On formally undecidable propositions in nondeterministic languages

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.