Higher-order and modal logic as a framework for explanation-based generalization

Abstract

Certain tasks, such as formal program development and theorem proving, fundamentally rely upon the manipulation of higher-order objects such as functions and predicates. Computing tools intended to assist in performing these tasks are at present inadequate in both the amount of ‘knowledge’ they contain (i.e., the level of support they provide) and in their ability to ‘learn’ (i.e., their capacity to enhance that support over time). The application of a relevant machine learning technique—explanation-based generalization (EBG)—has thus far been limited to first-order problem representations. We extend EBG to generalize higher-order values, thereby enabling its application to higher-order problem encodings. Logic programming provides a uniform framework in which all aspects of explanation-based generalization and learning may be defined and carried out. First-order Horn logics (e.g., Prolog) are not, however, well suited to higher-order applications. Instead, we employ λProlog, a higher-order logic programming language, as our basic framework for realizing higher-order EBG. In order to capture the distinction between domain theory and training instance upon which EBG relies, we extend λProlog with the necessity operator □ of modal logic. We develop a meta-interpreter realizing EBG for the extended language, λ□Prolog, and provide examples of higher-order EBG.