{'fraction_non_alphanumeric': 0.04241281809613572, 'fraction_numerical': 0.013195098963242224, 'mean_word_length': 5.1985981308411215, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Modeling Multiple Strategies for Solving Geometric Analogy Problems

Modeling Multiple Strategies for Solving Geometric Analogy Problems Andrew Lovett (andrew-lovett@cs.northwestern.edu) Kenneth Forbus (forbus@northwestern.edu) Qualitative Reasoning Group, EECS Department, 2133 Sheridan Road Evanston, IL 60208USA In the following section, we provide some background on the geometric analogy task. We then present our computational model, which utilizes two strategies for performing the task. Afterwards, we compare the model against human performance and discuss the results. We close with related work and conclusions. Abstract We present an improved computational model for performing geometric analogy. The model combines two previously modeled strategies and makes explicit claims about when people will use one strategy or the other. We compare the model to human performance on a classic problem set. The model’s strategy shifts, along with working memory load, account for most of the variance in human reaction times. Background Keywords: geometric analogy; visual problem-solving; structure-mapping Geometric analogy involves comparing images to identify differences. However, researchers disagree on which comparisons people make. The debate encompasses two competing strategies. The first involves inserting each possible answer into the analogy to evaluate it. Consider Figure 1A. The strategy proceeds as follows: 1) Compare A and B to get Δ(A,B), the differences between A and B. Here there is a change from two overlapping objects to one object inside the other. 2) For each possible answer i, compare C to i to get Δ(C,i), the differences between C and that answer. Then, perform a second-order comparison: measure the similarity of Δ(A,B) to Δ(C,i). Whichever answer produces the most similar set of differences is chosen. Here answer 3 produces an identical set of differences to Δ(A,B), so it is chosen. The second strategy solves for the answer directly: I) Compare A and B to get Δ(A,B). II) Compare A and C to get the corresponding objects. In Figure 1A, the large leftmost shapes correspond, and the small rightmost shapes correspond. III) Apply the differences in Δ(A,B) to the corresponding objects in C to infer D’, a representation of the answer. Here the small rectangle in C would move inside of the larger shape. Pick the answer most similar to D’. Mulholland, Pellegrino, and Glaser (1980) call the first strategy infer-infer-compare and the second strategy infer- map-apply. However, this assumes that different processes are used to compare images in steps 1), 2), and II). We believe structure-mapping can determine differences, identify correspondences, and measure similarity. Therefore, we instead call the strategies second-order comparison and visual inference. Sternber (1977) argued that people use visual inference to perform geometric analogy. However, Mulholland, Pellegrino, and Glaser (1980) found evidence that second- order comparison was being used. Bethell-Fox, Lohman, and Snow (1984) suggested that individuals may adjust their strategy, depending on problem difficulty. Their eye- tracking data demonstrated that people typically use visual inference, solving directly for the answer. However, as problems become more difficult, people may abandon this Introduction Visual problem-solving has long been a popular tool for evaluating people’s cognitive abilities (Raven, Raven, & Court, 1998; Dehaene et al., 2006). Problem-solving tasks frequently involve a sequence of images (e.g., Figure 1). Individuals must compare the images, identifying some pattern across them. They must then apply this pattern, finding the answer that best completes (or violates) it. Visual problem-solving depends on a comparison process for identifying commonalities and differences in images. We have previously argued that structure mapping (Gentner, 1983), a theory of analogical comparison, may also explain concrete visual comparison in humans (Markman & Gentner, 1996; Lovett et al., 2009a; Sagi, Gentner, & Lovett, in press). According to structure mapping, people compare stimuli by aligning the common relational structure in symbolic, qualitative representations. We have posited that structure mapping may play a ubiquitous role, identifying commonalities and differences and estimating similarity. Based on this hypothesis, we have built models of three visual problem-solving tasks: geometric analogy (Lovett et al., 2009b), Raven’s Progressive Matrices (Lovett, Forbus, & Usher, 2010), and the oddity task (Lovett & Forbus, 2011a). Here, we complement our model of visual comparison with a model of visual inference. Visual inference explains how individuals apply a set of differences to one image to create a novel image representation. It plays a key role in tasks such as geometric analogy (Figure 1), where participants are asked “A is to B as C is to…?” We show how this leads to a new model for geometric analogy. Rather than assuming participants always infer the correct answer, the model makes explicit claims about when visual inference will succeed and when it will fail, requiring a shift to an alternate strategy. The model’s strategic shifts correlate well with human reaction times on a classic geometric analogy problem set (Evans, 1968).