Complexity of Problems of Commutative Grammars

Abstract

We consider commutative regular and context-free grammars, or, in otherwords, Parikh images of regular and context-free languages. By using linearalgebra and a branching analog of the classic Euler theorem, we show that,under an assumption that the terminal alphabet is fixed, the membership problemfor 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 freegrammars (do G_1 and G_2 generate the same language?) is in $\mathrm{\Pi_2^P}$