An Algebraic Approach to Learning and Grounding

We consider the problem of learning the semantics of composite algebraic expressions from examples. The outcome is a versatile framework for studying learning tasks that can be put into the following abstract form: The input is a partial algebra A and a finite set of samples ({\phi}1, O1), ({\phi}2, O2), ..., each consisting of an algebraic term {\phi}i and a set of objects Oi. The objective is to simultaneously fill in the missing algebraic operations in A and ground the variables of every {\phi}i in Oi, so that the combined value of the terms is optimised. We demonstrate the applicability of this framework through case studies in grammatical inference, picture-language learning, and the grounding of logic scene descriptions.