On some families of languages related to the Dyck language

• Authors: Maurice Nivat
• Venue: STOC '70 Proceedings of the second annual ACM symposium on Theory of computing
• Year: 1970

We recall that the Dyck language on the alphabet of 2n letters X &equil; {x1,...,xn,&xmarc;1,...,&xmarc;n} is the equivalence class of the empty word 1 &egr; X@@@@ modulo the Thue congruence generated on X@@@@ by the 2n relations xi&xmarc;i &equil; &xmarc;ixi &equil; 1 i &egr; {1,...,n}. Indeed for all n &egr; X@@@@ the equivalence class of n modulo this congruence is a non ambiguous algebraic (ie. context-free) language, the complement of which is also a non ambiguous algebraic language. The same situation is true for the congruence generated on X@@@@ by the n relations xi&xmarc;i &equil; 1 i &egr; {1,...,n}. The author convinced himself that many other congruences have the same property [Ni 1], and undertook the task of finding them systematically. Here is presented a part of the results of this undertaking. An important family of congruences is brought to light which have interesting decidability properties: in the constructions leading to these decidability properties we used as a guide line the nice paper of Mac Naughton [McN]. Sufficient conditions are given in order that the equivalence classes (and their complements) of such a congruence be an algebraic language. We do not study in this paper the properties of these languages, this will be done elsewhere [Ni 2].