Learning, Logic, and Topology in a Common Framework

Abstract

Many connections have been established between learning and logic, or learning and topology, or logic and topology. Still, the connections are not at the heart of these fields. Each of them is fairly independent of the others when attention is restricted to basic notions and main results. We show that connections can actually be made at a fundamental level, and result in a parametrized logic that needs topological notions for its early developments, and notions from learning theory for interpretation and applicability. One of the key properties of first-order logic is that the classical notion of logical consequence is compact. We generalize the notion of logical consequence, and we generalize compactness to β-weak compactness where β is an ordinal. The effect is to stratify the set of generalized logical consequences of a theory into levels, and levels into layers. Deduction corresponds to the lower layer of the first level above the underlying theory, learning with less than β mind changes to layer β of the first level, and learning in the limit to the first layer of the second level. Refinements of Borel-like hierarchies provide the topological tools needed to develop the framework.