Decidability for Sturmian words

Abstract

We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order expansions of Presburger arithmetic by a single Sturmian word are uniformly $\omega$-automatic, and then deduce the decidability of the theory of the class of such structures. Using an implementation of this decision algorithm called Pecan, we automatically reprove classical theorems about Sturmian words in seconds, and are able to obtain new results about antisquares and antipalindromes in characteristic Sturmian words.

References in corpus (6)

• Three Lectures on Automatic Structures
• Ostrowski numeration systems, addition and finite automata
• Pecan: An Automated Theorem Prover for Automatic Sequences using Büchi Automata
• Every nonnegative real number is an abelian critical exponent
• Presburger Arithmetic with algebraic scalar multiplications
• The Additive Structure of Integers with the Lower Wythoff Sequence

Cited by in corpus (1)

• Pecan: An Automated Theorem Prover for Automatic Sequences using Büchi Automata