Abstract not found

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 purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

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ively 1) if the sum of the binary digits of n is even (respectively odd). This number q can be then obtained as the unique positive solution of 1 = P 1 n=1 ffi n q \Gamman . It is equal to 1:787231650::: In the electronic abstract of [4], the authors ask whether the number q = 1:787231650::: in Theorem 1 above is irrational. The purpose of this note is to prove, as a simple consequence of a result of Mahler, that q is transcendental. Theorem 2 The number q = 1:787231650::: defined as the smallest number in (1; 2) for which there exists a unique expansion of 1 as 1 = P 1 n=1 ffi n q<F1

A sequence a = a1a2...an is said to be non-repetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is non-repetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long non-repetitive sequences. In this paper we consider a natural generalization of Thue’s sequences for colorings of graphs. A coloring of the set of edges of a given graph G is non-repetitive if the sequence of colors on any path in G is non-repetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G). The main problem we consider is the relation between the numbers π(G) and ∆(G). We show, by an application of the Lovász Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ c∆(G) 2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n − 3, and π(T) ≤ 4(∆(T) − 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.

The agent design problem is as follows: given a specification of an environment, together with a specification of a task, is it possible to construct an agent that can be guaranteed to successfully accomplish the task in the environment? In this article, we study the computational complexity of the agent design problem for tasks that are of the form "achieve this state of affairs" or "maintain this state of affairs". We consider three general formulations of these problems (in both non-deterministic and deterministic environments) that differ in the nature of what is viewed as an "acceptable" solution: in the least restrictive formulation, no limit is placed on the number of actions an agent is allowed to perform in attempting to meet the requirements of its specified task. We show that the resulting decision problems are intractable, in the sense that these are non-recursive (but recursively enumerable) for achievement tasks, and non-recursively enumerable for maintenance tasks. In the second formulation, the decision problem addresses the existence of agents that have satisfied their specified task within some given number of actions. Even in this more restrictive setting the resulting decision problems are either PSPACE-complete or NP-complete. Our final formulation requires the environment to be history independent and bounded. In these cases polynomial time algorithms exist: for deterministic environments the decision problems are NL-complete; in non-deterministic environments, P-complete.

A sequence u = u 1 u 2 :::u n is said to be nonrepetitive if no two adjacent blocks of u are exactly the same. For instance, the sequence abcbcba contains a repetition bcbc, while abcacbabcbac is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set fa; b; cg. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this

. I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorphisms, automaticity, and k-regular sequences. Key words. finite automata, automatic sequences, transcendence, automaticity AMS(MOS) subject classifications. Primary 11B85, Secondary 11A63 11A55 11J81 1. Introduction. In this paper, I survey some interesting connections between number theory and the theory of formal languages. This is a very large and rapidly growing area, and I focus on a few areas that interest me, rather than attempting to be comprehensive. (An earlier survey of this area, written in French, is [1].) I also give a number of open questions. Number theory deals with the properties of integers, and formal language theory deals with the properties of strings. At the interse...

We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, (non-)inversions, squares and p-repetitions. We give recurrence formulas in the general case, then deducing exact formulas for particular families of morphisms. Many (classical or new) examples are given illustrating the techniques and showing their interest.