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An "element-free" probability distribution is what remains of a probability distribution after we forget the elements to which the probabilities were assigned. These objects naturally arise in Bayesian statistics, in situations where elements are used as labels and their specific identity is not important. This paper develops the structural theory of element-free distributions, using multisets and category theory. We give operations for moving between element-free and ordinary distributions, and we show that these operations commute with multinomial sampling. We then exploit this theory to prove two representation theorems. These theorems show that element-free distributions provide a natural representation for key random structures in Bayesian nonparametric clustering: exchangeable random partitions, and random distributions parametrized by a base measure., Comment: To appear at LICS 2024

Premonoidal categories and Freyd categories provide an encompassing framework for the semantics of call-by-value programming languages. Premonoidal categories are a weakening of monoidal categories in which the interchange law for the tensor product may not hold, modelling the fact that effectful programs cannot generally be re-ordered. A Freyd category is a pair of categories with the same objects: a premonoidal category of general programs, and a monoidal category of 'effect-free' programs which do admit re-ordering. Certain recent innovations in semantics, however, have produced models which are not categories but bicategories. Here we develop the theory to capture such examples by introducing premonoidal and Freyd structure in a bicategorical setting. The second dimension introduces new subtleties, so we verify our definitions with several examples and a correspondence theorem (between Freyd bicategories and certain actions of monoidal bicategories) which parallels the categorical framework., Comment: In Proceedings ACT 2023, arXiv:2312.08138

We develop the theory of strong and commutative monads in the 2-dimensional setting of bicategories. This provides a framework for the analysis of effects in many recent models which form bicategories and not categories, such as those based on profunctors, spans, or strategies over games. We then show how the 2-dimensional setting provides new insights into the semantics of concurrent functional programs. We introduce concurrent pseudomonads, which capture the fundamental weak interchange law connecting parallel composition and sequential composition. This notion brings to light an intermediate level, strictly between strength and commutativity, which is invisible in traditional categorical models. We illustrate the concept with the continuation pseudomonad in concurrent game semantics. In developing this theory, we take care to understand the coherence laws governing the structural 2-cells. We give many examples and prove a number of practical and foundational results., Comment: LICS 2024 + Appendix

We introduce a bicategorical model of linear logic which is a novel variation of the bicategory of groupoids, profunctors, and natural transformations. Our model is obtained by endowing groupoids with additional structure, called a kit, to stabilize the profunctors by controlling the freeness of the groupoid action on profunctor elements. The theory of generalized species of structures, based on profunctors, is refined to a new theory of \emph{stable species} of structures between groupoids with Boolean kits. Generalized species are in correspondence with analytic functors between presheaf categories; in our refined model, stable species are shown to be in correspondence with restrictions of analytic functors, which we characterize as being stable, to full subcategories of stabilized presheaves. Our motivating example is the class of finitary polynomial functors between categories of indexed sets, also known as normal functors, that arises from kits enforcing free actions. We show that the bicategory of groupoids with Boolean kits, stable species, and natural transformations is cartesian closed. This makes essential use of the logical structure of Boolean kits and explains the well-known failure of cartesian closure for the bicategory of finitary polynomial functors between categories of set-indexed families and cartesian natural transformations. The paper additionally develops the model of classical linear logic underlying the cartesian closed structure and clarifies the connection to stable domain theory.

Two families of denotational models have emerged from the semantic analysis of linear logic: dynamic models, typically presented as game semantics, and static models, typically based on a category of relations. In this paper we introduce a formal bridge between a dynamic model and a static model: the model of thin concurrent games and strategies, based on event structures, and the model of generalized species of structures, based on distributors. A special focus of this paper is the two-dimensional nature of the dynamic-static relationship, which we formalize with double categories and bicategories. In the first part of the paper, we construct a symmetric monoidal oplax functor from linear concurrent strategies to distributors. We highlight two fundamental differences between the two models: the composition mechanism, and the representation of resource symmetries. In the second part of the paper, we adapt established methods from game semantics (visible strategies, payoff structure) to enforce a tighter connection between the two models. We obtain a cartesian closed pseudofunctor, which we exploit to shed new light on recent results in the theory of the lambda-calculus., Comment: LICS 2023 version replaced with extended version, with proofs and some additional material

