Abstract not found

Abstract not found

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.

We settle an open problem, the inclusion problem for pattern languages [1, 2]. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue [16] and is important also, for instance, in recent work concerning inductive inference and learning. Our results concern both erasing and nonerasing patterns. Categories and Subject Descriptors: F.4.3 [Mathematical Logic and Formal Languages ]: Formal Languages --- Decision problems, Algebraic language theory; F.4.1 [Mathe- matical Logic and Formal Languages]: Mathematical Logic --- Computability theory. General Terms: Theory, Formal Languages Additional Key Words and Phrases: Patterns, Inclusion problems, Equivalence problems, Descriptive patterns, Unavoidable patterns 1 Introduction. The main result Instead of an exhaustive definition for a language, [7], it is sometimes better to give more leeway in the defi...

A pattern is a finite string of constant and variable symbols. The language generated by a pattern is the set of all strings of constant symbols which can be obtained from the pattern by substituting non-empty strings for variables. We study the learnability of one-variable pattern languages in the limit with respect to the update time needed for computing a new single hypothesis and the expected total learning time taken until convergence to a correct hypothesis. Our results are as follows. First, we design a consistent and set-driven learner that, using the concept of descriptive patterns, achieves update time O(n 2 log n), where n is the size of the input sample. The best previously known algorithm for computing descriptive one-variable patterns requires time O(n 4 log n) (cf. Angluin [2]). Second, we give a parallel version of this algorithm that requires time O(log n) and O(n 3 = log n) processors on an EREW-PRAM. Third, using a modified version of the sequential algorithm a...

Abstract. Improved upper and lower bounds on the number of squarefree ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised Brinkhuis triples. The problem of finding such triples can essentially be reduced to a combinatorial problem, which can efficiently be treated by computer. In particular, it is shown that the number of square-free ternary words of length n grows at least as 65 n/40, replacing the previous best lower bound of 2 n/17. 1.

An optimal O(log log n) time concurrent-read concurrent-write parallel algorithm for detecting all squares in a string is presented. A tight lower bound shows that over general alphabets this is the fastest possible optimal algorithm. When p processors are available the bounds become \Theta(d n log n p e + log log d1+p=ne 2p). The algorithm uses an optimal parallel string-matching algorithm together with periodicity properties to locate the squares within the input string.

Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters. Résumé Il est possible de construire des mots de longueur infinie sans carré sur un alphabet ayant au moins trois lettres. Nous démontrons que l’entropie du langage des mots sans carré sur un tel alphabet est strictement positive et l’encadrons par des bornes inférieure et supérieure raisonnables. Enfin, nous donnons pour l’entropie une expression approchée qui est asymptotiquement correcte et converge rapidement lorsque le nombre de lettres de l’alphabet tend vers l’infini.

. For every string inclusion relation there are two optimization problems: nd a longest string included in every string of a given nite language, and nd a shortest string including every string of a given nite language. As an example, the two well-known pairs of problems, the longest common substring (or subsequence) problem and the shortest common superstring (or supersequence) problem, are interpretations of these two problems. In this paper we consider a class of opposite problems connected with string non-inclusion relations: nd a shortest string included in no string of a given nite language and nd a longest string including no string of a given nite language. The predicate \string is not included in string " is interpreted either as \ is not a substring of " or as \ is not a subsequence of ". The main purpose is to determine the complexity status of the string non-inclusion optimization problems. Using graph approaches we present polynomial-time algorith...

Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the two-letter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2-avoidable. From this we deduce the complete classification of unavoidable binary patterns. We also study the concept of avoidability by iterated morphisms and prove that there are a few 2-avoidable patterns which are not avoided by any iterated morphism. 1 Introduction The concept of unavoidable pattern was introduced by Bean, Ehrenfeucht & McNulty [2] and independently by Zimin [10]. They gave a characterization of unavoidable patterns, but there is still no characterization of k-unavoidable patterns, i.e. patterns that are unavoidable over a k-letter alphabet. However, we can try to find all k-unavoidable patterns that can be written with a given alphabet. The case of unary patterns (in other terms: powers of a ...