A guided tour of minimal indices and shortest descriptions

Abstract.

The set of minimal indices of a Gödel numbering $\varphi$ is defined as ${\rm MIN}_{\varphi} = \{e: (\forall i < e)[\varphi_i \neq \varphi_e]\}$. It has been known since 1972 that ${\rm MIN}_{\varphi} \equiv_{\mathrm{T}} \emptyset^{\prime \prime }$, but beyond this ${\rm MIN}_{\varphi}$ has remained mostly uninvestigated. This paper collects the scarce results on ${\rm MIN}_{\varphi}$ from the literature and adds some new observations including that ${\rm MIN}_{\varphi}$ is autoreducible, but neither regressive nor (1,2)-computable. We also study several variants of ${\rm MIN}_{\varphi}$ that have been defined in the literature like size-minimal indices, shortest descriptions, and minimal indices of decision tables. Some challenging open problems are left for the adventurous reader.