STACS 1987: STACS 87 pp 360-370

# On the complexity of containment, equivalence, and reachability for finite and 2-dimensional vector addition systems with states

• Rodney R. Howell
• Dung T. Huynh
• Louis E. Rosier
• Hsu-Chun Yen
Contributed Papers Net Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 247)

## Abstract

In this paper, we analyze the complexity of the reachability, containment, and equivalence problems for two classes of vector addition systems with states (VASSs): finite VASSs and 2-dimensional VASSs. Both of these classes are known to have effectively computable semilinear reachability sets (SLSs). By giving upper bounds on the sizes of the SLS representations, we achieve upper bounds on each of the aforementioned problems. In the case of finite VASSs, the SLS representation is simply a listing of the reachability set; therefore, we derive a bound on the norm of any reachable vector based on the dimension, number of states, and amount of increment caused by any move in the VASS. The bound we derive shows an improvement of two levels in the primitive recursive hierarchy over results previously obtained by McAloon, thus answering a question posed by Clote. We then show this bound to be optimal. In the case of 2-dimensional VASSs,we analyze an algorithm given by Hopcroft and Pansiot that generates a SLS representation of the reachability set. Specifically, we show that the algorithm operates in $$2^{2^{c*l*n} }$$ nondeterministic time, where l is the length of the binary representation of the largest integer in the VASS, n is the number of transitions, and c is some fixed constant. We also give examples for which this algorithm will take $$2^{2^{d*l*n} }$$ nondeterministic time for some positive constant d. Finally, we give a method of determinizing the algorithm in such a way that it requires no more than $$2^{2^{c*l*n} }$$ deterministic time. From this upper bound and special properties of the generated SLSs, we derive upper bounds of DTIME($$2^{2^{c*l*n} }$$) for the three problems mentioned above.

## References

1. [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. [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. [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. [4]
Chandra, A., Kozen, D. and Stockmeyer, L., Alternation, JACM 28, 1 (January 1981), pp. 114–133.Google Scholar
5. [5]
Clote, P., The Finite Containment Problem for Petri Nets, Theoret. Comp. Sci. 43 (1986), 99–106.Google Scholar
6. [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. [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. [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. [9]
Grabowski, J., The Decidability of Persistence for Vector Addition Systems, Information Processing Letters 11, 1 (1980), pp. 20–23.Google Scholar
10. [10]
Hack, M., The Equality Problem for Vector Addition Systems is Undecidable, Theoret. Comp. Sci. 2 (1976), pp. 77–95.Google Scholar
11. [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. [12]
Howell, R., Huynh, D., Rosier, L., and Yen, H., Some Complexity Bounds for Problems Concerning Finite and 2-Dimensional Vector Addition Systems with States, to appear in Theoret. Comp. Sci. Google Scholar
13. [13]
Huynh, D., The Complexity of Semilinear Sets, Elektronische Informationsverarbeitung und Kybernetik 18 (1982), pp. 291–338.Google Scholar
14. [14]
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
15. [15]
Huynh, D., A Simple Proof for the Σ2P Upper Bound of the Inequivalence Problem for Semilinear Sets, Elektronische Informationsverarbeitung und Kybernetik 22 (1986), pp. 147–156.Google Scholar
16. [16]
Karp, R. and Miller, R., Parallel Program Schemata, J. of Computer and System Sciences 3, 2 (1969), pp. 147–195.Google Scholar
17. [17]
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
18. [18]
Landweber, L. and Robertson, E., Properties of Conflict-Free and Persistent Petri Nets, JACM 25, 3 (1978), pp. 352–364.Google Scholar
19. [19]
Lipton, R., The Reachability Problem Requires Exponential Space, Yale University, Dept. of CS., Report No. 62, Jan., 1976.Google Scholar
20. [20]
Mayr, E., Persistence of Vector Replacement Systems is Decidable, Acta Informatica 15 (1981), pp. 309–318.Google Scholar
21. [21]
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
22. [22]
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
23. [23]
McAloon, K., Petri Nets and Large Finite Sets, Theoret. Comp. Sci. 32 (1984), pp. 173–183.Google Scholar
24. [24]
Mayr, E. and Meyer, A., The Complexity of the Finite Containment Problem for Petri Nets, JACM 28, 3 (1981), pp. 561–576.Google Scholar
25. [25]
Muller, 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
26. [26]
Muller, H., Weak Petri Net Computers for Ackermann Functions, to appear in Elektronische Informationsverarbeitung und Kybernetik.Google Scholar
27. [27]
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
28. [28]
Rosier, L. and Yen, H., Logspace Hierarchies, Polynomial Time and the Complexity of Fairness Problems Concerning ω-Machines, Proceedings of the 3rd Annual Symposium on Theoretical Aspects of Computer Science, LNCS 210 (1986), pp. 306–320. To appear in SIAM J. Comput.Google Scholar
29. [29]
Rosier, L. and Yen, H., On the Complexity of Deciding Fair Termination of Probabilistic Concurrent Finite-State Programs, Proceedings of the 13th International Colloquium on Automata, Languages and Programming, LNCS 226 (1986), 334–343.Google Scholar
30. [30]
Van Leeuwen, J., A Partial Solution to the Reachability Problem for Vector Addition Systems, Proceedings of the 6th Annual ACM Symposium on Theory of Computing (1974), pp. 303–309.Google Scholar
31. [31]
Valk, R. and Vidal-Naquet, G., Petri Nets and Regular Languages, J. of Computer and System Sciences 23 (1981), pp. 299–325.Google Scholar
32. [32]
Yamasaki, H., On Weak Persistency of Petri Nets, Information Processing Letters 13, 3 (1981), pp. 94–97.Google Scholar

## Authors and Affiliations

• Rodney R. Howell
• 1
• Dung T. Huynh
• 2
• Louis E. Rosier
• 1
• Hsu-Chun Yen
• 3
1. 1.Department of Computer SciencesUniversity of Texas at AustinAustin
2. 2.Computer Science DepartmentUniversity of Texas at DallasRichardson
3. 3.Department of Computer ScienceIowa State UniversityAmes