Language Learning from Texts: Degrees of Intrinsic Complexity and Their Characterizations

摘要

This paper deals with two problems: 1) what makes languages to be learnable in the limit by natural strategies of varying hardness; 2) what makes classes of languages to be the hardest ones to learn. To quantify hardness of learning, we use intrinsic complexity based on reductions between learning problems. Two types of reductions are considered: weak reductions mapping texts (representations of languages) to texts, and strong reductions mapping languages to languages. For both types of reductions, characterizations of complete (hardest) classes in terms of their algorithmic and topo-logical potentials have been obtained. To characterize the strong complete degree, we discovered a new and natural complete class capable of "coding" any learning problem using density of the set of rational numbers. We have also discovered and characterized rich hierarchies of degrees of complexity based on "core" natural learning problems. The classes in these hierarchies contain "multidimensional" languages, where the information learned from one dimension aids to learn other dimensions. In one formalization of this idea, the grammars learned from the dimensions 1, 2,..., k specify the "subspace" for the dimension k + 1, while the learning strategy for every dimension is predefined. In our other formalization, a "pattern" learned from the dimension k specifies the learning strategy for the dimension k + 1. A number of open problems is discussed.