Many machine systems acquire new domain rules by trying to derive a solution to a problem, reaching an impasse, guessing a new rule that resolves the impasse, and going on. If the new rule allows the derivation to be eventually completed, that is taken as justification for including it in the domain theory, at least provisionally. This chapter analyzes a particular derivation completion learner, Cascade, that guesses new rules by specializing overly general rules. The analysis concentrates on three issues: (1) How can a derivation completion learner intelligently decide which impasses should be resolved and learned from? (2) Can Cascade's learning method acquire any domain rule, or are there limits to its power? (3) What would a library of overly general rules look like for a particular task domain? Would it be adhoc or show some kind of structure?