ON THE FOUNDATIONS OF COMPUTABILITY THEORY

The principal motivation for this work is the observation that there are significant deficiencies in the foundations of conventional computability theory. This thesis examines the problems with conventional computability theory, including its failure to address discrepancies between theory and practice in computer science, semantic confusion in terminology, and limitations in the scope of conventional computing models. In light of these difficulties, fundamental notions are re-examined and revised definitions of key concepts such as “computer,” “computable,” and “computing power” are provided. A detailed analysis is conducted to determine desirable semantics and scope of applicability of foundational notions. The credibility of the revised definitions is ascertained by demonstrating by their ability to address identified problems with conventional definitions. Their practical utility is established through application to examples. Other related issues, including hidden complexity in computations, subtleties related to encodings, and the cardinalities of sets involved in computing, are examined. A resource-based meta-model for characterizing computing model properties is introduced. The proposed definitions are presented as a starting point for an alternate foundation for computability theory. However, formulation of the particular concepts under discussion is not the sole purpose of the thesis. The underlying objective of this research is to open discourse on alternate foundations of computability theory and to inspire re-examination of fundamental notions

The Typical Constructible Object

International audienceBaire Category is an important concept in mathematical analysis. It provides a way of identifying the properties of typical objects and proving the existence of objects with specified properties avoiding explicit constructions. For instance it has been extensively used to better understand and separate classes of real functions such as analytic and smooth functions. Baire Category proves very useful in computability theory and computable analysis, again to understand the properties of typical objects and to prove existence results. However it cannot be used directly when studying classes of computable or computably enumerable objects: those objects are atypical. Here we show how Baire Category can be adapted to such small classes, and how one can define typical computably enumerable sets or lower semicomputable real numbers for instance