Abstract: A Taxonomy of Inductive Problems Charles Kemp and Alan Jern {ckemp,ajern}@cmu.edu Department of Psychology, Carnegie Mellon University Abstract dress multiple problems [7]. For example, exemplar models that formalize objects as points in a multidimensional space have been used to account for several problems including identification, stimulus generalization, recognition, and cat- egorization [13]. Our taxonomy includes all of these prob- lems along with many others. Since we aim to characterize inductive problems rather than to describe the psychological mechanisms that allow them to be solved, we hope that our taxonomy will be useful to researchers from many different traditions, including modelers who pursue probabilistic, log- ical, or connectionist approaches. The taxonomy we describe can serve as a guide for future work, and future models and experiments can address the problems that it contains. Inductive inferences about objects, properties, categories, re- lations, and labels have been studied for many years but there are few attempts to chart the range of inductive problems that humans are able to solve. We present a taxonomy that includes more than thirty inductive problems. The taxonomy helps to clarify the relationships between familiar problems such as identification, stimulus generalization, and categorization, and introduces several novel problems including property identifi- cation and object discovery. Keywords: induction; semantic cognition; generalization; cat- egorization; identification; reasoning Attempts to systematize knowledge have proven useful in several fields. Mendeleev presented a periodic table of the chemical elements that helped to clarify relationships be- tween known elements and that made predictions about the existence of new elements. Adelson and Bergen [1] devel- oped a “periodic table” of early vision that maps out a space of visual computations and identifies several that had previ- ously received little attention. This paper aims to make a sim- ilar contribution to the study of inductive inference. We de- scribe a taxonomy of inductive problems that aims to clarify the relationships between different problems and to highlight problems that have previously been overlooked. An inductive inference goes beyond the information given and reaches a conclusion that is likely but not certain given the available evidence. Inferences of this kind are relevant to almost every area of cognition, and take place, for example, when humans predict the motion of an occluded object, guess the meaning of a novel word, or decide how to grasp an object that is encountered for the first time. We will not discuss vi- sion, language, or motor control, but instead focus on a cluster of problems from an area that has been called semantic cog- nition. Research in this area aims to capture knowledge about objects and their properties, categories or collections of ob- jects, relationships between objects, and word meanings. The relevant literature includes studies of property induction [7], categorization (both supervised [12] and unsupervised [2]), stimulus generalization [15], identification [12], and word learning [17]. Accounts of semantic cognition differ in many respects but most of them rely on six basic notions: objects, properties, categories, relations, labels, and truth values. Our taxonomy takes these six notions as a starting point and attempts to chart the space of inductive problems that can be posed given a commitment to these notions. Two familiar problems that be- long to this space are categorization and property induction, or deciding whether an object has an unobserved property. Most psychological work on inductive inference focuses on a single inductive problem, but some existing frameworks ad- A semantic repository Our approach proposes that semantic knowledge can be cap- tured in terms of objects, relations, labels and truth-values. Our goal is to characterize all inductive problems that can be formulated in terms of these notions. We assume that knowledge about objects, relations and la- bels can be organized into a semantic repository. Let O be a set of objects, L be a set of labels, and T be the set {1, 0} that includes two truth values. For most cases that we con- sider, set O will include individuals such as dogs, people, and chairs, and set L will include strings of phonemes such as “Fido,” “dog,” and “brown.” Here we discuss a running ex- ample where O and L include the four people and the seven labels shown in Figure 1. Sets O, L and T correspond to primitive types, and rela- tions are constructed out of these types. Any property can be formalized as a unary relation r : O → T that assigns a truth value (1 or 0) to each object depending on whether it has the property. Figure 1 shows a property r 1 (·) that includes three of the four objects and can be glossed as bearded(·). A cat- egory can also be formalized as a unary relation r : O → T where the truth values now indicate whether a given object belongs to the category. Figure 1 shows a category r 2 (·) that can be glossed as Sikh(·). Binary relations of the form r : O × O → T assign a truth value to each pair of objects. In Figure 1, for example, relation r 3 (·, ·) can be glossed as parent(·, ·), and assigns value 1 to all pairs (o i , o j ) such that o i is the parent of o j . Relations with three or more places can also be considered, but here we focus on unary and binary relations. Both objects and relations can be associated with labels. Object labels can be captured by a relation r : O×L → T that indicates for each pair (o i , l j ) whether l j is a label of object o i . Figure 1 shows, for example, that each of the four objects in the repository has a unique label. Labels for the unary
Publication Year: 2009
Publication Date: 2009-01-01
Language: en
Type: article
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Cited By Count: 9
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