Balanced-by-construction regular and ω -regular languages

Paren_n is the typical generalisation of the Dyck language to multiple types of parentheses. We generalise its notion of balancedness to allow parentheses of different types to freely commute. We show that balanced regular and ω -regular languages can be characterised by syntactic constraints on regular and ω -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 ω -regular language.