We argue that all building blocks of transformer models can be expressed with a single concept: combinatorial Hopf algebra. Transformer learning emerges as a result of the subtle interplay between the algebraic and coalgebraic operations of the combinatorial Hopf algebra. Viewed through this lens, the transformer model becomes a linear time-invariant system where the attention mechanism computes a generalized convolution transform and the residual stream serves as a unit impulse. Attention-only transformers then learn by enforcing an invariant between these two paths. We call this invariant Hopf coherence. Due to this, with a degree of poetic license, one could call combinatorial Hopf algebras "tensors with a built-in loss function gradient". This loss function gradient occurs within the single layers and no backward pass is needed. This is in stark contrast to automatic differentiation which happens across the whole graph and needs a explicit backward pass. This property is the result of the fact that combinatorial Hopf algebras have the surprising property of calculating eigenvalues by repeated squaring.

This article contains a proposal to add coinduction to the computational apparatus of natural language understanding. This, we argue, will provide a basis for more realistic, computationally sound, and scalable models of natural language dialogue, syntax and semantics. Given that the bottom up, inductively constructed, semantic and syntactic structures are brittle, and seemingly incapable of adequately representing the meaning of longer sentences or realistic dialogues, natural language understanding is in need of a new foundation. Coinduction, which uses top down constraints, has been successfully used in the design of operating systems and programming languages. Moreover, implicitly it has been present in text mining, machine translation, and in some attempts to model intensionality and modalities, which provides evidence that it works. This article shows high level formalizations of some of such uses. Since coinduction and induction can coexist, they can provide a common language and a conceptual model for research in natural language understanding. In particular, such an opportunity seems to be emerging in research on compositionality. This article shows several examples of the joint appearance of induction and coinduction in natural language processing. We argue that the known individual limitations of induction and coinduction can be overcome in empirical settings by a combination of the the two methods. We see an open problem in providing a theory of their joint use.

Reinforcement learning algorithms solve sequential decision-making problems in probabilistic environments by optimizing for long-term reward. The desire to use reinforcement learning in safety-critical settings inspires a recent line of work on formally constrained reinforcement learning; however, these methods place the implementation of the learning algorithm in their Trusted Computing Base. The crucial correctness property of these implementations is a guarantee that the learning algorithm converges to an optimal policy. This paper begins the work of closing this gap by developing a Coq formalization of two canonical reinforcement learning algorithms: value and policy iteration for finite state Markov decision processes. The central results are a formalization of Bellman's optimality principle and its proof, which uses a contraction property of Bellman optimality operator to establish that a sequence converges in the infinite horizon limit. The CertRL development exemplifies how the Giry monad and mechanized metric coinduction streamline optimality proofs for reinforcement learning algorithms. The CertRL library provides a general framework for proving properties about Markov decision processes and reinforcement learning algorithms, paving the way for further work on formalization of reinforcement learning algorithms.