Daniel Goc

AUTOMATIC THEOREM-PROVING IN COMBINATORICS ON WORDS

We describe a technique for mechanically proving certain kinds of theorems in combinatorics on words, using finite automata and a software package for manipulating them. We illustrate our technique by applying it to (a) solve an open problem of Currie and Saari on the lengths of unbordered factors in the Thue-Morse sequence; (b) verify an old result of Prodinger and Urbanek on the regular paperfolding sequence; (c) find an explicit expression for the recurrence function for the Rudin-Shapiro sequence; and (d) improve the avoidance bound in Leechs squarefree sequence. We also introduce a new measure of infinite words called condensation and compute it for some famous sequences. We follow up on the study of Currie and Saari of least periods of infinite words. We show that the characteristic sequence of least periods of a k-automatic sequence is (effectively) k-automatic. We compute the least periods for several famous sequences. Many of our results were obtained by machine computations.

On the Number of Unbordered Factors

We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length.

Automatic theorem-proving in combinatorics on words

We describe a technique for mechanically proving certain kinds of theorems in combinatorics on words, using finite automata and a package for manipulating them. We illustrate our technique by applying it to (a) solve an open problem of Currie and Saari on the lengths of unbordered factors in the Thue-Morse sequence; (b) verify an old result of Prodinger and Urbanek on the paperfolding sequence and (c) find an explicit expression for the recurrence function for the Rudin-Shapiro sequence. All results were obtained by machine computations.

Primitive Words and Lyndon Words in Automatic and Linearly Recurrent Sequences

We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the function counting the number of primitive factors (resp., Lyndon factors) of length n in a k-automatic sequence is k-regular. Finally, we show that the number of Lyndon factors of a linearly recurrent sequence is bounded.

Subword Complexity and k-Synchronization

We show that the subword complexity function ρ x (n), which counts the number of distinct factors of length n of a sequence x, is k-synchronized in the sense of Carpi if x is k-automatic. As an application, we generalize recent results of Goldstein. We give analogous results for the number of distinct factors of length n that are primitive words or powers. In contrast, we show that the function that counts the number of unbordered factors of length n is not necessarily k-synchronized for k-automatic sequences.

BitDrones: Towards Levitating Programmable Matter Using Interactive 3D Quadcopter Displays

In this paper, we present BitDrones, a platform for the construction of interactive 3D displays that utilize nano quadcopters as self-levitating tangible building blocks. Our prototype is a first step towards supporting interactive mid-air, tangible experiences with physical interaction techniques through multiple building blocks capable of physically representing interactive 3D data.

NONDETERMINISTIC STATE COMPLEXITY OF PROPORTIONAL REMOVALS

The language

State complexity of permutation on finite languages over a binary alphabet

\frac{1}{2}(L)

A New Approach to the Paperfolding Sequences

consists of first halfs of strings in L. Many other variants of a proportional removal operation have been considered in the literature and a characterization of remov...

Nondeterministic State Complexity of Proportional Removals

Abstract The set of all strings Parikh equivalent to a string in a language L is called the permutation of L. The permutation of a finite n-state DFA (deterministic finite automaton) language over a binary alphabet can be recognized by a DFA with n 2 − n + 2 2 states. We show that if the language consists of equal length binary strings the bound can be improved to f ( n ) = n 2 + n + 1 3 and for every n congruent to 1 modulo 3 there exists an n-state DFA A recognizing a set of equal length strings such that the minimal DFA for the permutation of L ( A ) needs f ( n ) states.