Algebraic properties of structured context-free languages: old approaches and novel developments

The historical research line on the algebraic properties of structured CF languages initiated by McNaughton's Parenthesis Languages has recently attracted much renewed interest with the Balanced Languages, the Visibly Pushdown Automata languages (VPDA), the Synchronized Languages, and the Height-deterministic ones. Such families preserve to a varying degree the basic algebraic properties of Regular languages: boolean closure, closure under reversal, under concatenation, and Kleene star. We prove that the VPDA family is strictly contained within the Floyd Grammars (FG) family historically known as operator precedence. Languages over the same precedence matrix are known to be closed under boolean operations, and are recognized by a machine whose pop or push operations on the stack are purely determined by terminal letters. We characterize VPDA's as the subclass of FG having a peculiarly structured set of precedence relations, and balanced grammars as a further restricted case. The non-counting invariance property of FG has a direct implication for VPDA too.