Learning of Boolean Functions Using Support Vector Machines

Abstract

This paper concerns the design of a Support Vector Machine (SVM) appropriate for the learning of Boolean functions. This is motivated by the need of a more sophisticated algorithm for classification in discrete attribute spaces. Classification in discrete attribute spaces is reduced to the problem of learning Boolean functions from examples of its input/output behavior. Since any Boolean function can be written in Disjunctive Normal Form (DNF), it can be represented as a weighted linear sum of all possible conjunctions of Boolean literals. This paper presents a particular kernel function called the DNF kernel which enables SVMs to efficiently learn such linear functions in the high-dimensional space whose coordinates correspond to all possible conjunctions. For a limited form of DNF consisting of positive Boolean literals, the monotone DNF kernel is also presented. SVMs employing these kernel functions can perform the learning in a high-dimensional feature space whose features are derived from given basic attributes. In addition, it is expected that SVMs’ well-founded capacity control alleviates overfitting. In fact, an empirical study on learning of randomly generated Boolean functions shows that the resulting algorithm outperforms C4.5. Furthermore, in comparison with SVMs employing the Gaussian kernel, it is shown that DNF kernel produces accuracy comparable to best adjusted Gaussian kernels.