On the Equivalence, Containment, and Covering Problems for the Regular and Context-Free Languages

We consider the complexity of the equivalence and containment problems for regular expressions and context-free grammars, concentrating on the relationship between complexity and various language properties. Finiteness and boundedness of languages are shown to play important roles in the complexity of these problems. An encoding into grammars of Tur ing machine computations exponential in the size of the grammar is used to prove several exponential lower bounds. These lower bounds include exponential time for testing equivalence of grammars generating finite sets, and exponential space for testing equivalence of non-self-embedding grammars. Several problems which might be complex because of this encoding are shown to simplify for linear grammars. Other problems considered include grammatical covering and structural equivalence for right-linear, linear, and arbitrary grammars.