A recurring problem in game semantics is to enforce uniformity in strategies. Informally, a strategy is uniform when the Player's behaviour does not depend on the particular indexing of moves chosen by the Opponent. In game semantics, uniformity is used to define a resource modality !, that can be exploited for the semantics of programming languages. In this paper we give a new account of uniformity for strategies on event structures. This work is inspired by an older idea due to Melli\`es, that uniformity should be expressed as "bi-invariance" with respect to two interacting group actions. We explore the algebraic foundations of bi-invariance, adapt this idea to the language of event structures and define a general notion of uniform strategy in this context. Finally we revisit an existing approach to uniformity, and show how this arises as a special case of our constructions., Comment: Proceedings of MFPS 2022 (Final version)

We show that streams and lazy data structures are a natural idiom for programming with infinite-dimensional Bayesian methods such as Poisson processes, Gaussian processes, jump processes, Dirichlet processes, and Beta processes. The crucial semantic idea, inspired by developments in synthetic probability theory, is to work with two separate monads: an affine monad of probability, which supports laziness, and a commutative, non-affine monad of measures, which does not. (Affine means that $T(1)\cong 1$.) We show that the separation is important from a decidability perspective, and that the recent model of quasi-Borel spaces supports these two monads. To perform Bayesian inference with these examples, we introduce new inference methods that are specially adapted to laziness; they are proven correct by reference to the Metropolis-Hastings-Green method. Our theoretical development is implemented as a Haskell library, LazyPPL., Comment: Accepted for POPL 2023

We explore links between the thin concurrent games of Castellan, Clairambault and Winskel, and the weighted relational models of linear logic studied by Laird, Manzonetto, McCusker and Pagani. More precisely, we show that there is an interpretationpreserving "collapse" functor from the former to the latter. On objects, the functor defines for each game a set of possible execution states. Defining the action on morphisms is more subtle, and this is the main contribution of the paper. Given a strategy and an execution state, our functor needs to count the witnesses for this state within the strategy. Strategies in thin concurrent games describe non-linear behaviour explicitly, so in general each witness exists in countably many symmetric copies. The challenge is to define the right notion of witnesses, factoring out this infinity while matching the weighted relational model. Understanding how witnesses compose is particularly subtle and requires a delve into the combinatorics of witnesses and their symmetries. In its basic form, this functor connects thin concurrent games and a relational model weighted by N $\cup$ {+$\infty$}. We will additionally consider a generalised setting where both models are weighted by elements of an arbitrary continuous semiring; this covers the probabilistic case, among others. Witnesses now additionally carry a value from the semiring, and our interpretation-preserving collapse functor extends to this setting.

We study the differential properties of higher-order statistical probabilistic programs with recursion and conditioning. Our starting point is an open problem posed by Hongseok Yang: what class of statistical probabilistic programs have densities that are differentiable almost everywhere? To formalise the problem, we consider Statistical PCF (SPCF), an extension of call-by-value PCF with real numbers, and constructs for sampling and conditioning. We give SPCF a sampling-style operational semantics a la Borgstrom et al., and study the associated weight (commonly referred to as the density) function and value function on the set of possible execution traces. Our main result is that almost-surely terminating SPCF programs, generated from a set of primitive functions (e.g. the set of analytic functions) satisfying mild closure properties, have weight and value functions that are almost-everywhere differentiable. We use a stochastic form of symbolic execution to reason about almost-everywhere differentiability. A by-product of this work is that almost-surely terminating deterministic (S)PCF programs with real parameters denote functions that are almost-everywhere differentiable. Our result is of practical interest, as almost-everywhere differentiability of the density function is required to hold for the correctness of major gradient-based inference algorithms.