We propose a (probably not new) definition. Let $\varphi\_e$ be an effective enumeration of the partial computable functions.


A total function $f$ is promptly non-computable (PNC) [or promptly non-recursive, PNR] if there exists a computable function $h$ such that, for all $\varphi\_e$, there is some $m\le h(e)$ with $\varphi\_e(m)\ne f(m)$. (Possibly because $\varphi\_e(m)$ diverges.) In other words, $f$ differs from $\varphi\_e$ by position $h(e)$.


This is a weakening of diagonal non-computability (DNC) [or diagonally non-recursive, DNR], where we say $f$ is DNC if, for all $\varphi\_e$, $\varphi\_e(e)\ne f(e)$. Naturally, DNC would be a special case of PNC, taking $h$ to be the identity.


Does every PNC function compute a DNC function? Or is DNC actually stronger than PNC?



Is DNC/DNR stronger than "prompt" non-computability?
The graph of the course-of-values variant
$$\{(x,(f(0),\dots,f(x))): x\in \mathbb N\}$$
of such a function would be [effectively immune](https://en.wikipedia.org/wiki/Simple_set). Namely, if we enumerate a subset of this graph then there is an associated partial recursive function. Therefore it is equivalent to DNR.


