Tradeoffs in the inductive inference of nearly minimal size programs

Abstract

Inductive inference machines are algorithmic devices which attempt to synthesize (in the limit) programs for a function while they examine more and more of the graph of the function. There are many possible criteria of success. We study the inference of nearly minimal size programs. Our principal results imply that nearly minimal size programs can be inferred (in the limit) without loss of inferring power provided we are willing to tolerate a finite, but not uniformly, bounded, number of anomalies in the synthesized programs. On the other hand, there is a severe reduction of inferring power in inferring nearly minimal size programs if the maximum number of anomalies allowed is any uniform constant. We obtain a general characterization for the classes of recursive functions which can be synthesized by inferring nearly minimal size programs with anomalies. We also obtain similar results for Popperian inductive inference machines. The exact tradeoffs between mind change bounds on inductive inference machines and anomalies in synthesized programs are obtained. The techniques of recursive function theory including the recursion theorem are employed.