Lexicographical Generation of a Generalized Dyck Language

Given two disjoint alphabets T and T ] and a relation R T T ] , the generalized Dyck language D R over T T ] consists of all words w 2 (T T ]) ? which are equivalent to the empty word " under the congruence deened by x y " mod for all (x; y) 2 R. In this paper we present an algorithm that generates all words of length 2n of the generalized Dyck language lexicograph-ically. Thereby, each Dyck word is computed from its predecessor according to the lexicographical order without any knowledge about the Dyck words generated before. Additionally, we introduce a condition on the relation R for the language to be simply generated, which means that an algorithm needs to read only the suux to be changed in order to compute the successor of a word according to the lexicographical order. Furthermore, we analyze the algorithm that generates the Dyck words. For arbitrary R, we compute the s-th moments of the random variable describing the length of the suux to be changed in the computation of the successor of a Dyck word according to the lexicographical order. 1 Overview and Deenitions In this section we introduce the generalization of the Dyck language, the lexico-graphical order needed for the lexicographical generation and present all deenitions { illustrated by several examples { for the whole paper. Further, we point out the contents of the following sections. In this paper we present an algorithm that generates all words of length 2n of the generalized Dyck language given in Deenition 1 lexicographically. The algorithm reads a word from right to left and changes a suux of that word in order to generate the next word according to the lexicographical order given in Deenition 2. be the set of opening (resp. closing) brackets. Let jSj be the cardinality of the set S, so T = t 1 and T ] = t 2. With T := T. T ] , where. denotes the disjoint union of sets, and a relation R T T ] we obtain the generalized Dyck language associated with R by D R := fw 2 T ? j w " mod g , where " denotes the empty word and is the congruence over T which is deened by (8((a ; ] b) 2 R)((a ] b " mod). Let length(w) be the number of symbols …