Abstract Visual Reasoning: An Algebraic Approach for Solving Raven's Progressive Matrices

We introduce algebraic machine reasoning, a new reasoning framework that is well-suited for abstract reasoning. Effectively, algebraic machine reasoning reduces the difficult process of novel problem-solving to routine algebraic computation. The fundamental algebraic objects of interest are the ideals of some suitably initialized polynomial ring. We shall explain how solving Raven's Progressive Matrices (RPMs) can be realized as computational problems in algebra, which combine various well-known algebraic subroutines that include: Computing the Gr\"obner basis of an ideal, checking for ideal containment, etc. Crucially, the additional algebraic structure satisfied by ideals allows for more operations on ideals beyond set-theoretic operations. Our algebraic machine reasoning framework is not only able to select the correct answer from a given answer set, but also able to generate the correct answer with only the question matrix given. Experiments on the I-RAVEN dataset yield an overall $93.2\%$ accuracy, which significantly outperforms the current state-of-the-art accuracy of $77.0\%$ and exceeds human performance at $84.4\%$ accuracy.

Dependently Typed Knowledge Graphs

Reasoning over knowledge graphs is traditionally built upon a hierarchy of languages in the Semantic Web Stack. Starting from the Resource Description Framework (RDF) for knowledge graphs, more advanced constructs have been introduced through various syntax extensions to add reasoning capabilities to knowledge graphs. In this paper, we show how standardized semantic web technologies (RDF and its query language SPARQL) can be reproduced in a unified manner with dependent type theory. In addition to providing the basic functionalities of knowledge graphs, dependent types add expressiveness in encoding both entities and queries, explainability in answers to queries through witnesses, and compositionality and automation in the construction of witnesses. Using the Coq proof assistant, we demonstrate how to build and query dependently typed knowledge graphs as a proof of concept for future works in this direction.