Can testing show the absence of bugs? $(n + 1)$ points are required to uniquely determine a polynomial of degree $n$; for instance, two points in a plane determine exactly one line.

How many points are required to uniquely determine a computable function $f : N \rightarrow N$, given the length of a program that computes $f$ in a fixed language? (i.e. a bound on the Kolmogorov complexity of $f$).

The idea is that, at least theoretically, one could prove the correctness of a program by making enough tests.

If one has a program $P$ of length $L$ that computes $f$, there is a bound on the number of functions that can be computed with a source length of at most $L$.

Therefore one would "only" need to prove that:


$f$ can be computed with a source length $\leq L$
$P$ does not compute any other function computable in $L$ bytes or less (by testing)


This idea has probably no practical consequences (the bounds are surely bound to be exponential).
(This was meant as a comment, but went long).
Very interesting question.
If you are willing to think about other complexity measures besides Kolmogorov's, then there are some answers in Learning theory that might satisfy you. I leave it for the experts in the area.

For example, if I'm not mistaken, in "A theory of the learnable" Valiant proved that a boolean function can be reconstructed given a polynomial number of "positive points" on the size of its k-CNF formula (for any fixed k, and I mean
with "positive points" those of the form $(x_1,\ldots,x_n,1)$).

In Knuth's TAOCP 7.2.1.6 it is shown in an astonishing way (using the Christmas tree pattern) that to reconstruct a monote boolean function (i.e. non-decreasing in each variable) you need exactly ${n+1 \choose \lfloor n/2\rfloor+1}$ points.