An undecidable property of context-free languages

TL;DR

This paper proves that it is impossible to algorithmically determine whether a context-free language is dense in lexicographic order or whether two such languages share the same order type, highlighting fundamental limits in formal language theory.

Contribution

It establishes the undecidability of key properties related to lexicographic orderings of context-free languages, revealing fundamental computational limitations.

Findings

No algorithm can decide language density in lexicographic order.

It is undecidable whether two context-free languages have the same lexicographic order type.

Highlights limits of algorithmic analysis in formal language properties.

Abstract

We prove that there exists no algorithm to decide whether the language generated by a context-free grammar is dense with respect to the lexicographic ordering. As a corollary to this result, we show that it is undecidable whether the lexicographic orderings of the languages generated by two context-free grammars have the same order type.