This talk is about algorithmic learning (or inference) of programs or other definitions for computational objects - from data about those objects. It provides a sampler of results in three settings (together with a list of other settings that might have been presented). The three settings:
1.   A machine inductively infers (or learns) a (computable) function iff the machine fed data from the graph of the function, after some trial and error, eventually outputs a definition or definitions of the function. Theorems are presented contrasting cases where the definitions are computer programs and cases where the definitions are slightly quantificationally more complex than computer programs. Interpretation of the results for philosophy of science provides an unexpected and subtle difficulty with Popper's Refutability Principle for science.

2.   A number of child cognitive development phenomena follow the U-shaped form of: first learning, then unlearning, and subsequent relearning. One can ask if U-shaped learning is an evolutionary accident or essential. For learning grammars (or r.e. indices) from positive data for r.e. languages, one can ask if there are classes which are trial and error learnable but not without U-shaped learning. The answer is that it depends on whether the convergence (in the limit) to success is to one, two, or three (or more) successful grammars.

3.   Investigated are some surprising and delicate tradeoffs between the generality of an algorithmic learning device and the quality of the successful programs it converges to. Two classes of results are presented. There are results to the effect that, with small increases in generality of the learning device, the computational complexity of some successfully learned programs is provably unalterably suboptimal. There are also results in which the complexity of successfully learned programs is optimal and the learning device is quite general, but some of those optimal, learned programs are provably unalterably information deficient - in fact, deficient as to safe, algorithmic extractability of even the fact that they are approximately optimal. For these results, the safe, algorithmic methods of information extraction will be by proofs in arbitrary, true, recursively axiomatizable extensions of Peano Arithmetic.