Universal Imitation Games

TL;DR

This paper introduces a categorical framework for analyzing various types of universal imitation games, extending Turing's original concept to include static, dynamic, and evolutionary settings using advanced mathematics.

Contribution

It develops a novel category-theoretic approach to characterize different classes of imitation games, including dynamic and evolutionary ones, and explores potential quantum extensions.

Findings

Categorical characterization of static, dynamic, and evolutionary UIGs.

Dynamic UIGs modeled as initial algebras over well-founded sets.

Evolutionary UIGs related to final coalgebras of non-well-founded sets.

Abstract

Alan Turing proposed in 1950 a framework called an imitation game to decide if a machine could think. Using mathematics developed largely after Turing -- category theory -- we analyze a broader class of universal imitation games (UIGs), which includes static, dynamic, and evolutionary games. In static games, the participants are in a steady state. In dynamic UIGs, "learner" participants are trying to imitate "teacher" participants over the long run. In evolutionary UIGs, the participants are competing against each other in an evolutionary game, and participants can go extinct and be replaced by others with higher fitness. We use the framework of category theory -- in particular, two influential results by Yoneda -- to characterize each type of imitation game. Universal properties in categories are defined by initial and final objects. We characterize dynamic UIGs where participants are learning by inductive inference as initial algebras over well-founded sets, and contrast them with participants learning by conductive inference over the final coalgebra of non-well-founded sets. We briefly discuss the extension of our categorical framework for UIGs to imitation games on quantum computers.