A Structure-Mapping Model of Raven's Progressive Matrices

A Structure-Mapping Model of Raven’s Progressive Matrices Andrew Lovett (andrew-lovett@northwestern.edu) Kenneth Forbus (forbus@northwestern.edu) Jeffrey Usher (usher@cs.northwestern.edu) Qualitative Reasoning Group, Northwestern University, 2133 Sheridan Road Evanston, IL 60208 USA Abstract We present a computational model for solving Raven’s Progressive Matrices. This model combines qualitative spatial representations with analogical comparison via structure- mapping. All representations are automatically computed by the model. We show that it achieves a level of performance on the Standard Progressive Matrices that is above that of most adults, and that the problems it fails on are also the hardest for people. Keywords: Analogy, Spatial Cognition, Problem-Solving Introduction There is increasing evidence that visual comparison may rely on the same structural alignment processes used to perform conceptual analogies (Markman & Gentner, 1996; Lovett et al., 2009a; Lovett et al., 2009b). An excellent task for exploring this is the Raven’s Progressive Matrices (RPM) (Raven, Raven, & Court, 2000) In RPM problems (Figure 1), a test-taker is presented with a matrix of images in which the bottom right image is missing, and asked to pick the answer that best completes the matrix. Though RPM is a visual task, performance on it correlates highly with other assessment tasks, many of them non-visual (e.g., Snow & Lohman, 1989; see Raven, Raven, & Court, 2000, for a review). Thus, RPM appears to tap into important, basic cognitive abilities beyond spatial reasoning, such as the ability to perform analogies. This paper presents a computational model that uses analogy to perform the RPM task, building on existing cognitive models of visual representation and analogical comparison. Our claims are: 1) Tasks such as RPM rely heavily on qualitative, structural representations of space (e.g., Biederman, 1987; Forbus, Nielsen, & Faltings, 1991). These representations describe relations between objects in a visual scene, such as their relative location. Importantly, these representations are hierarchical (Palmer, 1977); they can also describe larger-scale relations between groups of objects or smaller- scale relations between parts of an object. 2) Spatial representations are compared via structure- mapping (Gentner, 1983), a process of structural alignment first proposed to explain how people perform analogies. Structure-mapping is used here to compute the similarity of two images, to identify corresponding objects in the images, and to generate abstractions based on commonalities and differences. We previously (Lovett, Forbus, & Usher, 2007) described a model based on these principles that achieved human adult-level performance on two sections of the Standard Progressive Matrices test. That model was unable to handle the more difficult sections of the test because it only considered differences between pairs of images. This paper describes a more advanced model which performs at an above-average level on the hardest four sections of the test. It remains grounded in the same principles but is able to identify patterns of differences across rows of images. Like before, all inputs are automatically computed from vectorized input. We first discuss Carpenter, Just, and Shell’s (1991) computational model of the RPM. We then describe our model and its results on the Standard Progressive Matrices test. We end with conclusions and future work. Background The best-established model of Raven’s Progressive Matrices was developed by Carpenter, Just, and Shell (1991). It was based on both analysis of the test and psychological studies of human performance. The analysis led to the observation that all but two of the problems in the Advanced Progressive Matrices, the hardest version of the test, could be solved via the application of a set of six rules (see Figure 1 for examples). Each rule describes how a set of corresponding objects vary across the three images in a row. The simplest, Constant in a Row, says that the objects stay the same. Quantitative Pairwise Progression (Figure 1A) says that one of the object’s attributes or relations gradually changes. The other rules are more complex, requiring the individual to align objects with different shapes (Distribution of Three), or to find objects that only exist in two of the three images (Figure Addition or Subtraction, Distribution of Two). The psychological studies suggested that most people solved the problems by studying the top row, incrementally generating hypotheses about how the objects varied across that row, and then looking at the middle row to test those hypotheses. This process began by comparing consecutive pairs of images in a row. Armed with their observations, Carpenter et al. built two computational models to solve the Advanced Progressive Matrices: FAIRAVEN and BETTERAVEN. Both models used hand-coded input representations. They solved a problem by: 1) identifying which of the six rules applied to the first two rows, and 2) computing a mapping between those two rows and the bottom row to determine how to apply the same rules in that row.