Complexity of Problems of Commutative Grammars

TL;DR

This paper investigates the computational complexity of problems related to commutative grammars, showing that membership for regular grammars is in P and equivalence for context-free grammars is in a2_2^P under fixed alphabet assumptions.

Contribution

It introduces a novel algebraic approach to analyze the complexity of commutative grammar problems, providing complexity classifications for membership and equivalence.

Findings

Membership problem for regular grammars is in P.

Equivalence problem for context-free grammars is in a2_2^P.

Uses linear algebra and Euler theorem analogs for analysis.

Abstract

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 G_1 and G_2 generate the same language?) is in $\mathrm{\Pi_2^P}$.