Embodied Enumeration : Appealing to Activities for Mathematical Explanation

A computational cognitive agent capable of mathematical reasoning should be able to: (1) perform mental operations over abstract internal representations (e.g., counting numbers) and (2) perform physical operations over concrete external objects (e.g., counting apples). In a context of justification, both of these types of activity can contribute to mathematical explanations. In this paper, we focus on an agent’s ability to enumerate external objects, the knowledgelevel representational requirements (and side-effects) for such an ability, and the applicability of the enumeration result during an explanation. The SNePS knowledge-representation, reasoning, and acting system is used for a preliminary implementation of Cassie, our enumeration agent. The GLAIR architecture, which is used to model Cassie’s embodiment, is also discussed. Introduction: Computational Math Cognition This paper reports on some results from a larger research project whose ultimate goal is a computational characterization of (developmentally) early mathematical cognition. Mathematical cognition is an interdisciplinary investigation into the cognitive foundations of mathematical reasoning. For the most part, this investigation has been concerned with human mathematical cognition (Butterworth 1999; Dehaene 1997; Gelman & Gallistel 1978; Lakoff & Núñez 2000; Piaget 1965; Wiese 2003), with fewer (but nevertheless important) contributions involving computational models of human mathematical reasoning (Bobrow 1968; Davis & Lenat 1982; Dellarosa 1985; Fletcher 1985; Ohlsson & Rees 1991). Besides telling us something about the human mind, the development of AI agents with mathematical cognition can benefit the agents themselves. Even an early human ability such as ordinal numeracy gives an agent a tool which is applicable to a variety of reasoning tasks. Our specific interest in the field of mathematical cognition concerns the nature of mathematical understanding (as opposed to learning or the acquisition of concepts) and the ∗I would like to thank William Rapaport, Stuart Shapiro, Douglas Clements, and the members of the SNePS research group for their comments on earlier versions of this work. Copyright c © 2006, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. extent to which a computational agent can demonstrate such understanding. It will therefore be useful to state some of our general views on mathematical understanding and how these views impact the agent requirements: • Mathematical understanding can be demonstrated by a question-and-answer dialogue: An agent must be capable of explaining its procedural choices and justifying its conceptual beliefs. This demands a system in which the a “questioner” can trigger agent inference after (and perhaps during) a mathematical activity. The Turing Test (Turing 1950), usually taken to be a measure of naturallanguage understanding, can also measure mathematical understanding. • Mathematical understanding is activity-driven: An agent must be capable of acting and expanding its knowledge during action. An episodic memory of the activity should be accessible to the agent so that it can recall its experience. Simply put, mathematics involves doing something, not just thinking about things. • Mathematical understanding is impacted by agent embodiment: An agent must be capable of interacting with entities external to its “mind” and aligning such entities with mental symbols. • Mathematical understanding requires semantics: Mathematical concepts and procedures must be understood in terms of other concepts and procedures. The basis for this recursive understanding is an internalized symbolic representation of concrete external entities (Rapaport 1995). In the hopes of meeting the challenge of these additional requirements, we take a synthetic, bottom-up approach to agent design in which the (developmentally) early abilities of counting and arithmetic are the first provided for the agent. In this paper, we identify a situation where, in the course of an explanation, an agent may appeal to the mathematical activity of enumeration for the justification of a result. We briefly discuss the nature of abstract counting routines and the multi-modal aspect of enumeration. SNePS, the implementation platform for our enumeration agent Cassie, is described along with the GLAIR architecture for embodied SNePS agents. Taking the act of visually enumerating apples as a motivating example, we spend the rest of the paper detailing a preliminary implementation of the enumeration ability in Cassie. Throughout, we will consider the impact of the implementation on Cassie’s knowledge-level representations (since this is the level with which Cassie will produce mathematical explanations). Mathematical Explanations In the following dialogue, a questioner tries to probe the addressee’s understanding of the fact the greatest common divisor (GCD) of 8 and 6 is 2: Q1: Why is 2 the greatest common divisor of 8 and 6? A1: 2 is the greatest of the common divisors of 8 and 6. Q2: What makes 2 the greatest of the common divisors? A2: 2 is said later in the counting sequence than any of the other