis, G.: Improving flexibility of workflow management systems. In: van der Aalst, W., Desel, J., Oberweis, A. (eds.) Business Process Management. LNCS, vol. 1806, pp. 218–234. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45594-9_14
Badouel, E., Darondeau, P.: Trace nets and process automata. Acta Informatica 32(7), 647–679 (1995)
Badouel, E., Bernardinello, L., Darondeau, P.: The synthesis problem for elementary net systems is NP-complete. Theor. Comput. Sci. 186(1–2), 107–134 (1997)
Badouel, E., Bernardinello, L., Darondeau, P.: Petri Net Synthesis. Texts in Theoretical Computer Science. An EATCS Series, pp. 1–325. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47967-4. ISBN 978-3-662-47966-7
Best, E., Erofeev, E., Schlachter, U., Wimmel, H.: Characterising Petri net solvable binary words. In: Kordon, F., Moldt, D. (eds.) PETRI NETS 2016. LNCS, vol. 9698, pp. 39–58. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-39086-4_4
Cortadella, J.: Private correspondence (2017)
Cortadella, J., Kishinevsky, M., Kondratyev, A., Lavagno, L., Yakovlev, A.: Complete state encoding based on the theory of regions. In: International Symposium on Advanced Research in Asynchronous Circuits and Systems, pp. 36–47. IEEE (1996)
Hiraishi, K.: Some complexity results on TS and elementary net systems. Theor. Comput. Sci. 135(2), 361–376 (1994)
Kleijn, J., Koutny, M., Pietkiewicz-Koutny, M., Rozenberg, G.: Step semantics of Boolean nets. Acta Informatica 50(1), 15–39 (2013)
Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Discret. Comput. Geom. 26(4), 573–590 (2001)
Rosenke, C., Tredup, R.: The hardness of synthesizing elementary net systems from highly restricted inputs. arXiv:1711.00220 (2017)
Schmitt, V.: Flip-flop nets. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 515–528. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-60922-9_42
Yakovlev, A.V., Koelmans, A.M.: Petri nets and digital hardware design. In: Reisig, W., Rozenberg, G. (eds.) ACPN 1996. LNCS, vol. 1492, pp. 154–236. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-65307-4_49
OMG. Business Process Model and Notation (BPMN). Object Management Group (2016)
Scheer, A.-W.: Business Process Engineering, Reference Models for Industrial Enterprises. Springer, Heidelberg (1994). https://doi.org/10.1007/978-3-642-79142-0
UML. Unified Modeling Language (UML). Object Management Group (2016)
Acknowledgements
We thank the anonymous reviewers for their helpful suggestions. The first author thanks Eric Badouel for the inspiring correspondence about the synthesis of ENS.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Tredup, R., Rosenke, C., Wolf, K. (2018). Elementary Net Synthesis Remains NP-Complete Even for Extremely Simple Inputs. In: Khomenko, V., Roux, O. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2018. Lecture Notes in Computer Science(), vol 10877. Springer, Cham. https://doi.org/10.1007/978-3-319-91268-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-91268-4_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91267-7
Online ISBN: 978-3-319-91268-4
eBook Packages: Computer ScienceComputer Science (R0)