On Families of Full Trios Containing Counter Machine Languages

TL;DR

This paper investigates families of automata with reversal-bounded counters constrained by fixed patterns, providing characterizations and comparisons that connect to key language classes and decidability properties.

Contribution

It introduces a framework for analyzing automata with pattern-specified counter behaviors, identifying minimal full trios containing important language classes.

Findings

Characterization of the smallest full trio containing bounded semilinear languages

Identification of the smallest full trio containing bounded context-free languages

Analysis of decidability properties within this framework

Abstract

We look at nondeterministic finite automata augmented with multiple reversal-bounded counters where, during an accepting computation, the behavior of the counters is specified by some fixed pattern. These patterns can serve as a useful "bridge" to other important automata and grammar models in the theoretical computer science literature, thereby helping in their study. Various pattern behaviors are considered, together with characterizations and comparisons. For example, one such pattern defines exactly the smallest full trio containing all the bounded semilinear languages. Another pattern defines the smallest full trio containing all the bounded context-free languages. The "bridging" to other families is then applied, e.g. to certain Turing machine restrictions, as well as other families. Certain general decidability properties are also studied using this framework.