On the equivalence and containment problems for context-free languages

Abstract

LetG andG ₀ be context-free grammars. Necessary and sufficient conditions onG ₀ are obtained for the decidability ofL(G ₀)$\subseteq$ L((G) It is also shown that it is undecidable for whichG ₀,L(G) $\subseteq$ is decidable. Furthermore, given thatL(G) $\subseteq$ is decidable for a fixedG ₀, there is no effective procedure to determine the algorithm which decidesL(G) $\subseteq$ IfL(G ₀) is a regular set,L(G) = L(G ₀) is decidable if and only ifL(G ₀) is bounded. However, there exist non-regular, unboundedL(G ₀) for whichL(G) = L(G ₀) is decidable.