On formally undecidable propositions in nondeterministic languages

TL;DR

This paper explores the limitations of nondeterministic language classes, revealing that certain powerful languages contain undecidable propositions that are not represented in their nondeterministic counterparts, challenging foundational assumptions.

Contribution

It demonstrates that some languages in deterministic classes contain undecidable propositions absent from their nondeterministic counterparts, exposing fundamental inconsistencies.

Findings

Undecidable propositions exist in deterministic languages but not in nondeterministic ones.

Nondeterministic language definitions may be inherently inconsistent.

Certain questions on nondeterministic time complexity are ill-posed.

Abstract

Any class of languages $\mathbf{L}$ accepted in time $\mathbf{T}$ has a counterpart $\mathbf{NL}$ accepted in nondeterministic time $\mathbf{NT}$. It follows from the definition of nondeterministic languages that $\mathbf{L} \subseteq \mathbf{NL}$. This work shows that every sufficiently powerful language in $\mathbf{L}$ contains a string corresponding to G\"{o}del's undecidable proposition, but this string is not contained in its nondeterministic counterpart. This inconsistency in the definition of nondeterministic languages shows that certain questions regarding nondeterministic time complexity equivalences are irrevocably ill-posed.