A Characterization of Parenthesis Languages

A parenthesis language is a context-free language possessing a grammar in which each application of a production introduces a unique pair of parentheses, delimiting the scope of that production. Parenthesis languages are nontrivial since only one kind of parenthesis is used. In this paper it is shown that algorithms exist to determine if a context-free language is a parenthesis language, or if it is equal to the language defined by a given parenthesis grammar. A possible merit of these results lies in the fact that parenthesis languages are the most general class of languages for which such problems are now known to be solvable; in fact, other problems which are very similar to the one solved here are known to be recursively unsolvable.