Completeness results for conflict-free vector replacement systems
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 [\textit{N. D. Jones}, \textit{L. H. Landweber} and \textit{Y. E. Lien}, Theor. Comput. Sci., 277-299 (1977; Zbl 0357.68048)], it follows that the problem is NP- complete. Next, we show as our main result that the containment and equivalence problems are \(\Pi^ p_ 2\)-complete, where \(\Pi^ p_ 2\) 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 \textit{Thiet-Dung Huynh} [Electron. Informationsverarbeitung Kybernetik 18, 291-338 (1982; Zbl 0519.68060)] (concerning SLSs) to complete our upper bound proof.
conflict-free vector replacement systems
scientific article; zbMATH DE number 4080930
scientific article