Pac-learning Recursive Logic Programs: Negative Results

TL;DR

This paper demonstrates that generalizations of certain recursive logic programs are computationally hard to learn, establishing boundaries for efficient PAC-learning of recursive clauses.

Contribution

It provides negative results showing the cryptographic hardness of learning extended classes of recursive logic programs, defining limits of PAC-learnability.

Findings

Learning programs with unbounded recursive clauses is cryptographically hard.

Adding certain recursive clauses makes learning as hard as learning boolean DNF.

These results delineate the boundary of efficiently learnable recursive logic classes.

Abstract

In a companion paper it was shown that the class of constant-depth determinate k-ary recursive clauses is efficiently learnable. In this paper we present negative results showing that any natural generalization of this class is hard to learn in Valiant's model of pac-learnability. In particular, we show that the following program classes are cryptographically hard to learn: programs with an unbounded number of constant-depth linear recursive clauses; programs with one constant-depth determinate clause containing an unbounded number of recursive calls; and programs with one linear recursive clause of constant locality. These results immediately imply the non-learnability of any more general class of programs. We also show that learning a constant-depth determinate program with either two linear recursive clauses or one linear recursive clause and one non-recursive clause is as hard as learning boolean DNF. Together with positive results from the companion paper, these negative results establish a boundary of efficient learnability for recursive function-free clauses.