Universal Imitation Games

TL;DR

The paper builds a unified, category-theoretic framework for universal imitation games (UIGs) by casting interactions as functors and employing Yoneda-based universality to unify static, dynamic, and evolutionary imitation. It interprets learning through inductive (initial algebras) and coinductive (final coalgebras) lenses, extends these ideas to non-well-founded sets, and connects reinforcement learning, causal inference, and variational inequalities within a coalgebraic setting. Key contributions include a taxonomy of UIGs, a coinductive learning paradigm (coidentification in the limit), and a metric-coinduction approach to solving VIs, with explicit links to network economics of generative AI and evolutionary dynamics. The framework aims to provide a versatile, mathematically principled toolbox for analyzing AI systems, causal models, RL, and evolving strategic interactions across classical and quantum computational substrates.

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.

Figures (38)

• Figure 1: Alan Turing proposed imitation games as a way to pose the problem of whether machines could think.
• Figure 2: In Turing's imitation game, the identity of a participant is determined by a series of questions and answers (the specific questions above are from Turing's original paper turing. We model interactions as ${\cal C}(-, x): {\cal C}^{op} \rightarrow {\cal D}$ and ${\cal C}(x, -): {\cal C} \rightarrow {\cal D}$ as contravariant and covariant functors, respectively, from a category ${\cal C}$ into another category ${\cal D}$ (such as Sets or any other enriched category). Two key results by Yoneda provide the theoretical foundation for our paper. The Yoneda Lemma shows objects can be defined purely in terms of these contravariant and covariant functors. In addition, Yoneda investigated bivalent functors $F: {\cal C}^{op} \times {\cal C} \rightarrow {\cal D}$ that combine both contravariant and covariant actions, and proposed a categorical "integral calculus" over coends and ends, which reveal deep similarities between generative probabilistic models, distance-based models, kernel methods, optimal transport and topological representations.
• Figure 3: The theoretical foundation of our approach to universal imitation games is based on two celebrated results of Yoneda. The first (top row) shows that Yoneda embeddings $\text{H}(x) = C(-, x)$ are universal representers of objects in a category. The second (middle row) is based on Yoneda's categorical "integral calculus" using coends and ends that unify diverse approaches to solving imitation games studied in AI over the past six decades. The bottom row shows that universal properties defined by initial and final objects provide a unified way to characterize approaches to dynamic UIGs using passive learning (inductive inference) as well as active learning (causal inference and reinforcement learning).
• Figure 4: A classification of universal imitation games based on the characteristics of the participants and the framework employed to discriminate among them.
• Figure 5: In the study of dynamic UIGs, we contrast the approach of inductive inference, which we associate with initial algebras over well-founded sets, with the framework of coinductive inference, which relates to final coalgebras over non-well-founded sets. Algebras can be defined as mappings $F(X) \rightarrow X$, for some functor $F$ on a category ${\cal C}$ (e.g., of Sets), where $X$ is some object in ${\cal C}$, whereas coalgebras go in the opposite direction $X \rightarrow F(X)$.

Theorems & Definitions (154)

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