On Bounded Semilinear Languages, Counter Machines, and Finite-Index ET0L

Abstract

We show that for every trio $\mathcal{L}$ containing only semilinear languages, all bounded languages in $\mathcal{L}$ can be accepted by one-way nondeterministic reversal-bounded multicounter machines ($\textsf {NCM}$), and in fact, even by the deterministic versions of these machines $(\textsf {DCM})$. This implies that for every semilinear trio (where these properties are effective), it is possible to decide containment, equivalence, and disjointness concerning its bounded languages. We also provide a relatively simple condition for when the bounded languages in a semilinear trio coincide exactly with those accepted by $\textsf {DCM}$ machines. This is applied to finite-index $\textsf {ET0L}$ systems, where we show that the bounded languages generated by these systems are exactly the bounded languages accepted by $\textsf {DCM}$. We also define, compare, and characterize several other types of languages that are both bounded and semilinear.

The research of O.H. Ibarra was supported, in part, by NSF Grant CCF-1117708. The research of I. McQuillan was supported, in part, by Natural Sciences and Engineering Research Council of Canada Grant 327486-2010.