In a companion paper it was shown that the class of constant-depth determinate k-ary recursive clauses is e ciently 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 paclearnability. 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 e cient learnability for recursive function-free clauses.
As noted above, a companion paper exists which presents negative learnability results for several natural generalizations of the language kd-MaxRecLang REF .
Pac-learning Recursive Logic Programs: Negative Results
Journal of Artificial Intelligence Research, Vol 2, (1995), 541-573
Computer Science Mathematics