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 $(\varphi_1, O_1), (\varphi_2, O_2), \ldots$, each consisting of an algebraic term $\varphi_i$ and a set of objects~$O_i$. The objective is to simultaneously fill in the missing algebraic operations in $\alg$ and ground the variables of every $\varphi_i$ in $O_i$, 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.