Pac-Learning Non-Recursive Prolog Clauses

Recently there has been an increasing amount of research on learning concepts expressed in subsets of Prolog; the term inductive logic programming (ILP) has been used to describe this growing body of research. This paper seeks to expand the theoretical foundations of ILP by investigating the pac-learnability of logic programs. We focus on programs consisting of a single function-free non-recursive clause, and focus on generalizations of a language known to be pac-learnable: namely, the language of determinate function-free clauses of constant depth. We demonstrate that a number of syntactic generalizations of this language are hard to learn, but that the language can be generalized to clauses of constant locality while still allowing pac-learnability. More speciically, we rst show that determinate clauses of log depth are not pac-learnable, regardless of the language used to represent hypotheses. We then investigate the eeect of allowing indeterminacy in a clause, and show that clauses with k indeterminate variables are as hard to learn as DNF. We next show that a more restricted language of clauses with bounded indeterminacy is learnable using k-CNF to represent hypotheses, and that restricting the \locality" of a clause to a constant allows pac-learnability even if an arbitrary amount of indeterminacy is allowed. This last result is also shown to be a strict generalization of the previous result for determinate function-free clauses of constant depth. Finally, we present some extensions of these results to logic programs with multiple clauses.