# Normal and sinkless Petri nets

## Abstract

We examine both the modeling power of normal and sinkless Petri nets and the computational complexities of various classical decision problems with respect to these two classes. We argue that although neither normal nor sinkless Petri nets are strictly more powerful than persistent Petri nets, they nonetheless are both capable of modeling a more interesting class of problems. On the other hand, we give strong evidence that normal and sinkless Petri nets are easier to analyze than persistent Petri nets. In so doing, we apply techniques originally developed for conflict-free Petri nets — a class defined solely in terms of the structure of the net — to sinkless Petri nets — a class defined in terms of the behavior of the net. As a result, we give the first comprehensive complexity analysis of a class of potentially unbounded Petri nets defined in terms of their behavior.

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## References

- [Bak73]H. Baker.
*Rabin's Proof of the Undecidability of the Reachability Set Inclusion Problem of Vector Addition Systems*. Memo 79, MIT Project MAC, Computer Structure Group, 1973.Google Scholar - [BT76]I. Borosh and L. Treybig. Bounds on positive integral solutions of linear Diophantine equations.
*Proc. AMS*, 55:299–304, March 1976.Google Scholar - [CLM76]E. Cardoza, R. Lipton, and A. Meyer. Exponential space complete problems for Petri nets and commutative semigroups. In
*Proceedings of the 8th Annual ACM Symposium on Theory of Computing*, pages 50–54, 1976.Google Scholar - [CM75]S. Crespi-Reghizzi and D. Mandrioli. A decidability theorem for a class of vector addition systems.
*Information Processing Letters*, 3:78–80, 1975.CrossRefGoogle Scholar - [Gra80]J. Grabowski. The decidability of persistence for vector addition systems.
*Information Processing Letters*, 11:20–23, 1980.CrossRefGoogle Scholar - [GY80]A. Ginzburg and M. Yoeli. Vector addition systems and regular languages.
*J. of Computer and System Sciences*, 20:277–284, 1980.CrossRefGoogle Scholar - [Hac75]M. Hack.
*Petri Net Languages*. Memo 124, MIT Project MAC, Computer Structure Group, 1975.Google Scholar - [Hac76]M. Hack. The equality problem for vector addition systems is undecidable.
*Theoret. Comp. Sci.*, 2:77–95, 1976.CrossRefGoogle Scholar - [HP79]J. Hopcroft and J. Pansiot. On the reachability problem for 5-dimensional vector addition systems.
*Theoret. Comp. Sci.*, 8:135–159, 1979.CrossRefGoogle Scholar - [HR88a]R. Howell and L. Rosier. Completeness results for conflict-free vector replacement systems.
*J. of Computer and System Sciences*, 37:349–366, 1988.CrossRefGoogle Scholar - [HR88b]R. Howell and L. Rosier. On questions of fairness and temporal logic for conflict-free Petri nets. In G. Rozenberg, editor,
*Advances in Petri Nets 1988*, pages 200–226, Springer-Verlag, Berlin, 1988. LNCS 340.Google Scholar - [HRHY86]R. Howell, L. Rosier, D. Huynh, and H. Yen. Some complexity bounds for problems concerning finite and 2-dimensional vector addition systems with states.
*Theoret. Comp. Sci.*, 46:107–140, 1986.CrossRefGoogle Scholar - [HRY87]R. Howell, L. Rosier, and H. Yen. An O(n
^{1.5}) algorithm to decide boundedness for conflict-free vector replacement systems.*Information Processing Letters*, 25:27–33, 1987.CrossRefGoogle Scholar - [HRY88]R. Howell, L. Rosier, and H. Yen.
*Normal and Sinkless Petri Nets*. Technical Report TR-CS-88-14, Kansas State University, Manhattan, Kansas 66506, 1988.Google Scholar - [HU79]J. Hopcroft and J. Ullman.
*Introduction to Automata Theory, Languages, and Computation*. Addison-Wesley, Reading, Mass., 1979.Google Scholar - [Huy82]D. Huynh. The complexity of semilinear sets.
*Elektronische Informationsverarbeitung und Kybernetik*, 18:291–338, 1982.Google Scholar - [Huy85]D. Huynh. The complexity of the equivalence problem for commutative semigroups and symmetric vector addition systems. In
*Proceedings of the 17th Annual ACM Symposium on Theory of Computing*, pages 405–412, 1985.Google Scholar - [Huy86]D. Huynh. A simple proof for the σ
_{2}^{P}upper bound of the inequivalence problem for semilinear sets.*Elektronische Informationsverarbeitung und Kybernetik*, 22:147–156, 1986.Google Scholar - [JLL77]N. Jones, L. Landweber, and Y. Lien. Complexity of some problems in Petri nets.
*Theoret. Comp. Sci.*, 4:277–299, 1977.CrossRefGoogle Scholar - [Kos82]R. Kosaraju. Decidability of reachability in vector addition systems. In
*Proceedings of the 14th Annual ACM Symposium on Theory of Computing*, pages 267–280, 1982.Google Scholar - [Lam87]J. Lambert. Consequences of the decidability of the reachability problem for Petri nets. In
*Proceedings of the Eighth European Workshop on Application and Theory of Petri Nets*, pages 451–470, 1987. To appear in*Theoret. Comp. Sci.*Google Scholar - [Lip76]R. Lipton.
*The Reachability Problem Requires Exponential Space*. Technical Report 62, Yale University, Dept. of CS., Jan. 1976.Google Scholar - [LR78]L. Landweber and E. Robertson. Properties of conflict-free and persistent Petri nets.
*JACM*, 25:352–364, 1978.CrossRefGoogle Scholar - [May81]E. Mayr. Persistence of vector replacement systems is decidable.
*Acta Informatica*, 15:309–318, 1981.CrossRefGoogle Scholar - [May84]E. Mayr. An algorithm for the general Petri net reachability problem.
*SIAM J. Comput.*, 13:441–460, 1984. A preliminary version of this paper was presented at the*13th Annual Symposium on Theory of Computing*, 1981.CrossRefGoogle Scholar - [MM81]E. Mayr and A. Meyer. The complexity of the finite containment problem for Petri nets.
*JACM*, 28:561–576, 1981.CrossRefGoogle Scholar - [MM82]E. Mayr and A. Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals.
*Advances in Mathematics*, 46:305–329, 1982.CrossRefGoogle Scholar - [Mul81]H. Müller. On the reachability problem for persistent vector replacement systems.
*Computing, Suppl.*, 3:89–104, 1981.Google Scholar - [Pet81]J. Peterson.
*Petri Net Theory and the Modeling of Systems*. Prentice Hall, Englewood Cliffs, NJ, 1981.Google Scholar - [Rac78]C. Rackoff. The covering and boundedness problems for vector addition systems.
*Theoret. Comp. Sci.*, 6:223–231, 1978.CrossRefGoogle Scholar - [Rei85]W. Reisig.
*Petri Nets: An Introduction*. Springer-Verlag, Heidelberg, 1985.Google Scholar - [Sto77]L. Stockmeyer. The polynomial-time hierarchy.
*Theoret. Comp. Sci.*, 3:1–22, 1977.CrossRefGoogle Scholar - [VV81]R. Valk and G. Vidal-Naquet. Petri nets and regular languages.
*J. of Computer and System Sciences*, 23:299–325, 1981.CrossRefGoogle Scholar - [Yam84]H. Yamasaki. Normal Petri nets.
*Theoret. Comp. Sci.*, 31:307–315, 1984.CrossRefGoogle Scholar