Abstract not found

We introduce equality up-to constraints over finite trees and investigate their expressiveness. Equality up-to constraints subsume equality constraints, subtree constraints, and one-step rewriting constraints.

Makanin's algorithm [Ma77] shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions [P'e81, Ab87, Sc90, Ja90]. In this paper we present a version of the algorithm which probably cannot be further simplified without fundamentally new insights which exceed Makanin's original ideas. We give a transformation which is efficient, conceptually simple and applies to arbitrary generalized equations. No further subprocedure is needed for the generation of the search tree. Particular attention is then given to the proof that proper generalized equations are transformed into proper generalized equations. This point, which is important for the termination argument, was treated erroneously in other papers. We also show that a combination of the basic algorithm for string-unification (see [Pl72, Le72, Si75, Si78]...

this paper, it is shown that the subword complexity of a D0L language is bounded by cn (resp. cn log n, cn) if the morphism that generates the languages is arbitrary (resp. growing, uniform). This result was extended in [Pan84a]: Theorem 6.7. The subword complexity of an in nite word generated by iterating a morphism is of one of the following types: (n), (n log n), (n log n log n), (n ), or (1)

We prove the following two variants of the defect theorem. Let X be a finite set of words over a finite alphabet. Then if a nonperiodic bi-infinite word w has two X-factorizations, then the combinatorial rank of X is at most card(X) \Gamma 1, i.e. there exists a set F such that X ` F

. We introduce the notion of equation on trees, generalizing the corresponding notion for words, and we develop the first steps of a theory of tree equations. The main result of the paper states that, if a pair of trees is the solution of a tree equation with two indeterminates, then the two trees are both powers of the same tree. As an application, we show that a tree can be expressed in a unique way as a power of a primitive tree. This extends a basic result of combinatorics on words to trees. Some open problems are finally proposed. 1 Introduction In this paper we are mainly concerned with k-ary trees whose vertices are labeled by letters of an alphabet A. We look at a labeled k-ary tree as a generalization of a word, in the sense that words correspond to the particular case of k = 1, i.e. to unary trees. a a b c b a b a c b A binary tree A word Given that combinatorics on words is a well developed theory, with several applications to computer science, it appears to be a natural ...

Abstract. For 30 years the Lempel–Ziv factorization LZx of a string x = x[1..n] has been a fundamental data structure of string processing, especially valuable for string compression and for computing all the repetitions (runs) in x. Traditionally the standard method for computing LZx was based on Θ(n)-time (or, depending on the measure used, O(n log n)-time) processing of the suffix tree STx of x. Recently Abouelhoda et al. proposed an efficient Lempel–Ziv factorization algorithm based on an “enhanced ” suffix array – that is, a suffix array SAx together with supporting data structures, principally an “interval tree”. In this paper we introduce a collection of fast spaceefficient algorithms for LZ factorization, also based on suffix arrays, that in theory as well as in many practical circumstances are superior to those previously proposed; one family out of this collection achieves true Θ(n)-time alphabet-independent processing in the worst case by avoiding tree structures altogether. Mathematics Subject Classification (2000). 68W05. Keywords. Lempel–Ziv factorization, suffix array, suffix tree, LZ factorization.

Contents 1 Prerequisites in Combinatorics on words 1 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Codes and the defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Stable, left unitary and right unitary monoids . . . . . . . . . . . . 4 1.2.2 The defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 An algorithm for the code problem . . . . . . . . . . . . . . . . . . 6 1.3 Word Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Generalities on trees 11 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations between k-ary trees . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Tree codes 25 3.1 Factorizations on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Stable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

We prove that for any finite prefix code X with n elements, the non special subgroups in the syntactic monoid of X # have degree at most n 1. This implies in particular that the groups in the syntactic monoid of X # are all cyclic when X is a prefix code with three elements.