Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

TL;DR

This paper investigates the state complexity of automata recognizing arithmetic relations in Fibonacci representation, introduces new automata constructions, and improves existing bounds in the field of Fibonacci-automatic sequences.

Contribution

It presents new automata constructions for Fibonacci-automatic sequences, analyzes their state complexity, and improves bounds on related automata recognition problems.

Findings

Automata recognizing Fibonacci-based relations have bounded state complexity.

New automata constructions improve previous bounds in the literature.

The paper discusses both theoretical and practical aspects of automata recognition in Fibonacci representation.

Abstract

We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of B\"uchi arithmetic to actually construct some of the studied automata recognizing relations.