# Recent results on the complexity of problems related to Petri nets

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## Abstract

In this paper, we examine the complexity of the boundedness, containment, equivalence, and reachability problems for certain subclasses of Petri nets (PNs) (equivalently vector addition systems (VASs), vector addition systems with states (VASSs), or vector replacement systems (VRSs)). Specifically, we consider the complexity of the boundedness problem for general VASSs, fixed dimensional VASSs, and conflict-free VRSs. We consider the complexity of the remaining problems for bounded VASSs, 2-dimensional VASSs, and conflict-free VRSs. Instances in each of these classes are known to have effectively computable semilinear reachability sets (SLSs). In each case, our results are derived by showing how to obtain succinct and sometimes special representations of the associated SLSs. The results discussed here constitute a summary of results obtained elsewhere by the authors. No proofs appear in this document, although we do strive to outline the general strategies involved. Readily available sources for the detailed proofs are indicated.

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## References

- [1]Baker, H., Rabin's Proof of the Undecidability of the Reachability Set Inclusion Problem of Vector Addition Systems, MIT Project MAC. CSGM 79, Cambridge, MA, 1973.Google Scholar
- [2]Borosh, I. and Treybig, L., Bounds on Positive Integral Solutions of Linear Diophantine Equations,
*Proc. AMS 55*, 2 (March 1976), pp. 299–304.Google Scholar - [3]Cardoza, E., Lipton. R. and Meyer, A., Exponential Space Complete Problems for Petri Nets and Commutative Semigroups,
*Proceedings of the 8th Annual ACM Symposium on Theory of Computing*(1976), pp. 50–54.Google Scholar - [4]Chan, T., The Boundedness Problem for 3-Dimensional Vector Addition Systems with States, Centre of Computer Studies and Applications, University of Hong Kong, Pokfulam Road, Hong Kong.Google Scholar
- [5]Clote, P., The Finite Containment Problem for Petri Nets,
*Theoret. Comp. Sci. 43*(1986), 99–106.Google Scholar - [6]Crespi-Reghizzi, S. and Mandrioli, D., A Decidability Theorem for a Class of Vector Addition Systems,
*Information Processing Letters 3*, 3 (1975), pp. 78–80.Google Scholar - [7]Davis, M., Matijasevic, Y. and Robinson, J., Hilbert's Tenth Problem. Diophantine Equations: Positive Aspects of a Negative Solution,
*Proceedings of Symposium on Pure Mathematics 28*(1976), pp. 323–378.Google Scholar - [8]Ginzburg, A. and Yoeli, M., Vector Addition Systems and Regular Languages,
*J. of Computer and System Sciences 20*(1980), pp. 277–284.Google Scholar - [9]Grabowski, J., The Decidability of Persistence for Vector Addition Systems,
*Information Processing Letters 11*, 1 (1980), pp. 20–23.Google Scholar - [10]Hack, M., The Equality Problem for Vector Addition Systems is Undecidable,
*Theoret. Comp. Sci. 2*(1976), pp. 77–95.Google Scholar - [11]Hopcroft, J. and Pansiot, J., On the Reachability Problem for 5-Dimensional Vector Addition Systems,
*Theoret. Comp. Sci. 8*(1979), pp. 135–159.Google Scholar - [12]Howell, R., Rosier, L., Huynh, D., and Yen, H., Some Complexity Bounds for Problems Concerning Finite and 2-Dimensional Vector Addition Systems with States,
*Theoret. Comp. Sci. 46*(1986), 107–140.Google Scholar - [13]Howell, R., and Rosier, L., Completeness Results for Conflict-Free Vector Replacement Systems, to be presented at the
*14th International Colloquium on Automata, Languages, and Programming*, July, 1987, Karlsruhe, F.R.G. Also Rep. 86-21, The University of Texas at Austin, Austin, Texas, 78712, 1986.Google Scholar - [14]Howell, R., Rosier, L., and Yen, H., An O(n
^{1.5}) Algorithm to Decide Boundedness for Conflict-Free Vector Replacement Systems, to appear in*Information Processing Letters*.Google Scholar - [15]Huynh, D., The Complexity of Semilinear Sets,
