Completeness results for conflict-free vector replacement systems

摘要

In this paper, we give completeness results for the reachability, containment, and equivalence problems for conflict-free vector replacement systems (VRSs). We first give an NP algorithm for deciding reachability, thus giving the first primitive recursive algorithm for this problem. Since Jones, Landweber, and Lien have shown this problem to be NP-hard, it follows that the problem is NP-complete. Next, we show as our main result that the containment and equivalence problems are ∏2P-complete, where ∏2P is the set of all languages whose complements are in the second level of the polynomial-time hierarchy. In showing the upper bound, we first show that the reachability set has a semilinear set (SLS) representation that is exponential in the size of the problem description, but which has a high degree of symmetry. We are then able to utilize in part a strategy introduced by Huynh (concerning SLSs) to complete our upper bound proof.