Learning small programs with additional information

Abstract

This paper was inspired by [FBW 94]. An arbitrary upper bound on the size of some program for the target function suffices for the learning of some program for this function. In [FBW 94] it was discovered that if “learning” is understood as “identification in the limit,” then in some programming languages it is possible to learn a program of size not exceeding the bound, while in some other programming languages this is not possible.

We have studied three other learning types, namely, “finite identification,” “co-learning” and “confidence-learning.” These three types are very different. Co-learning with the considered additional information in the form “an arbitrary upper bound for the size of the minimal program” allows the learning of the class of all recursive functions. “Finite identification” does not allow this. “Confidence-learning” is strong enough to learn the class of all recursive functions even without the additional information. However, the results of our paper show exactly the opposite rating for the capabilities of learning programs not exceeding the size given by the bound.

For finite identification it is still possible to identify small programs with additional information in some programming languages but not in all of them. For co-learning it is not possible in any programming language. These results contrast to the result in [FKS 94] showing that an arbitrary class of recursive functions is co-learnable if and only if it is identifiable in the limit. Finally, for confidence-learning it is in general not possible to identify small programs with additional information.

This work was facilitated by an international agreement under NSF Grants 9119540 and 9421640

Supported by Latvian Science Council Grant No.96.0282

Supported by Latvian Science Council Grant No.96.0282

Supported in part by NSF Grants 9020079 and 9301339