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

## 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.

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