arXiv subjects

Jules Hedges

Publications and source records attributed to Jules Hedges.

At least 19 recordsLinked to original sources

arXiv2025

Canonical bidirectional typechecking

We demonstrate that the checkable/synthesisable split in bidirectional typechecking coincides with existing dualities in polarised System L, also known as polarised $\mu\tilde{\mu}$-calculus. Specifically, positive terms and negative coterms are checkable, and negative terms and positive coterms are synthesisable. This combines a standard formulation of bidirectional typechecking with Zeilberger's `cocontextual' variant. We extend this to ordinary `cartesian' System L using Mc Bride's co-de Bruijn formulation of scopes, and show that both can be combined in a linear-nonlinear style, where linear types are positive and cartesian types are negative. This yields a remarkable 3-way coincidence between the shifts of polarised System L, LNL calculi, and bidirectional calculi.

cs.PL↗

arXiv2024

Rendering string diagrams recursively

String diagrams are a graphical language used to represent processes that can be composed sequentially or in parallel, which correspond graphically to horizontal or vertical juxtaposition. In this paper we demonstrate how to compute the layout of a string diagram by folding over its algebraic representation in terms of sequential and parallel composition operators. The algebraic representation can be seen as a term of a free monoidal category or a proof tree for a small fragment of linear logic. This contrasts to existing non-compositional approaches that use graph layout techniques. The key innovation is storing the diagrams in binary space-partition trees, maintaining a right-trapezoidal shape for the diagram's outline as an invariant. We provide an implementation in Haskell, using an existing denotational graphics library called Diagrams. Our renderer also supports adding semantics to diagrams to serve as a compiler, with matrix algebra used as an example.

math.CT↗

arXiv2024

Reinforcement Learning in Categorical Cybernetics

We show that several major algorithms of reinforcement learning (RL) fit into the framework of categorical cybernetics, that is to say, parametrised bidirectional processes. We build on our previous work in which we show that value iteration can be represented by precomposition with a certain optic. The outline of the main construction in this paper is: (1) We extend the Bellman operators to parametrised optics that apply to action-value functions and depend on a sample. (2) We apply a representable contravariant functor, obtaining a parametrised function that applies the Bellman iteration. (3) This parametrised function becomes the backward pass of another parametrised optic that represents the model, which interacts with an environment via an agent. Thus, parametrised optics appear in two different ways in our construction, with one becoming part of the other. As we show, many of the major classes of algorithms in RL can be seen as different extremal cases of this general setup: dynamic programming, Monte Carlo methods, temporal difference learning, and deep RL. We see this as strong evidence that this approach is a natural one and believe that it will be a fruitful way to think about RL in the future.

cs.LG↗

arXiv2023

Bayesian open games

This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.

cs.GT↗

arXiv2023

Value Iteration is Optic Composition

Dynamic programming is a class of algorithms used to compute optimal control policies for Markov decision processes. Dynamic programming is ubiquitous in control theory, and is also the foundation of reinforcement learning. In this paper, we show that value improvement, one of the main steps of dynamic programming, can be naturally seen as composition in a category of optics, and intuitively, the optimal value function is the limit of a chain of optic compositions. We illustrate this with three classic examples: the gridworld, the inverted pendulum and the savings problem. This is a first step towards a complete account of reinforcement learning in terms of parametrised optics.

math.CT↗

arXiv2023

The Compositional Structure of Bayesian Inference

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 observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference.

math.CT↗

arXiv2023

Composing games into complex institutions

Game theory is used by all behavioral sciences, but its development has long centered around tools for relatively simple games and toy systems, such as the economic interpretation of equilibrium outcomes. Our contribution, compositional game theory, permits another approach of equally general appeal: the high-level design of large games for expressing complex architectures and representing real-world institutions faithfully. Compositional game theory, grounded in the mathematics underlying programming languages, and introduced here as a general computational framework, increases the parsimony of game representations with abstraction and modularity, accelerates search and design, and helps theorists across disciplines express real-world institutional complexity in well-defined ways. Relative to existing approaches in game theory, compositional game theory is especially promising for solving game systems with long-range dependencies, for comparing large numbers of structurally related games, and for nesting games into the larger logical or strategic flows typical of real world policy or institutional systems.

cs.GT↗

arXiv2022

Towards Foundations of Categorical Cybernetics

We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach to compositional game theory closely related to open games, in which the controller is a composite of game-theoretic agents. We believe that 'cybernetic' is an appropriate name for the processes that can be described in this framework.

math.CT↗

arXiv2022

Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion

We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian inversion.

math.CT↗

arXiv2022

Compositional modelling of network games

The analysis of games played on graph-like structures is of increasing importance due to the prevalence of social networks, both virtual and physical, in our daily life. As well as being relevant in computer science, mathematical analysis and computer simulations of such distributed games are vital methodologies in economics, politics and epidemiology, amongst other fields. Our contribution is to give compositional semantics of a family of such games as a well-behaved mapping, a strict monoidal functor, from a category of open graphs (syntax) to a category of open games (semantics). As well as introducing the theoretical framework, we identify some applications of compositionality.

