Ostrowski numeration systems, addition and finite automata

We present an elementary three pass algorithm for computing addition in Ostrowski numeration systems. When $a$ is quadratic, addition in the Ostrowski numeration system based on $a$ is recognizable by a finite automaton. We deduce that a subset of $X\subseteq \mathbb{N}^n$ is definable in $(\mathbb{N},+,V_a)$, where $V_a$ is the function that maps a natural number $x$ to the smallest denominator of a convergent of $a$ that appears in the Ostrowski representation based on $a$ of $x$ with a non-zero coefficient, if and only if the set of Ostrowski representations of elements of $X$ is recognizable by a finite automaton. The decidability of the theory of $(\mathbb{N},+,V_a)$ follows

Decidability for Sturmian Words

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 ?-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

Decision Algorithms for Ostrowski-Automatic Sequences

We extend the notion of automatic sequences to a broader class, the Ostrowski-automatic sequences. We develop a procedure for computationally deciding certain combinatorial and enumeration questions about such sequences that can be expressed as predicates in first-order logic. In Chapter 1, we begin with topics and ideas that are preliminary to this work, including a small introduction to non-standard positional numeration systems and the relationship between words and automata. In Chapter 2, we define the theoretical foundations for recognizing addition in a generalized Ostrowski numeration system and formalize the general theory that develops our decision procedure. Next, in Chapter 3, we show how to implement these ideas in practice, and provide the implementation as an integration to the automatic theorem-proving software package -- Walnut. Further, we provide some applications of our work in Chapter 4. These applications span several topics in combinatorics on words, including repetitions, pattern-avoidance, critical exponents of special classes of words, properties of Lucas words, and so forth. Finally, we close with open problems on decidability and higher-order numeration systems and discuss future directions for research

A Fibonacci analogue of the two's complement numeration system

Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of the two's complement notation and we show that addition in this numeration system can be performed by a deterministic finite-state transducer. The result is based on the Berstel adder, which performs addition of the usual Fibonacci representations of nonnegative integers and for which we provide a new constructive proof. Moreover, we characterize the Fibonacci-equivalent of the two's complement notation as an increasing bijection between $\mathbb{Z}$ and a particular language.Comment: v3: 21 pages, 3 figures, 3 tables. v4: 24 pages, added a new section characterizing the Fibonacci's complement numeration system as an increasing bijection. v5: changes after revie

A General Approach to Proving Properties of Fibonacci Representations via Automata Theory

We provide a method, based on automata theory, to mechanically prove the correctness of many numeration systems based on Fibonacci numbers. With it, long case-based and induction-based proofs of correctness can be replaced by simply constructing a regular expression (or finite automaton) specifying the rules for valid representations, followed by a short computation. Examples of the systems that can be handled using our technique include Brown's lazy representation (1965), the far-difference representation developed by Alpert (2009), and three representations proposed by Hajnal (2023). We also provide three additional systems and prove their validity.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Numeration systems: a bridge between formal languages and number theory

Considering an integer base b, any integer is represented by a word over a finite digit-set, its base-b expansion. In theoretical computer science, one is interested in syntactical properties of words or languages, i.e., sets of words. In this introductory talk, I will present recognizable sets of numbers : the set of their representations is accepted by a finite automaton. We will see that this property strongly depends on the choice of the numeration system. We will therefore review some fundamental questions and introduce automatic sequences. Thanks to Büchi-Bruyère theorem, first order logic and decidable theories may be used to produce automatic proofs and in particular solve, in an automated way, arithmetical problems. I will not assume any knowledge from the audience about formal languages theory

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper