Relationships Between Bounded Languages, Counter Machines, Finite-Index Grammars, Ambiguity, and Commutative Regularity

TL;DR

This paper explores the relationships between bounded languages, counter machines, and grammar systems, revealing conditions under which languages are regular, ambiguous, or commutatively regular, and establishing equivalences among various language classes.

Contribution

It establishes that bounded languages in semilinear trios can be accepted by deterministic reversal-bounded multicounter machines and characterizes when these languages coincide with those accepted by finite-index grammars.

Findings

Bounded languages in semilinear trios are accepted by DCM.

Many finite-index grammar systems generate or accept counting or commutative regular languages.

Finite-index ETOL and matrix grammars can generate inherently ambiguous languages.

Abstract

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 and finite-index ETOL -- have identical bounded languages as DCM. Then connections between ambiguity, counting regularity, and commutative regularity are made, as many machines and grammars that are unambiguous can only generate/accept counting regular or commutatively regular languages. Thus, such a system that can generate/accept a non-counting regular or non-commutatively regular language implies the existence of inherently ambiguous languages over that system. In addition, it is shown that every language generated by an unambiguous finite-index matrix grammar has a rational characteristic series in commutative variables, and is counting regular. This result plus the connections are used to demonstrate that finite-index matrix grammars and finite-index ETOL can generate inherently ambiguous languages (over their grammars), as do several machine models. It is also shown that all bounded languages generated by these two grammar systems (those in any semilinear trio) can be generated unambiguously within the systems. Finally, conditions on languages generated by finite-index matrix grammars and finite-index ETOL implying commutative regularity are obtained. In particular, it is shown that every finite-index EDOL language is commutatively regular.