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

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.

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