Analogies and their use in problem-solving

Abstract

The thesis deals with the meaning of "analogy" and how it can be extracted and exploited for the following two classes of problems: CLASS 1: How can one transfer the theorems of one domain (in mathematics) to analogous theorems of another domain? CLASS 2: Given a pair, A, of patterns, how does one establish the relationship between the two patterns and use this relationship to identify analogous patterns in another set, B, of patterns? A partial solution to these problems is proposed.

Solution for CLASS 1 An Analogical Theorem Prover (ATP) is designed and developed to use proofs of theorems in group theory to generate proofs of similar theorems in ring theory, based on discovered analogies between group and ring theories. The ATP uses:

First-order predicate calculus for knowledge representation Robinson’s resolution rule for inference Transformation maps for representation of analogies

Solution for CLASS 2 A Geometrical Analogy Problem Solver (GAPS) is designed to solve a sub-class of geometry analogy problems, first considered by Evans [5]. Like ATP, GAPS uses:

First-order predicate calculus for representing figural relationships Analogies are computed between various problem statements and evaluated to find solutions

Comparison of ATP and GAPS Both systems share:

A common scheme for knowledge representation Kling’s [9] matching procedures for computing analogies

However, they differ in their goals:

ATP aims to compute analogies and apply them to transfer the knowledge of a known proof to generate a proof of a similar theorem GAPS focuses on the discovery of analogies

Main Contributions of the Thesis

A proof procedure for knowledge transfer using analogies A method for using logical language to analyze geometrical analogies

Organization of the Thesis

Chapter 1: Reviews the role of analogies in human and AI problem-solving. Presents the motivation, description of the work, and a literature survey.

Chapter 2: Defines analogies and their representation. Formulates a class of analogous problems in group and ring theories. Describes the ATP system and presents its results and conclusions.

Chapter 3: Deals with geometry analogy problems. Reviews Evans’ ANALOGY program. Formulates problems using first-order predicate calculus and discusses the GAPS system. Presents results and conclusions. Chapter 4: Summarizes the main results of the thesis and suggests directions for future work.