6 research outputs found
Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences
We prove that the property of being closed (resp., palindromic, 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., palindromic, privileged, rich, trape- zoidal, 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, the
period-doubling sequence, and the Fibonacci sequence. Finally, we also show
that the function counting the total number of palindromic factors in a prefix
of length n of a k-automatic sequence is not k-synchronized
Repetitions in infinite palindrome-rich words
Rich words are characterized by containing the maximum possible number of
distinct palindromes. Several characteristic properties of rich words have been
studied; yet the analysis of repetitions in rich words still involves some
interesting open problems. We address lower bounds on the repetition threshold
of infinite rich words over 2 and 3-letter alphabets, and construct a candidate
infinite rich word over the alphabet with a small critical
exponent of . This represents the first progress on an open
problem of Vesti from 2017.Comment: 12 page
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