Complexity of Linear Subsequences of Fibonacci-Automatic Sequences
🤖 AI Summary
This study investigates the state complexity and runtime complexity of automata recognizing linear subsequences and associated arithmetic relations within Fibonacci-automatic sequences. Building upon Zeckendorf representations and Büchi arithmetic, the work constructs automata that recognize fundamental arithmetic relations and establishes a refined framework for complexity analysis. The authors improve upon recent upper bounds proposed by Bosma and Don, providing precise characterizations of the state complexity for several classes of arithmetic-relation automata. Through concrete examples, the paper demonstrates the effectiveness and superiority of the proposed methodology in automaton construction, offering both theoretical advancements and practical validation in the domain of automatic sequences and their computational representations.
Application Category
📝 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 Büchi arithmetic to actually construct some of the studied automata recognizing relations.
Problem
Research questions and friction points this paper is trying to address.
Fibonacci-automatic sequences
state complexity
linear subsequences
Zeckendorf representation
automata
Innovation
Methods, ideas, or system contributions that make the work stand out.
Fibonacci-automatic sequences
state complexity
Zeckendorf representation
Büchi arithmetic
automata construction
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