The paper uses the notion of a learning machine for (computable) functions. This is an algorithmic device with the elements of a set as step-by-step input and with programs as output (from time to time). We say, e.g., that the machine identifies \(f\) if almost all output programs compute \(f\) when the input is the graph of \(f\). Two infinite hierarchies of (learning) criteria for identification are recalled and the relations (inclusions) of the relevant classes \((\text{Ex}^{\text{a}}, \text{Bc}^{\text{a}})\) of sets of recursive functions are summarized.
Then the authors recall limiting programs and introduce generator programs; e.g., a total program is a \(0\)-generator for \(f\) if almost all its outputs are programs computing \(f\). As a part of motivation, they show that some global properties (e.g. monotonicity of \(f\)) cannot be proved (in a first-order extension of Peano arithmetic) from any (ordinary) program for \(f\) while they can be proved from a higher-order (i.e. limiting or generator) program.
The main results are concerned with comparing the learning power in the cases of higher-order programs and ordinary programs; i.e. the classes \(\text{Lim Ex}^{\text{a}}\), \(\text{Lim Bc}^{\text{a}}\), \(\text{Gen Ex}^{\text{a}}\), \(\text{Gen Bc}^{\text{a}}\) are compared with \(\text{Ex}^{\text{a}}\), \(\text{Bc}^{\text{a}}\) and among themselves. Some of the relevant problems (inclusions) are left open.