Complexity of Problems for Commutative Grammars

TL;DR

This paper investigates the computational complexity of problems related to Parikh images of languages accepted by automata and grammars, providing tight bounds and algorithms for key decision problems in the commutative setting.

Contribution

It establishes tight complexity bounds and polynomial algorithms for membership and disjointness, and proves Pi2P-completeness for equivalence of context-free grammars in the commutative case.

Findings

Polynomial algorithms for membership and disjointness in automata Parikh images.

Pi2P-completeness of equivalence for context-free grammars.

Tight complexity bounds for key decision problems in commutative language analysis.

Abstract

We consider Parikh images of languages accepted by non-deterministic finite automata and context-free grammars; in other words, we treat the languages in a commutative way --- we do not care about the order of letters in the accepted word, but rather how many times each one of them appears. In most cases we assume that the alphabet is of fixed size. We show tight complexity bounds for problems like membership, equivalence, and disjointness. In particular, we show polynomial algorithms for membership and disjointness for Parikh images of non-deterministic finite automata over fixed alphabet, and we show that equivalence is Pi2P complete for context-free grammars over fixed terminal alphabet.