Complexity of Problems of Commutative Grammars

We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v?) is P, and that the equivalence problem for context free grammars (do G1 and G2 generate the same language?) is in Π P 2 .