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 alg and a finite set of examples (varphi1, O1), (varphi2, O2), ldots, each consisting of an algebraic term varphii and a set of objects~Oi. The objective is to simultaneously fill in the missing algebraic operations in alg and ground the variables of every varphii 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.