Abstract
Yen proposed a construction for a semilinear representation of the reachability set of BPP-Petri nets which can be used to decide the equivalence problem of two BPP-PNs in doubly exponential time. We first address a gap in this construction which therefore does not always represent the reachability set. We propose a solution which is formulated in such a way that a large portion of Yen’s construction and proof can be retained, preserving the size of the semilinear representation and the double exponential time bound (except for possibly larger values of some constants). In the second part of the paper, we propose very efficient algorithms for several variations of the boundedness and liveness problems of BPP-PNs. For several more complex notions of boundedness, as well as for the covering problem, we show NP-completeness. To demonstrate the contrast between BPP-PNs and a slight generalization regarding edge multiplicities, we show that the complexity of the classical boundedness problem increases from linear time to coNP-hardness. Our results also imply corresponding complexity bounds for related problems for process algebras and (commutative) context-free grammars.
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References
Christensen, S.: Distributed bisimularity is decidable for a class of infinite state-space systems. In: Cleaveland, W.R. (ed.) CONCUR 1992. LNCS, vol. 630, pp. 148–161. Springer, Heidelberg (1992)
Christensen, S., Hirshfeld, Y., Moller, F.: Bisimulation equivalence is decidable for basic parallel processes. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 143–157. Springer, Heidelberg (1993)
Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundamenta Informaticae 31(1), 13–25 (1997)
Esparza, J., Rossmanith, P., Schwoon, S.: A uniform framework for problems on context-free grammars. EATCS Bulletin 72, 169–177 (2000)
Huynh, D.T.: The complexity of semilinear sets. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 324–337. Springer, Heidelberg (1980)
Huynh, D.T.: Commutative grammars: The complexity of uniform word problems. Information and Control 57(1), 21–39 (1983)
Huynh, D.T.: A simple proof for the sum upper bound of the inequivalence problem for semilinear sets. Elektronische Informationsverarbeitung und Kybernetik, 147–156 (1986)
Karp, R.M., Miller, R.E.: Parallel program schemata. Journal of Computer and System Sciences 3(2), 147–195 (1969)
Kučera, A.: Regularity is decidable for normed PA processes in polynomial time. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 111–122. Springer, Heidelberg (1996)
Landweber, L.H., Robertson, E.L.: Properties of conflict-free and persistent Petri nets. J. ACM 25, 352–364 (1978)
Mayr, E.W.: An algorithm for the general Petri net reachability problem. In: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, STOC 1981, pp. 238–246. ACM, New York (1981)
Mayr, E.W., Weihmann, J.: Results on equivalence, boundedness, liveness, and covering problems of BPP-Petri nets. Technical Report TUM-I1325, Institut für Informatik, TU München (March 2013)
Mayr, R.: Tableau methods for PA-processes. In: Galmiche, D. (ed.) TABLEAUX 1997. LNCS, vol. 1227, pp. 276–290. Springer, Heidelberg (1997)
Mayr, R.: Decidability and Complexity of Model Checking Problems for Infinite-State Systems. PhD thesis, Technische Universitt Mijnchen (1998)
Mayr, R.: On the complexity of bisimulation problems for basic parallel processes. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 329–341. Springer, Heidelberg (2000)
Murata, T.: Petri nets: Properties, analysis and applications. Proceedings of the IEEE 77(4), 541–580 (1989)
Pottier, L.: Minimal solutions of linear diophantine systems: bounds and algorithms. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, pp. 162–173. Springer, Heidelberg (1991)
Tarjan, R.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)
Yen, H.-C.: On reachability equivalence for BPP-nets. Theoretical Computer Science 179(1-2), 301–317 (1997)
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Mayr, E.W., Weihmann, J. (2013). Results on Equivalence, Boundedness, Liveness, and Covering Problems of BPP-Petri Nets. In: Colom, JM., Desel, J. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2013. Lecture Notes in Computer Science, vol 7927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38697-8_5
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DOI: https://doi.org/10.1007/978-3-642-38697-8_5
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