Automated Reasoning with Uncertainties

In this work we assume that uncertainty is a multifaceted concept which admits several di erent measures, and present a system for automated reasoning with multiple representations of uncertainty. Our focus is on problems which present more than one of these facets and therefore in which a multivalued representation of uncertainty and the study of its possibility of computational realisation are important for designing and implementing knowledge-based systems. We present a case study on developing a computational language for reasoning with uncertainty, starting with a semantically sound and computationally tractable language and gradually extending it with specialised syntactic constructs to represent measures of uncertainty, preserving its unambiguous semantic characterisation and computability properties. Our initial language is the language of normal clauses with SLDNF as the inference rule, and we select three facets of uncertainty, which are not exhaustive but cover many situations found in practical problems: vagueness, statistics and degrees of belief. To each of these facets we associate a speci c measure: fuzzy measures to vagueness, probabilities on the domain to statistics and probabilities on possible worlds to degrees of belief. The resulting language is semantically sound and computationally tractable, and admits relatively e cient implementations employing pruning and caching. ii Acknowledgements My supervisors were Paul Chung (during the rst two years), Jane Hesketh (during the nal year) and Dave Robertson (all the way through it). I cannot overstate how important, useful and pleasant their guidance and friendship was to develop this project. Didier Dubois and Henri Prade kindly accepted me as a visiting student at the IRIT Universit e Paul Sabatier (Toulouse), during November 1990. This research owes much to the technical advice and stimulating discussions with both Dubois and Prade, and with J erôme Lang and Sandra Sandri. Many people have generously o ered me important advice and suggestions along the development of this work. I would like to thank especially Fahiem Bacchus, John Fox and V.S. Subrahmanian. Others have been patient enough to read early drafts of this thesis and make many insightful comments and suggestions. Those were Alvaro Fernandes, Alex Gammerman and Tom Kane. Thanks to my examiners J. Paris and A. Bundy for being so severe as they were with their corrections and suggestions for change, consequently forcing me to change signi cantly the nal quality of this work. It was once said that the best of all presents is the presence of our friends. It is a pleasure to have so much to thank to so many friends. In special, I would like to thank Brian Ross, Alvaro, Regina; Dave, Margot, Jane, Paul, Anissa; Alan Black, Dave Mo at, Ian Lewin, Suresh, Am, Joke, Emilio, Blanca, Franziska, Rob Scott, Keiichi, Carla, Nelson, Sheila Rock and Lais. Thanks to my family. The nancial support for the development of this PhD project came from CNPq Conselho Nacional de Desenvolvimento Cient i co e Tecnol ogico (Brazil) grant nr. 203004/89.2. iii Declaration I declare that I composed this thesis myself in its entirety and that it describes my own research. Fl avio Soares Corrêa da Silva Edinburgh March 22, 1993 iv