Characterizations of the Dyck Sets

— Two représentations of the Dyck sets, using Utngucige-iJworeni opérations, are presented. Two représentations of Dyck sets in terms of simpler languages are given here. First, the Dyck set on k letters (k ^ 2) is shown to be definable from the Dyck set on one letter by use of language-theoretic opérations. Second, membership in the Dyck set on one letter is expressed in terms of the language { 0"l" : // ^ 0 }. In both cases complémentation is used, along with some of the AFL opérations ; use of complémentation is necessary. In Section 1, some basic définitions from formai language theory are reviewed, including the définition of the Dyck sets. In Section 2, the Dyck set on two letters is shown to be equal to a certain language that is defined from the Dyck set on one letter by application of the opérations of inverse homomorphism, product, union and complémentation (Theorem 2.3). The method of définition generalizes easily to the Dyck set on k letters for any k ^ 2. Deterministic automata that accept the Dyck sets have been described elsewhere [9, 13]. The représentation of the Dyck set on two letters that will be given here can be used to simplify the mode of opération of such automata. One result of these constructions is a représentation of the context-free languages using two-way counter machines and length-preserving homomorphism. In Section 3 the Dyck set on one letter is considered. By making use of a known characterization of that set, it is shown that the Dyck set on one letter can be defined from { 0"T : n ^ 0 } by application of the Boolean and AFL opérations. The results are summarized in Theorem 3.2. (*) Received : December 1975. Revised : June 1976. l) This research was supported in part by the National Science Foundation under Grants DCR-75-15945, and MCS-76-05744. () Department of System Science, University of California, Los Angeles, California, USA and Computer Systems Laboratory, University of California, Santa Barbara, California, USA. R.A.I.R.O. Informatique théorique/Theoretical Computer Science, vol. 11, n° 1, 1977