[Paper Review] Closed, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences.

Luke Schaeffer, Jeffrey Shallit|arXiv (Cornell University)|Aug 9, 2015

semigroups and automata theory3 citations

TL;DR

This paper establishes that properties like being closed, rich, privileged, trapezoidal, or balanced in automatic sequences can be expressed in first-order logic, implying their characteristic functions are automatic. It introduces a new characterization of privileged words and computes these characteristic functions for key sequences such as Thue-Morse, Rudin-Shapiro, and Fibonacci.

ABSTRACT

We prove that the property of being closed (resp., rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those n for which an automatic sequence x has a closed (resp., privileged, rich, trapezoidal, balanced) factor of length n is automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, period-doubling sequence, and the Fibonacci sequence.

Motivation & Objective

• To determine whether fundamental combinatorial properties of automatic sequences can be logically formalized.
• To establish that the set of lengths n for which an automatic sequence has a closed, rich, privileged, trapezoidal, or balanced factor is itself automatic.
• To develop a new logical characterization of the privileged property to enable its expressibility in first-order logic.
• To compute the characteristic functions of these properties for prominent automatic sequences, including Thue-Morse and Rudin-Shapiro.

Proposed method

• Expressing the properties of being closed, rich, privileged, trapezoidal, and balanced using first-order logic formulas over automatic sequences.
• Introducing a novel characterization of privileged words that enables their logical expressibility in first-order logic.
• Leveraging the closure properties of automatic sequences under first-order definable predicates to show that the characteristic functions of such lengths are automatic.
• Applying known algorithms for first-order definable sets in automatic sequences to compute the characteristic functions for specific sequences.
• Using structural properties of well-known sequences—such as Thue-Morse, Rudin-Shapiro, and Fibonacci—to compute their respective characteristic functions.
• Validating the method by computing and verifying the characteristic functions for each target sequence.

Experimental results

Research questions

• RQ1 Can the property of being closed in an automatic sequence be expressed in first-order logic?
• RQ2 Is the set of lengths n for which an automatic sequence contains a rich factor of length n automatic?
• RQ3 Can the privileged property be logically characterized in a way that allows first-order expressibility in automatic sequences?
• RQ4 Are the characteristic functions of trapezoidal and balanced factors in automatic sequences automatic?
• RQ5 What are the explicit characteristic functions for prominent sequences like Thue-Morse and Rudin-Shapiro?

Key findings

• The characteristic function of lengths n for which an automatic sequence has a closed factor of length n is automatic.
• The characteristic function for lengths n with a rich factor in an automatic sequence is automatic.
• A new logical characterization of privileged words enables their expressibility in first-order logic over automatic sequences.
• The characteristic function for lengths n with a privileged factor in an automatic sequence is automatic.
• The method successfully computes the characteristic functions for the Thue-Morse, Rudin-Shapiro, ordinary paperfolding, period-doubling, and Fibonacci sequences.
• The results confirm that the logical expressibility of combinatorial properties leads to automaticity of their corresponding characteristic functions.