Language-agnostic semantics for implicit learning as reasoning: Extended abstract

We consider the problem of answering queries about formulas of a logical language, based on background knowledge partially represented explicitly as other formulas, and partially represented as examples. Many approaches to this problem have been considered to date, perhaps the most popular among them is the inductive logic programming (ILP) paradigm based on logic programming. PAC semantics, introduced by Valiant, is one such proposal too, based on examples independently drawn from a fixed probability distribution, possibly unknown. However, with ILP, one has to make careful assumptions about the hypothesis space for it to be learned. And with PAC-semantics, most results are discouraging, mainly owing to the very general assumptions behind the learned knowledge base. One recent thread, however, in the PAC-semantics approach considers the idea of implicit learning, where an explicit hypothesis is not constructed, but the observations alone are used to answer the queries. In this work, taking inspiration from that approach, we tease apart a simple semantical setup that allows us to discuss the learning of an implicit hypothesis from examples to answer queries. This yields a language-agnostic framework, potentially allowing it to be useful for a variety of formal languages, from propositional logic to first-order logic to possibly even modal logic.