A Structure Mapping Model for Solving Geometric Analogy Problems

A Structure Mapping Model for Solving Geometric Analogy Problems Emmett Tomai Andrew Lovett Kenneth D. Forbus Jeffrey Usher Qualitative Reasoning Group Northwestern University 1890 Maple Avenue, Evanston, IL, 60201 USA etomai@northwestern.edu Abstract Evan's 1968 ANALOGY system was the first computer model of analogy. This paper demonstrates that structure-mapping, when combined with high-level visual processing and qualitative representations, can do the same kinds of problems with hand-drawn sketched inputs. Importantly, the bulk of the computations are not particular to the model of this task but are general purpose: we sketch the problems using our existing sketch understanding system, sKEA, which computes visual structure that is used by our existing analogical matcher, SME. We show how SME can be used to facilitate high-level visual matching and how second-order analogies over differences computed via analogies between sketches provide a more elegant model of this task. A B C Figure 1: Sketched input of a GMAT problem Like Evans, we view these problems as non-trivial and useful for the exercising of internal descriptions. Evans was starting from scratch. Our goal is to show that the progress in analogical processing, qualitative spatial reasoning, and other areas of cognitive science allows us to use general-purpose simulation models to construct a model that solves this same task. We start by briefly reviewing the essentials of SME, our model of analogical matching, followed by a a summary of the sketching Knowledge Entry Associate (sKEA) [10], the open-domain sketch understanding system used in these experiments, focusing on representation of sketches and the visual structures computed over them. This includes the use of SME to help recognize visual rotations and reflections, a novel extension. We then describe how we use second-order analogies between differences found via analogies between sketches to solve these problems. We walk through an analysis of one example in detail to illustrate the processing and then summarize how our model performs on the Evans corpus. Finally, we discuss plans for future work. 1. Introduction One of the mysteries of human cognition is how we make sense of the world around us. We have powerful visual systems, and it appears that part of their job is to compute descriptions of visual structure (cf. [15,20]) which can be used for recognition and understanding. We have argued previously that qualitative spatial reasoning plays an important role in medium and high-level visual processing [8]. Qualitative spatial representations provide a bridge between vision and cognition, since they seem to be computed via visual processes, but take functional constraints into account. We have been exploring this idea by research on sketching. Understanding sketches is a useful approach to understanding qualitative visual structure because starting with digital ink lets us focus on processes of perceptual organization and ignore image processing issues. Previously we have described techniques for imposing human-like visual structure on sketches and how that structure enables our software to better model human similarity judgments [9]. Building on that work, we describe here a computational model of the classic Miller Geometric Analogies task. We use the same set of problems used by Evans in his pioneering work on the system ANALOGY [3]. Figure 1 provides an example. The form of the question is, “A is to B as C is to 2. Overview of SME We rely on the Structure-Mapping Engine (SME) [4], an implementation of Gentner’s structure mapping theory [12] to provide human-like analogical processing. SME takes a pair of structured representations, the base and the target, and returns a set of directed mappings between them. Each mapping consists of entity correspondences between entities in the two representations, expression correspondences that form the multi-level structure of support for the entity matches and candidate inferences that project unmatched relationships and features from the base into the target [4]. The mappings are assigned a