Categorical Logic as a Foundation for Reasoning Under Uncertainty

Categorical logic is a sophisticated mathematical theory which has been successfully applied in computer science, applied mathematics, and in mathematical foundations. Formal languages called logics are interpreted in mathematical structures called categories. Sets, fuzzy sets, probability spaces, and a vast array of other objects have been studied as categories. Theorems relating fragments of first and higher order logics to categories which can be used as models have been developed by the research community. In particular, there is a body of research on fuzzy sets and probability spaces which has not yet been employed in industry. Metron proposes to use these technologies to integrate Bayesian networks, Dempster-Shafer theory, fuzzy logic, and other models of inference under uncertainty. We anticipate discovering new perspectives on machine learning algorithms in this categorical framework. We further propose to investigate the efficiency of the functional programming languages ML and Haskell in the design and execution of our algorithms.