arXiv subjects

Toby St. Clere Smithe

Publications and source records attributed to Toby St. Clere Smithe.

6 recordsLinked to original sources

arXiv2026

First Steps towards Categorical Algebraic Artificial Chemistry

We construct a functor that gives a dynamics to an algebraic model of interacting components. The construction generalises a computational model of Fontana and Buss in the field of artificial life known as AlChemy, in which molecules and their chemical interactions are emulated by lambda calculus terms and their application and subsequent reduction. We discuss future directions for the application of category theory to algebraic artificial chemistry as an organisational tool, with a focus on formalising the connection between the algebraic and the dynamical facets of such models.

cs.CE↗

arXiv2023

Open Dynamical Systems as Coalgebras for Polynomial Functors, with Application to Predictive Processing

We present categories of open dynamical systems with general time evolution as categories of coalgebras opindexed by polynomial interfaces, and show how this extends the coalgebraic framework to capture common scientific applications such as ordinary differential equations, open Markov processes, and random dynamical systems. We then extend Spivak's operad Org to this setting, and construct associated monoidal categories whose morphisms represent hierarchical open systems; when their interfaces are simple, these categories supply canonical comonoid structures. We exemplify these constructions using the 'Laplace doctrine', which provides dynamical semantics for active inference, and indicate some connections to Bayesian inversion and coalgebraic logic.

math.CT↗

arXiv2022

Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines

We develop the compositional theory of active inference by introducing activity, functorially relating statistical games to the dynamical systems which play them, using the new notion of approximate inference doctrine. In order to exhibit such functors, we first develop the necessary theory of dynamical systems, using a generalization of the language of polynomial functors to supply compositional interfaces of the required types: with the resulting polynomially indexed categories of coalgebras, we construct monoidal bicategories of differential and dynamical ``hierarchical inference systems'', in which approximate inference doctrines have semantics. We then describe ``externally parameterized'' statistical games, and use them to construct two approximate inference doctrines found in the computational neuroscience literature, which we call the `Laplace' and the `Hebb-Laplace' doctrines: the former produces dynamical systems which optimize the posteriors of Gaussian models; and the latter produces systems which additionally optimize the parameters (or `weights') which determine their predictions.

nlin.AO↗

arXiv2022

Compositional Active Inference I: Bayesian Lenses. Statistical Games

We introduce the concepts of Bayesian lens, characterizing the bidirectional structure of exact Bayesian inference, and statistical game, formalizing the optimization objectives of approximate inference problems. We prove that Bayesian inversions compose according to the compositional lens pattern, and exemplify statistical games with a number of classic statistical concepts, from maximum likelihood estimation to generalized variational Bayesian methods. This paper is the first in a series laying the foundations for a compositional account of the theory of active inference, and we therefore pay particular attention to statistical games with a free-energy objective.

math.ST↗

arXiv2021

Some Notions of (Open) Dynamical System on Polynomial Interfaces

We define indexed categories of (open) dynamical system and random dynamical system over polynomial interfaces, where time is given by an arbitrary monoid $\mathbb{T}$. We consider the case of open random dynamical systems over both open and closed noise sources, and the case where the interface of the random system is `nested' over the interface of its noise source. We show that, in discrete time, our categories of dynamical systems over polynomial interfaces $p$ are equivalent to Spivak's categories $p$-$\mathbf{Coalg}$ of $p$-coalgebras. We then define a notion of generalized $pT$-coalgebra for a monad $T$, thereby extending the coalgebraic notion of dynamical system to general time, and show that this construction bestows a notion of open Markov process when the monad $T$ is a probability monad. Finally, we list some further connections and open questions.

math.DS↗

arXiv2020

Bayesian Updates Compose Optically

Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may ask whether composing the inversions of the component processes gives the same belief update as the inversion of the whole. We answer this question affirmatively, showing that the relevant compositional structure is precisely that of the lens pattern, and that we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a corresponding fibred category. We define a general notion of (mixed) Bayesian lens, and discuss the (un)lawfulness of these lenses when their contravariant components are exact Bayesian inversions. We prove our main result both abstractly and concretely, for both discrete and continuous states, taking care to illustrate the common structures.

math.CT↗