Example
Suppose FOIL's task is to learn the concept grandfather(X,Y) given the relations father(X,Y) and parent(X,Y). Furthermore, suppose our current Body consists of grandfather(X,Y) ← parent(X,Z). This can be extended by conjoining Body with any of the literals father(X,X), father(Y,Z), parent(U,Y), or many others - to create this literal, the algorithm must choose both a predicate name and a set of variables for the predicate (at least one of which is required to be present already in an unnegated literal of the clause). If FOIL extends a clause grandfather(X,Y) ← true by conjoining the literal parent(X,Z), it is introducing the new variable Z. Positive examples now consist of those values <X,Y,Z> such that grandfather(X,Y) is true and parent(X,Z) is true; negative examples are those where grandfather(X,Y) is true but parent(X,Z) is false.
On the next iteration of FOIL after parent(X,Z) has been added, the algorithm will consider all combinations of predicate names and variables such that at least one variable in the new literal is present in the existing clause. This results in a very large search space. Several extensions of the FOIL theory have shown that additions to the basic algorithm may reduce this search space, sometimes drastically.
Read more about this topic: First Order Inductive Learner
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