On specifying Boolean functions by labelled examples Original Research Article

We say a function t in a set H of 0, 1-valued functions defined on a set X is specified by S⊆ X if the only function in H which agrees with t on S is t itself. The specification number of t is the least cardinality of such an S. For a general finite class of functions, we show that the specification number of any function in the class is at least equal to a parameter from Romanik and Smith (1990) known as the testing dimension of the class. We investigate in some detail the specification numbers of functions in the set of linearly separable Boolean functions of n variables — those functions f such that f−1(0) and f−1(1) can be separated by a hyperplane. We present general methods for finding upper bounds on these specification numbers and we characterise those functions which have largest specification number. We obtain a general lower bound on the specification number and we show that for all nested functions, this lower bound is attained. We give a simple proof of the fact that for any linearly separable Boolean function, there is exactly one set of examples of minimal cardinality which specifies the function. We discuss those functions which have limited dependence, in the sense that some of the variables are redundant (that is, there are irrelevant attributes), giving tight upper and lower bounds on the specification numbers of such functions. We then bound the average, or expected, number of examples needed to specify a linearly separable Boolean function. In the final section of the paper, we address the complexity of computing specification numbers and related parameters.