A Note on the Complexity of Boolean Concepts

Abstract

What is the relationship between the degree of learning difficulty of a Boolean concept (i.e., a category defined by logical rules expressed in terms of Boolean operators) and the complexity of its logical description? Feldman [(2000). Minimization of Boolean complexity in human concept learning. Nature, 407(October), 630–633] investigated this question experimentally by defining the complexity of a Boolean formula (that logically describes a concept) as the length of the shortest formula logically equivalent to it. Using this measure as the independent variable in his experiment, he concludes that in general, the subjective difficulty of learning a Boolean concept is well predicted by Boolean complexity. Moreover, he claims that one of the landmark results and benchmarks in the human concept learning literature, the Shepard, Hovland, and Jenkins learning difficulty ordering, is precisely predicted by this hypothesis. However, in what follows, we introduce a heuristic procedure for reducing Boolean formulae, based in part on the well-established minimization technique from Boolean algebra known as the Quine–McCluskey (QM) method, which when applied to the SHJ Boolean concept types reveals that some of their complexity values are notably different from the approximate values obtained by Feldman. Furthermore, using the complexity values for these simpler expressions fails to predict the correct empirical difficulty ordering of the SHJ concept types. Motivated by these findings, this note includes a brief tutorial on the QM method and concludes with a brief discussion on some of the challenges facing the complexity hypothesis.