Convex spaces as an order-theoretic basis for problem-solving

Teow-Hin Ngair, University of Pennsylvania


The ability to represent and use a body of knowledge or information is fundamental to all problem solvers. It is well recognized that different problems require different representations so that the corresponding algorithms can be implemented efficiently. This thesis advocates that ordered structures play an important role in many problem domains. In particular, we focus on the study of an ordered structure called a convex space. We demonstrate that this particular ordered structure helps answer some of the important questions concerning problem solving tasks in the fields of Artificial Intelligence and Database Systems that are of major interest from both theoretical and practical points of view. Specifically, we study how convex spaces can be used to formulate the algorithms related to version spaces, querying independent databases, Assumption-based Truth Maintenance Systems (ATMS), and the generation of prime implicates. This results in a unifying framework for studying and understanding the representation used in these seemingly unrelated areas. Consequently, we derive general admissibility criteria for version space learning, a semantics for describing a useful database merging procedure in addition to some standard relational database operations, and a consistent semantics for the basic and extended ATMS's. Moreover, the order-theoretic study leads to tile derivation and implementation of better algorithms to replace some of the existing algorithms. Most noteworthy are a more general prime implicate generation algorithm that can also function as very efficient abductive reasoner and theorem prover, an ordered-theoretic based extended ATMS algorithm which eliminates the necessity of inefficient resolution procedures, and a focused ATMS algorithm that supports an efficient diagnostic system. The complexity issues of various algorithms are discussed and some empirical results are presented.

Subject Area

Computer science|Mathematics|Artificial intelligence

Recommended Citation

Ngair, Teow-Hin, "Convex spaces as an order-theoretic basis for problem-solving" (1992). Dissertations available from ProQuest. AAI9235180.