Visual Analogy: Reexamining Analogy as a Constraint Satisfaction Problem

Visual Analogy: Reexamining Analogy as a Constraint Satisfaction Problem Patrick W. Yaner (yaner@cc.gatech.edu) Ashok K. Goel (goel@cc.gatech.edu) Artificial Intelligence Laboratory College of Computing, Georgia Institute of Technology Atlanta, GA 30332-0280 Abstract Holyoak and Thagard proposed that the retrieval and mapping tasks of analogy can be viewed as constraint satisfaction prob- lems, and described a connectionist implementation of their proposal. In this paper, we describe another constraint satis- faction method for the two tasks in the context of visual anal- ogy: in our method, the source cases are organized in a dis- crimination tree, and all the source cases are searched at once. We also present an evaluation of the method for retrieval and mapping of 2-D line drawings from an external memory. The evaluation is based on structural constraints, and uses subgraph isomorphism as the similarity measure. One result is that a de- composition of the retrieval task into feature-based reminding and structure-based selection appears to provide little compu- tational benefit over just selection. Introduction Holyoak and Thagard proposed that the retrieval (Thagard, Holyoak, Nelson, & Gochfeld, 1990) and mapping (Holyoak & Thagard, 1989) tasks of analogy can be productively viewed as constraint satisfaction problems. Their proposal incorporated structural, semantic and pragmatic constraints and used graph isomorphism as the primary similarity mea- sure. Their mapping system, called ACME, and the com- plementary retrieval system, named ARCS, provided connec- tionist implementations of their proposal. In ACME, nodes are constructed for each map hypothesis (between a source element and a target element), with inhibitory and excitatory links between different nodes, and the network is run until it reaches quiescence. The work described here builds on Holyoak and Thagard’s proposal but seeks a different solution to the retrieval and mapping tasks. While we also view the re- trieval and mapping tasks as constraint satisfaction problems (CSPs), our method for addressing the tasks (i) organizes the source cases in a discrimination tree, (ii) uses (general- purpose) heuristics to guide the search, (iii) performs a back- tracking search, and (iv) searches all the source cases at once. The goal of our current work is to develop a computational theory of visual analogy. Analogies transfer relational knowl- edge from a source (or base) case to a target problem. De- pending on the nature of the target and the source, the knowl- edge transferred in an analogy may pertain to different kinds of relations, for example, causal, functional or teleological relations. In visual analogy, the pertinent relations are spa- tial relations among visual elements. In a different part of the project, we have developed a technique for transfer of spatial knowledge, given a target problem and a source case and given a mapping between the two (Davies & Goel, 2001, 2003). In the part described in this paper, we focus on the retrieval and mapping tasks. Our methodology is to start with simple problems and in- crementally add complexity to them. This incremental na- ture of the methodology is manifested in three ways: firstly, visual knowledge can be of many forms, such as depictive bit-mapped representations, sketches, or animations, but our work deals specifically with diagrammatic knowledge rep- resented symbolically as discrete geometric elements and the spatial relations between them; secondly, though visual analogies, like analogies more generally (as proposed by Holyoak and Thagard), can involve semantic and pragmatic constraints, we start with just the structural constraints im- posed by requiring source and target to match structures; and thirdly, from a graph theoretic perspective, there may be more than one sort of graph isomorphism measure that may be the ideal measure, such as maximal common subgraph, but we begin with subgraph isomorphism as our metric. The retrieval task, in this work, assumes a computer-based library of 2D line drawings, takes as input a query (target) in the form of a drawing (and no other information), and gives as output the source drawings that are most similar to the target. The mapping task takes as input a target problem and a source case, and gives as output correspondences between the basic elements of the source case and the target problem. Retrieval Following earlier work on analogical retrieval—e.g., MAC/FAC (Forbus, Gentner, & Law, 1995)—our retrieval architecture supports a two-stage process for diagram retrieval: reminding (or initial recall), and selection. The ar- chitecture consists of (up to) six basic components: an initial stage generating feature vectors, a process that generates a semantic network describing the contents (spatial structure in this case) of an drawing, a process that matches a target’s description (semantic network) to source descriptions from memory, a working memory with potential sources to match with the target, and finally, an interface to the rest of the analogy system in which this retrieval would be taking place. The reminding task takes as input a target example and re- turns as output references to stored drawings whose feature vectors match that of the target. The stored drawings are indexed by feature vectors describing their spatial elements; the feature vector for the target is constructed dynamically. References to those drawings with sufficiently similar feature vectors (according to some appropriate criteria, as explained below) are brought into the working memory. In the selec-