Star-Freeness, First-Order Definability and Aperiodicity of Structured Context-Free Languages

A classic result in formal language theory is the equivalence among aperiodic finite automata, star-free regular expressions, and first-order logic on words. Extending these results to structured subclasses of context-free languages, such as tree languages, did not work as smoothly: there are star-free tree languages that are counting. We argue that investigating the same properties within the family of operator precedence languages (OPLs) by going back to string languages rather than tree languages may lead to equivalences that perfectly match those on regular languages. We define operator precedence expressions; we show that they define exactly the class of OPLs and that, when restricted to the star-free subclass, coincide with first-order definable OPLs and are aperiodic. Since operator precedence languages strictly include other classes of structured languages such as visibly pushdown languages, the same results given in this paper hold as trivial corollary for that family too.