A classification approach for automated reasoning systems—a case study in graph theory

Abstract

Reasoning systems which create classifications of structured objects face problem of how object descriptions can be used to reflect their components as well as relations among these components. Current reasoning systems on graph theory do not adequately provide models to discover complex relations among mathematical concepts (eg: relations involving subgraphs) mainly due to inability to solve this problem. This thesis presents an approach to construct a knowledge-based system, GC (Graph Classification), which overcomes this difficulty in performing automated reasoning in graph theory. We describe graph concepts based on an attribute called Linear Recursive Constructivity (LRC). LRC defines classes by an algebraic formula supported by background knowledge of graph types. We use subsumption checking on decomposed algebraic expressions of graph classes as a major proof method. The search is guided by case-split-based inferencing. Using approach GC has generated proofs for many theorems such as any two distinct cycles (closed paths) having a common edge e contain a cycle not traversing e, cycle C1 contains edges e1, e2, and cycle C2 contains edges e2, e3, then there exists a cycle that contains e1 and e3 and the union of a tree and a path is a tree if they have only a single common vertex. The main contributions of this thesis are: (1) Development of a classification-based knowledge representation and a reasoning approach for graph concepts, thus providing a simple model for structured mathematical objects. (2) Development of an algebraic theory for simplifying and decomposing graph concepts. (3) Development of a proof search and a case-splitting technique with guidance of graph type knowledge. (4) Development of a proving mechanism that can be generate constructive proofs by manipulating only simple linear formalization of theorems.