Bounded Languages Described by GF(2)-grammars

TL;DR

This paper explores GF(2)-grammars, a novel class with unique algebraic properties, establishing conditions for certain language subsets and proving inherent ambiguity of specific languages using algebraic methods.

Contribution

It introduces strong algebraic conditions for GF(2)-grammars and proves the inherent ambiguity of particular languages, resolving longstanding open questions.

Findings

Established necessary conditions for GF(2)-grammars to describe certain language subsets.

Proved the inherent ambiguity of the language {a^n b^m c^k | n != m or m != k}.

Provided a new algebraic proof of ambiguity for {a^n b^m c^k | n = m or m = k}.

Abstract

GF(2)-grammars are a recently introduced grammar family with some unusual algebraic properties. They are closely connected to unambiguous grammars. By using the method of formal power series, we establish strong conditions that are necessary for subsets of a^* b^* and a^* b^* c^* to be described by some GF(2)-grammar. By further applying the established results, we settle the long-standing open question of proving inherent ambiguity of the language {a^n b^m c^k | n != m or m != k}$, as well as give a new purely algebraic proof of the inherent ambiguity of the language {a^n b^m c^k}{n = m or m = k}.