Balanced-by-Construction Regular and ω-Regular Languages

Abstract

Parenn is the typical generalization of the Dyck language to multiple types of parentheses. We generalize its notion of balancedness to allow parentheses of different types to freely commute. We show that balanced regular and [Formula: see text]-regular languages can be characterized by syntactic constraints on regular and [Formula: see text]-regular expressions and, using the shuffle on trajectories operator, we define grammars for balanced-by-construction expressions with which one can express every balanced regular and [Formula: see text]-regular language.

Balanced-By-Construction Regular and $$\omega $$-Regular Languages

“…We note that all finite words and balanced regular languages are bounded by default; boundedness is only a restriction on infinite words and ω-languages. 3 Boundedness…”

“…The grammar in Figure 2 can be extended with ω as in (1) to obtain an expression grammar Ω ¡ for balanced ω-regular languages [3,Appendix B].…”