Linearly bounded infinite graphs

TL;DR

This paper introduces linearly bounded graphs as a new class of infinite automata that capture the behavior of linearly bounded Turing machines, providing structural insights and comparisons with rational graphs.

Contribution

It defines linearly bounded graphs, explores their properties and characterizations, and compares them to rational graphs, establishing a hierarchy in the context-sensitive language acceptance.

Findings

Linearly bounded graphs accept the same languages as linearly bounded machines.

In bounded-degree cases, rational graphs form a strict subset of linearly bounded graphs.

The paper offers alternative characterizations via rewriting systems and transductions.

Abstract

Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs.