  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.

Formal Languages and Automata Theory