Solving Geometric Proportional Analogies with the Analogy Model HDTP

Solving Geometric Proportional Analogies with the Analogy Model HDTP Angela Schwering, Helmar Gust, Kai-Uwe Kuhnberger, Ulf Krumnack (aschweri|hgust|kkuehnbe|krumnack@uni-osnabrueck.de) Institute of Cognitive Science, Albrechtstr. 28, 49076 Osnabruck, Germany Institute for Geoinformatics, Weselerstr. 258, 48151 Munster, Germany Abstract Intelligence tests often use geometric proportional analogies to examine the intelligence quotient of humans. Completing such a series of geometric figures can be a cognitively demanding task, because the solution requires a suitable conceptualization of the geometric figures. Furthermore, there might exist several, equally correct solutions depending on the conceptualization. In this paper, we demonstrate how the symbolic analogy model HDTP solves such analogies: HDTP uses Gestalt principles and qualitative spatial reasoning to compute a psychologically preferred representation of the figures, adapts these representations if necessary, and constructs a solution based on an analogical mapping. Keywords: analogy; geometric proportional analogies; re-representation Introduction and Motivation Analogical reasoning is considered to be fundamental in human cognition and human problem solving (Gust et al., 2008; Hofstadter, 2001). Geometric proportional analogies (GPA) are a special form of analogous problems: A GPA consists of a serious of four geometric figures A, B, C, and D, where the same relation holds between figure A and B as between figure C and D. Such analogies are commonly used in intelligence tests to measure the intelligence quotient. Figure 1 shows an example for a GPA where the figure D is missing. One has to establish an analogous mapping between figures A and C and analyze the relation between A and B, which is afterwards transferred and applied to figure C to construct the missing figure D. The black elements of figure C are repeated in the middle. The second solution can be explained by grouping the top elements in figure A (respectively C) and construct B (respectively D) by moving the top elements one unit down along the y-axis. Solution 3 can be explained by grouping the middle elements and repeating them with flipped colors. A B C solution 1: 35% of participants solution 2: 31% of participants solution 3: 17% of participants Figure 2: Human subject tests have revealed several different solutions for figure D, of which three preferred solutions are shown in the picture. In this paper, we extend earlier work (Schwering et al., 2007) and present a computational model to analyze and detect different plausible solutions to GPAs. We show how the analogy model Heuristic-Driven Theory Projection (HDTP) uses Gestalt principles and qualitative spatial reasoning to detect cognitively preferred representations of geometric figures, adapts these representations if necessary, and constructs a solution based on an analogical mapping. The paper is structured as follows: after this introduction, we give an overview of related work on analogy models for GPAs. In section three we explain the basics of HDTP and show how geometric figures are formally represented in HDTP. In section four we give details on the analogy- making process illustrated with an example. We conclude the paper with a discussion and directions for future work. Related Work Figure 1: Example for a geometric proportional analogy (GPA) with several possible solutions for figure D. The difficulty in solving this analogy lies in its ambiguity. It is possible to construct several, equally correct solutions depending on the conceptualization of the geometric figures. We investigated experimentally different solution strategies (Schwering et al., 2008). Figure 2 illustrates three preferred solutions for the running example: Solution 1 can be explained by grouping the black elements in figure A and C. Figure B repeats the black elements of figure A in the middle. Applying the analogous strategy leads to figure D: Proportional analogies were studied in various domains such as the natural-language domain (Indurkhya, 1989, 1992), the string domain (Hofstadter & Mitchell, 1995), analogical spatial arrangement at a table top scale (French, 2002), and in the domain of geometric figures. In (1962), Evans developed a heuristic program to solve GPAs. Before the actual mapping process, the program computes meaningful components consisting of several line segments in each figure. Evan’s analogy machine determines the relation between A-B, computed a mapping between A-C based on rotation, scaling, or mirroring, and selected an appropriate solution from a list of possible solutions. In contrast to our approach, the representation and the mapping phase are sequentially separated from each