Solving Tree Problems with Category Theory

TL;DR

This paper introduces a category-theoretic framework for formalizing and establishing equivalences between tree-structured problems and solutions, aiming to uncover principles underlying general problem-solving in AI.

Contribution

It develops a novel category-theoretic approach to model tree problems and demonstrates the existence of functors linking problem categories to solution categories, including metric-based equivalences.

Findings

Proves the existence of a functor between tree problem and solution categories.

Introduces a metric to quantify equivalences of tree problem categories.

Formalizes analogies across different problem domains using category theory.

Abstract

Artificial Intelligence (AI) has long pursued models, theories, and techniques to imbue machines with human-like general intelligence. Yet even the currently predominant data-driven approaches in AI seem to be lacking humans' unique ability to solve wide ranges of problems. This situation begs the question of the existence of principles that underlie general problem-solving capabilities. We approach this question through the mathematical formulation of analogies across different problems and solutions. We focus in particular on problems that could be represented as tree-like structures. Most importantly, we adopt a category-theoretic approach in formalising tree problems as categories, and in proving the existence of equivalences across apparently unrelated problem domains. We prove the existence of a functor between the category of tree problems and the category of solutions. We also provide a weaker version of the functor by quantifying equivalences of problem categories using a metric on tree problems.