State Complexity of Shifts of the Fibonacci Word
🤖 AI Summary
This work investigates the state complexity of automata that generate shifted Fibonacci words \( f_{i+c} \) when the input index \( i \) is given in Zeckendorf representation. Considering both most-significant-digit-first (msd-first) and least-significant-digit-first (lsd-first) input orders, the authors combine state complexity analysis with techniques from Diophantine approximation to construct automata requiring only \( O(\log c) \) states for any fixed shift \( c \). This bound closely approaches the information-theoretic lower bound for aperiodic sequences, thereby revealing deep structural regularities inherent in shifted Fibonacci words under Zeckendorf encoding. The result not only provides a constructive proof of the feasibility of such compact automata but also offers a theoretical foundation for optimizing automaton-based generation of these symbolic sequences.
Application Category
📝 Abstract
The Fibonacci infinite word ${\bf f} = (f_i)_{i \geq 0} = 01001010\cdots$ is one of the most celebrated objects in combinatorics on words. There is a simple $5$-state automaton that, given $i$ in lsd-first Zeckendorf representation, computes its $i$'th term $f_i$, and a $2$-state automaton for msd-first. In this paper we consider the state complexity of the automaton generating the shifted sequence $(f_{i+c})_{i \geq 0}$, and show that it is $O(\log c)$ for both msd-first and lsd-first input. This is close to the information-theoretic minimum for an aperiodic sequence. The techniques involve a mixture of state complexity techniques and Diophantine approximation.
Problem
Research questions and friction points this paper is trying to address.
state complexity
Fibonacci word
shifted sequence
Zeckendorf representation
automaton
Innovation
Methods, ideas, or system contributions that make the work stand out.
state complexity
Fibonacci word
Zeckendorf representation
Diophantine approximation
automatic sequences
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