Similarity: A Transformational Approach

Similarity: A Transformational Approach Ulrike Hahn (HahnU@Cardiff.ac.uk) School of Psychology, Cardiff University, PO Box 901, Cardiff, CF10 3YG, UK. Lucy B. C. Richardson (RichardsonLB@Cardiff.ac.uk) School of Psychology, Cardiff University, PO Box 901, Cardiff, CF10 3YG, UK. Nick Chater (Nick.Chater@Warwick.ac.uk) Institute for Applied Cognitive Science, Department of Psychology, University of Warwick, Coventry, CV4 7AL, UK. Abstract Representational Distortion is a new account of similarity in which the transformation distance between representations determines similarity: entities that are perceived to be similar have representations that are readily transformed into one another, whereas dissimilar entities require numerous transformations. Here we present experimental evidence in favour of this viewpoint. Introduction The breadth of cognitive and social contexts in which similarity is invoked as an explanatory construct is vast. Similarity forms part of the explanations of memory retrieval, categorization, visual search, problem solving, learning, linguistic knowledge and processing, reasoning, and social judgment. The two classical approaches to similarity are the spatial account (e.g., Nosofsky, 1986), which represents similarity in terms of distance in psychological space, and Tversky‘s (1977) contrast model which views similarity as a ﬁmction of common and distinctive features of the entities under comparison. Both of these accounts have been used successfully in cognitive modeling, however both also suffer from fundamental limitations (see Hahn & Chater, 1997, 1998a, 1998b). These models are restricted in scope by the fact that they define similarity over very speciﬁc and simple types of representation: points in space or feature sets. However, the representation of complex real-world stimuli, from faces to auditory scenes, is typically assumed to require structured representations, that can explicitly describe objects, their parts, properties and the relations between them. Relational information of this kind cannot readily be encoded using lists of features or dimensional values (Hahn & Chater, 1998a). The present paper considers a recent theoretical approach to similarity, Representational Distortion (henceforth, RD; Hahn & Chater, 1997; Chater & Hahn, 1997), which aims to provide a theoretical framework applying to similarity judgements. According to RD, the similarity between two entities is a function of the “complexity” required to “distort” or “transform” the representation of one into the representation of the other. The simpler the necessary transformation, the more similar they are assumed to be. How can the complexity of the transformation between two representations be measured? At a theoretical level, Hahn and Chater draw on a branch of mathematics, Kolmogorov complexity theory (Li & Vitanyi, 1997) that provides a rigorous and general way of measuring the complexity of representations and transformations between them. In intuitive terms, according to Kolmogorov complexity theory, the complexity of a representation is the length of the shortest computer program that can generate that representation. The idea is that representations that can be generated by a short program are simple; those that require longer programs are complex. We will not consider the virtues of this measure of complexity here, except to note that it supports substantial applications in the cognitive and computing sciences (Chater, 1999). Kolmogorov complexity has a natural application as a measure of similarity between representations. The simplest measure is the length of the shortest program that “distorts” one representation into the other. According to this viewpoint, the degree to which two representations are similar is determined by how many instructions must be followed to transform one into the other. For example, the conditional Kolmogorov complexity between the sequences 1 2 3 4 5 and 2 3 4 5 6 is small, because the simple instructions add 1 to each digit and subtract 1 from each digit sufﬁce to transform one into the other. In the same way, 1 2 3 4 5 and 2 4 6 8 10 (multiply/divide each digit by 2) are presumed to be similar. On the other hand, 1 2 3 4 5 and 3 5 7 9 11 are viewed as less similar, because two operations are required to transform one into the other (e.g., multiply by 2 and add 1). Finally, two entirely unrelated representations will be maximally dissimilar because there will be no efﬁcient way of transforming