Relationships between bounded languages, counter machines, finite-index grammars, ambiguity, and commutative regularity

It is shown that for every language family that is a trio containing only semilinear languages, all bounded languages in it can be accepted by one-way deterministic reversal-bounded multicounter machines (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. A condition is also provided for when the bounded languages in a semilinear trio coincide exactly with those accepted by DCM machines, and it is used to show that many grammar systems of finite index - such as finite-index matrix grammars (M-fin) and finite-index ETOL (ETOLfin) - have identical bounded languages as DCM.