Non-closure under complementation for unambiguous linear grammars

TL;DR

This paper proves that the class of unambiguous linear languages is not closed under complementation by providing a specific example and showing its complement cannot be generated by any context-free grammar.

Contribution

It offers an explicit example and an alternative proof demonstrating the non-closure of unambiguous linear languages under complementation.

Findings

Unambiguous linear languages are not closed under complementation.

A specific unambiguous linear grammar's complement is not context-free.

Provides an alternative proof to a classical non-closure result.

Abstract

The paper demonstrates the non-closure of the family of unambiguous linear languages (that is, those defined by unambiguous linear context-free grammars) under complementation. To be precise, a particular unambiguous linear grammar is presented, and it is proved that the complement of this language is not defined by any context-free grammar. This also constitutes an alternative proof for the result of Hibbard and Ullian ("The independence of inherent ambiguity from complementedness among context-free languages", J.ACM, 1966) on the non-closure of the unambiguous languages under complementation.