*Elektronische Informationsverarbeitung und Kybernetik 18*(1982), pp. 291–338.Google Scholar - [16]Huynh, D., The Complexity of the Equivalence Problem for Commutative Semigroups and Symmetric Vector Addition Systems,
*Proceedings of the 17th Annual ACM Symposium on Theory of Computing*(1985), pp. 405–412.Google Scholar - [17]Huynh, D., Complexity of the Word Problem for Commutative Semigroups of Fixed Dimension,
*Acta Informatica 22*(1985), pp. 421–432.Google Scholar - [18]Huynh, D., A Simple Proof for the
*Σ*_{2}^{P}Upper Bound of the Inequivalence Problem for Semilinear Sets,*Elektronische Informationsverarbeitung und Kybernetik 22*(1986), pp. 147–156.Google Scholar - [19]Jones, N., and Laaser, W., Complete Problems for Deterministic Polynomial Time,
*Theoret. Comp. Sci. 3*(1977), pp. 105–117.Google Scholar - [20]Jones, N., Landweber, L. and Lien, Y., Complexity of Some Problems in Petri Nets,
*Theoret. Comp. Sci. 4*(1977), pp. 277–299.Google Scholar - [21]Karp, R. and Miller, R., Parallel Program Schemata,
*J. of Computer and System Sciences 3*, 2 (1969), pp. 147–195.Google Scholar - [22]Kasai, T., and Iwata, S., Gradually Intractable Problems and Nondeterministic Log-Space Lower Bounds,
*Math. Systems Theory 18*(1985), 153–170.Google Scholar - [23]Keller, R.M.,
*Vector Replacement Systems: A Formalism for Modelling Asynchronous Systems*, TR 117, (Princeton University, CSL, 1972).Google Scholar - [24]Kosaraju, R., Decidability of Reachability in Vector Addition Systems,
*Proceedings of the 14th Annual ACM Symposium on Theory of Computing*(1982), pp. 267–280.Google Scholar - [25]Landweber, L. and Robertson, E., Properties of Conflict-Free and Persistent Petri Nets,
*JACM 25*, 3 (1978), pp. 352–364.Google Scholar - [26]Lipton, R., The Reachability Problem Requires Exponential Space, Yale University, Dept. of CS., Report No. 62, Jan., 1976.Google Scholar
- [27]Mayr, E. and Meyer, A., The Complexity of the Finite Containment Problem for Petri Nets,
*JACM 28*, 3 (1981), pp. 561–576.Google Scholar - [28]Mayr, E. and Meyer, A., The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals,
*Advances in Mathematics 46*(1982), pp. 305–329.Google Scholar - [29]Mayr, E., An Algorithm for the General Petri Net Reachability Problem,
*SIAM J. Comput. 13*, 3 (1984), pp. 441–460. A preliminary version of this paper was presented at the*13th Annual Symposium on Theory of Computing*, 1981.Google Scholar - [30]Mayr, E., Persistence of Vector Replacement Systems is Decidable,
*Acta Informatica 15*(1981), pp. 309–318.CrossRefGoogle Scholar - [31]McAloon, K., Petri Nets and Large Finite Sets,
*Theoret. Comp. Sci. 32*(1984), pp. 173–183.Google Scholar - [32]Minsky, M.,
*Computation: Finite and Infinite Machines,*(Prentice Hall, Englewood Cliffs, NJ, 1967).Google Scholar - [33]Müller, H., Decidability of Reachability in Persistent Vector Replacement Systems,
*Proceedings of the 9th Symposium on Mathematical Foundations of Computer Science*, LNCS 88 (1980), pp. 426–438.Google Scholar - [34]Müller, H., Weak Petri Net Computers for Ackermann Functions,
*Elektronische Informationsverarbeitung und Kybernetik 21*(1985), 236–244.Google Scholar - [35]Peterson, J.,
*Petri Net Theory and the Modeling of Systems,*(Prentice Hall, Englewood Cliffs, NJ, 1981).Google Scholar - [36]Rackoff, C., The Covering and Boundedness Problems for Vector Addition Systems,
*Theoret. Comp. Sci. 6*(1978), pp. 223–231.Google Scholar - [37]Rosier, L. and Yen, H., A Multiparameter Analysis of the Boundedness Problem for Vector Addition Systems,
*J. of Computer and System Sciences 32*, 1 (February 1986), pp. 105–135.Google Scholar - [38]Savitch, W., Relationships between Nondeterministic and Deterministic Tape Complexities,
*J. of Computer and System Sciences 4*(1970), 177–192.Google Scholar - [39]Stockmeyer, L., The Polynomial-Time Hierarchy,
*Theoret. Comp. Sci. 3*(1977), 1–22.Google Scholar - [40]Valk, R. and Vidal-Naquet, G., Petri Nets and Regular Languages,
*J. of Computer and System Sciences 23*(1981), pp. 299–325.CrossRefGoogle Scholar