Subjective Complexity of Categories Defined over Three-Valued Features

Subjective Complexity of Categories Defined over Three-Valued Features Cordelia D. Aitkin (cdaitkin@ruccs.rutgers.edu) Department of Psychology, Rutgers University – New Brunswick, 152 Frelinghuysen Rd. Piscataway, NJ 08854 USA Jacob Feldman (jacob@ruccs.rutgers.edu) Department of Psychology, Center for Cognitive Science, Rutgers University – New Brunswick, 152 Frelinghuysen, Rd. Piscataway, NJ 08854 USA number of conjunctions or dimensions involved in the concept. The variety of possible conceptual forms in Boolean spaces ranges far beyond the six classes used by Shepard et al. (1961). An extensive study of learning difficulty in our laboratory (Feldman, 2000) considered 35 more such classes, testing the accuracy with which subjects could learn each classification after a fixed duration of learning. Each of these concept forms, like Shepard et al.’s famous types, involved a combination of Boolean features that is not isomorphic in logical form (congruent) to any of the other classes. And also like the famous set of six, the classes studied represented an exhaustive survey of one part of the space of Boolean forms: in the Shepard et al. set, all concepts in 3-dimensional Boolean space with four positive examples; in the Feldman (2000) set, all concepts in 3- or 4- dimensional Boolean space with two, three, or four positive examples (see Feldman, 2003). Considering this wide array of types, a simple empirical trend emerged: the subjective learning difficulty of each concept type was well-predicted by its Boolean complexity. The Boolean complexity of a Boolean concept is simply the length of the shortest propositional formula equivalent to the original concept regarded as a propositional formula, measured in literals (mentions of a variable name). Like its more famous cousin Kolmogorov complexity (see Li & Vitanyi, 1997), Boolean complexity reflects the inherent incompressibility of the concept in question. The ease of learning is well predicted by the degree to which the concept can be faithfully compressed into a more compact form. This finding thus represents a kind of simplicity principle at work in human learning (cf. Pothos & Chater, 2002). The implication is that learners seek to compress the examples they have observed into a more compact representation, and learn more effectively to the extent that the training examples are, in fact, faithfully compressible. As has often been remarked in the literature, though, Boolean features represent a particularly artificial space in which to study categorization. One would like to extend this simplicity principle to a wider and perhaps more natural type of feature space. The notion of Boolean complexity, however, does not extend easily to non-Boolean features. Features with three or more discrete values (e.g., shape = {square, circle, triangle} or s p e c i e s = {dog, elephant, llama}), while intuitively seeming like a simple extension of the Boolean features studied in the 1960s, do not lend themselves to a simple propositional representation. Abstract Many studies in the last four decades have investigated the relative difficulty in learning of concepts defined by various logical forms. Historically, most such studies have used Boolean concepts, that is, categories formed by logical combinations of binary-valued variables. In previous work, we have found that in such categories subjective difficulty is well predicted by Boolean complexity, that is, the length of the shortest propositional formula equivalent to the concept. However, this formalization does not extend easily to concepts defined using features with more than two values. Such categories are of particular interest because they are not easily handled by any contemporary theories based on continuous metric spaces. In more recent work, we have developed a representational formalism suitable for representing such categories. This theory provides a measure of conceptual complexity for such categories, called algebraic complexity. Here we report an experiment testing the learnability of a set of a set of categories defined over two three-valued features. The results show that algebraic complexity gives a good account of the subjective learnability of these concepts. Keywords: Categorization, complexity, learning Introduction Categories help us to organize a complex world by grouping objects and entities in a coherent fashion. But when it comes to coherence, not all categories are created equal. Some collections (pet cats) seem well-organized and coherent; others (snips, snails, and puppy-dog tails) seem disjoint, heterogeneous, and incoherent. Studies since the 1950s have investigated the factors influencing subjective coherence, as manifested in the ease with which categories with various types of logical form can be learned by subjects. Simple conjunctive categories (big red things) can be learned accurately from few examples, while disjunctive categories (big red or small blue things) require more examples before subjects can acquire them (see Bourne, 1970, for a summary of this extensive literature). However the spectrum of categorical coherence cannot be reduced to the simple dichotomy of conjunction vs. disjunction. In a more complex setting involving three Boolean features, Shepard, Hovland and Jenkins (1961) found a reliable ordering of learning difficulty among six logical forms, whose structural differences cannot be described solely in terms of the