It always has been bugging me that we (humans) know pretty easily when most programs we write halt or not, but the halting problem is still undecidable.

I have just thought of a variant approximation-like extension to this problem, that tries to “maximize” the correct answers an algorithm can make on this problem.

Formally speaking, let us say that an turing machine $A$ that **always halts**, is $r$-solving the halting problem, for $r:mathbb{N}rightarrowmathbb{N}$, if $r(n)le |{xin Sigma^*mid xtext{ is a TM}spacelandspace|x|le nspace landspace A(x) text{ is correct}}|$ where $A(x)$ is said to be correct $iff$ $A(x)$ outputs $true$ for $xin HALT$ and outputs $false$ otherwise.

Equivalently, we can define $r$ to be a function $r:mathbb{N}rightarrow (0,1)$ and then the requirement will change to $ncdot r(n) le cdots$

**My question is:** What is the largest $r$ function that we have an $r$-solving algorithm for? Or, what is the highest $r$ function that we can have an $r$-solving efficient (e.g. polynomial) algorithm for? Or are there other known things about the behavior of $r$-solving algorithms?

This can be easily adapted to any other problem, so I would be glad for any answer that explains this on any interesting language, and not necessarily the halting problem