Language equations with symmetric difference

Abstract

Systems of language equations used by Ginsburg and Rice ("Two families of languages related to ALGOL", JACM, 1962) to represent context-free grammars are modified to use the symmetric difference operation instead of union. Contrary to a natural expectation that these two types of equations should have incomparable expressive power, it is shown that equations with symmetric difference can express every recursive set by their unique solutions, every recursively enumerable set by their least solutions and every co-recursively-enumerable set by their greatest solutions. The solution existence problem is II 1 -complete, the existence of a unique, a least or a greatest solution is ∏ 2 -complete, while the existence of finitely many solutions is Σ 3 -complete. © Springer-Verlag Berlin Heidelberg 2006.