On the learnability of Boolean formulae

Abstract

We study the computational feasibility of learning boolean expressions from examples. Our goals are to prove results and develop general techniques that shed light on the boundary between the classes of expressions that are learnable in polynomial time and those that are apparently not. The elucidation of this boundary, for boolean expressions and possibly other knowledge representations, is an example of the potential contribution of complexity theory to artificial intelligence. We employ the distribution-free model of learning introduced in /lo]. A more complete discussion and justification of this model can be found in [4,10,11,12]. [4] includes some discussion that is relevant more particularly to infinite representations, such as geometric ones, rather than the finite case of boolean functions. For other recent related work see [1,2,7,&g]. The results of this paper fall into three categories: closure properties of learnable classes, negative results, and distribution-specific positive results. The closure properties are of two kinds. In section 3 we discuss closure under boolean operations on the members of the learnable classes. The assumption that the classes are learnable from positive or negative ex-