Tree-based language complexity of Thompson's group F

TL;DR

This paper investigates the language complexity of Thompson's group F by defining a normal form based on tree structures, demonstrating it can be recognized by a finite automaton with counters, and establishing a graph automatic structure.

Contribution

It introduces a novel normal form for Thompson's group F using caret types and shows it admits a 3-counter graph automatic structure, advancing understanding of its computational properties.

Findings

Normal forms based on caret types are accepted by a finite state machine with 2 counters.

Thompson's group F has a 3-counter graph automatic structure.

The approach extends the concept of graph automatic groups to F.

Abstract

The definition of graph automatic groups by Kharlampovich, Khoussainov and Miasnikov and its extension to C-graph automatic by Murray Elder and the first author raise the question of whether Thompson's group F is graph automatic. We define a language of normal forms based on the combinatorial "caret types" which arise when elements of F are considered as pairs of finite rooted binary trees, which we show to be accepted by a finite state machine with 2 counters, and forms the basis of a 3-counter graph automatic structure for the group.