cs.GT↗

arXiv2021

Fibre optics

Lenses, optics and dependent lenses (or equivalently morphisms of containers, or equivalently natural transformations of polynomial functors) are all widely used in applied category theory as models of bidirectional processes. From the definition of lenses over a finite product category, optics weaken the required structure to actions of monoidal categories, and dependent lenses make use of the additional property of finite completeness (or, in case of polynomials, even local cartesian closure). This has caused a split in the applied category theory literature between those using optics and those using dependent lenses. The goal of this paper is to unify optics with dependent lenses, by finding a definition of fibre optics admitting both as special cases.

math.CT↗

arXiv2020

Sheet diagrams for bimonoidal categories

Bimonoidal categories (also known as rig categories) are categories with two monoidal structures, one of which distributes over the other. We formally define sheet diagrams, a graphical calculus for bimonoidal categories that was informally introduced by Staton. Sheet diagrams are string diagrams drawn on a branching surface, which is itself an extruded string diagram. Our main result is a soundness and completeness theorem of the usual form for graphical calculi: we show that sheet diagrams form the free bimonoidal category on a signature.

math.CT↗

arXiv2020

The game semantics of game theory

We use a reformulation of compositional game theory to reunite game theory with game semantics, by viewing an open game as the System and its choice of contexts as the Environment. Specifically, the system is jointly controlled by $n \geq 0$ noncooperative players, each independently optimising a real-valued payoff. The goal of the system is to play a Nash equilibrium, and the goal of the environment is to prevent it. The key to this is the realisation that lenses (from functional programming) form a dialectica category, which have an existing game-semantic interpretation. In the second half of this paper, we apply these ideas to build a compact closed category of `computable open games' by replacing the underlying dialectica category with a wave-style geometry of interaction category, specifically the Int-construction applied to the cartesian monoidal category of directed-complete partial orders.

cs.GT↗

arXiv2019

Foundations of brick diagrams

We discuss the foundations of 2-dimensional graphical languages, with a view towards their computer implementation in a 'compiler' for monoidal categories. In particular, we discuss the close relationship between string diagrams, pasting diagrams, linear logic proof trees and k-d trees, the last being a data structure from computational geometry and computer graphics that recursively partitions a space. We introduce a minor variant of cubical pasting diagrams, which we call 'brick diagrams', which are used in the Statebox visual programming language. This paper is intended as a discussion and literature review, and contains little mathematics.

math.CT↗

arXiv2019

Limits of bimorphic lenses

Bimorphic lenses are a simplification of polymorphic lenses that (like polymorphic lenses) have a type defined by 4 parameters, but which are defined in a monomorphic type system (i.e. an ordinary category with finite products). We show that the category of bimorphic lenses is complete when the base category is complete, cocomplete and cartesian closed, and so symmetric bimorphic lenses can be defined as spans of ordinary bimorphic lenses. This is in contrast to monomorphic lenses, which do not have pullbacks, and for which the category of spans can be defined in an ad-hoc way only when the lenses satisfy a certain axiom (the put-get law). This is a step towards a theory of symmetric polymorphic lenses. Bimorphic lenses additionally play an essential role in compositional game theory, and spans of bimorphic lenses are a step towards a compact closed category of open games.

math.CT↗

arXiv2019

Sequential games and nondeterministic selection functions

This paper analyses Escardó and Oliva's generalisation of selection functions over a strong monad from a game-theoretic perspective. We focus on the case of the nondeterminism (finite nonempty powerset) monad $\mathcal{P}$. We use these nondeterministic selection functions of type $\mathcal{J}^{\mathcal{P}}_R X = (X \rightarrow R) \rightarrow \mathcal{P} (X)$ to study sequential games, extending previous work linking (deterministic) selection functions to game theory. Similar to deterministic selection functions, which compute a subgame perfect Nash equilibrium play of a game, we characterise those non-deterministic selection functions which have a clear game-theoretic interpretation. We show, surprisingly, no non-deterministic selection function exists which computes the set of all subgame perfect Nash equilibrium plays. Instead we show that there are selection functions corresponding to sequential versions of the iterated removal of strictly dominated strategies.

cs.GT↗

arXiv2019

From open learners to open games

The categories of open learners (due to Fong, Spivak and Tuyéras) and open games (due to the present author, Ghani, Winschel and Zahn) bear a very striking and unexpected similarity. The purpose of this short note is to prove that there is a faithful symmetric monoidal functor from the former to the latter, which means that any supervised neural network (without feedback or other complicating features) can be seen as an open game in a canonical way. Roughly, each parameter is controlled by a different player, and the game's best response relation encodes the dynamics of gradient descent. We suggest paths for further work exploiting the link.

math.CT↗

arXiv2018

Towards Functorial Language-Games

In categorical compositional semantics of natural language one studies functors from a category of grammatical derivations (such as a Lambek pregroup) to a semantic category (such as real vector spaces). We compositionally build game-theoretic semantics of sentences by taking the semantic category to be the category whose morphisms are open games. This requires some modifications to the grammar category to compensate for the failure of open games to form a compact closed category. We illustrate the theory using simple examples of Wittgenstein's language-games.

cs.LO↗