Jeffrey Shallit

The Ubiquitous Prouhet-Thue-Morse Sequence

We discuss a well-known binary sequence called the Thue-Morse sequence, or the Prouhet-Thue-Morse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The Prouhet-Thue-Morse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.

The ring of k -regular sequences, II

In this paper, we continue our study of k-regular sequences begun in 1992. We prove some new results, give many new examples from the literature, and state some open problems.

A Second Course in Formal Languages and Automata Theory

Intended for graduate students and advanced undergraduates in computer science, A Second Course in Formal Languages and Automata Theory treats topics in the theory of computation not usually covered in a first course. After a review of basic concepts, the book covers combinatorics on words, regular languages, context-free languages, parsing and recognition, Turing machines, and other language classes. Many topics often absent from other textbooks, such as repetitions in words, state complexity, the interchange lemma, 2DPDAs, and the incompressibility method, are covered here. The author places particular emphasis on the resources needed to represent certain languages. The book also includes a diverse collection of more than 200 exercises, suggestions for term projects, and research problems that remain open.

UNARY LANGUAGE OPERATIONS, STATE COMPLEXITY AND JACOBSTHAL'S FUNCTION

In this paper we give the cost, in terms of states, of some basic operations (union, intersection, concatenation, and Kleene star) on regular languages in the unary case (where the alphabet contains only one symbol). These costs are given by explicitly determining the number of states in the noncyclic and cyclic parts of the resulting automata. Furthermore, we prove that our bounds are optimal. We also present an interesting connection to Jacobsthals function from number theory.

A lower bound technique for the size of nondeterministic finite automata

In this note, we prove a simple theorem that provides a lower bound on the size of nondeterministic finite automata which accept a given regular language.

Numeration systems, linear recurrences, and regular sets

A numeration system based on a strictly increasing sequence of positive integers u0=1, u1u2,... expresses a non-negative integer n as a sum n=∑ j=0 i ajuj. In this case we say the string a i a i −1 ...a1 a0 is a representation for n.

Characterizing Regular Languages with Polynomial Densities

A language L is said to have a polynomial density if the function pL.(n)=¦L∩∑n¦ of L is bounded by a polynomial. We show that the function p R(n) of a regular language R is O(n k ), for some k≥0, if and only if R can be represented as a finite union of the regular expressions of the form xy 1 * z1 ...y t * zt with a nonnegative integer t≤k+1, where x,y 1,z 1,..., yt, zt are all strings in ∑*.

On the iteration of certain quadratic maps over GF(p)

We consider the properties of certain graphs based on iteration of the quadratic maps x->x^2 and x->x^2-2 over a finite field GF(p).

Polynomial versus exponential growth in repetition-free binary words

It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7/3. More precisely, there are only polynomially many binary words of length n that avoid 7/3-powers, but there are exponentially many binary words of length n that avoid 7/3+-powers. This answers an open question of Kobayashi from 1986.

Automaticity I

Let?and?be nonempty alphabets with?finite. Letfbe a function mapping?* to?. We explore the notion ofautomaticity, which attempts to model how “close”fis to a finite-state function. Formally, the automaticity offis a functionAf(n) which counts the minimum number of states in any deterministic finite automaton that computesfcorrectly on all strings of length ?n(and its behavior on longer strings is not specified). We defineAL(n) for languagesLto beA?L(n), where?Lis the characteristic function ofL. The same or similar notions were examined previously by Trakhtenbrot, Grinberg and Korshunov, Karp, Breitbart, Gabarr?, Dwork and Stockmeyer, and Kaneps and Freivalds. Karp proved that ifL??* is not regular, thenAL(n)?(n+3)/2 infinitely often. We prove that the lower bound is best possible. We obtain results on the growth rate ofAf(n). If |?|=k?2 and |?|=l 0 we haveAf(n)>(1??)Ckn+1/nfor all sufficiently largen. We also obtain bounds onNL(n), the nondeterministic automaticity function. This is similar toAf(n), except that it counts the number of states in the minimal NFA, and it is defined for languagesL??*. For |?|=k?2, we haveNL(n)=O(kn/2). Also, for almost all languagesLand every?>0 we have formula]for all sufficiently largen. We prove some incomparability results between the automaticity measure and those defined earlier by Gabarr? and others. Finally, we examine the notion of automaticity as applied to sequences.