arXiv subjects

Sean Moss

Publications and source records attributed to Sean Moss.

9 recordsLinked to original sources

arXiv2025

A causal Markov category with Kolmogorov products

In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an example where the deterministic subcategory is the category of Stone spaces (i.e. the dual of the category of Boolean algebras) and the kernels correspond to a restricted class of Kleisli arrows for the Radon monad. We look at this from two perspectives. First via pro-completions and Stone spaces directly. Second via duality with Boolean and algebras and effect algebras.

math.CT↗

arXiv2025

Theory-Grounded Evaluation of Human-Like Fallacy Patterns in LLM Reasoning

We study logical reasoning in language models by asking whether their errors follow established human fallacy patterns. Using the Erotetic Theory of Reasoning (ETR) and its open-source implementation, PyETR, we programmatically generate 383 formally specified reasoning problems and evaluate 38 models. For each response, we judge logical correctness and, when incorrect, whether it matches an ETR-predicted fallacy. Two results stand out: (i) as a capability proxy (Chatbot Arena Elo) increases, a larger share of a model's incorrect answers are ETR-predicted fallacies $(\rho=0.360, p=0.0265)$, while overall correctness on this dataset shows no correlation with capability; (ii) reversing premise order significantly reduces fallacy production for many models, mirroring human order effects. Methodologically, PyETR provides an open-source pipeline for unbounded, synthetic, contamination-resistant reasoning tests linked to a cognitive theory, enabling analyses that focus on error composition rather than error rate.

cs.CL↗

arXiv2024

Scoped Effects as Parameterized Algebraic Theories

Notions of computation can be modelled by monads. Algebraic effects offer a characterization of monads in terms of algebraic operations and equational axioms, where operations are basic programming features, such as reading or updating the state, and axioms specify observably equivalent expressions. However, many useful programming features depend on additional mechanisms such as delimited scopes or dynamically allocated resources. Such mechanisms can be supported via extensions to algebraic effects including scoped effects and parameterized algebraic theories. We present a fresh perspective on scoped effects by translation into a variation of parameterized algebraic theories. The translation enables a new approach to equational reasoning for scoped effects and gives rise to an alternative characterization of monads in terms of generators and equations involving both scoped and algebraic operations. We demonstrate the power of our fresh perspective by way of equational characterizations of several known models of scoped effects.

cs.PL↗

arXiv2023

Denotational semantics for languages for inference: semirings, monads, and tensors

Computational effects are commonly modelled by monads, but often a monad can be presented by an algebraic theory of operations and equations. This talk is about monads and algebraic theories for languages for inference, and their connections to semirings and tensors. A basic class of examples of algebraic theories comes from considering the theory of modules for a semiring, e.g. the theory of unnormalized distributions, where the semiring is that of the non-negative real numbers. We propose that an interesting perspective is given by studying theories via semirings, and to this end explore several examples of subtheories of module theories, mostly relating to probability. Our main contribution concerns the commutative combination of effects, as studied by Hyland, Plotkin and Power: we observe that while the semiring tensor does not in general determine the tensor of subtheories of module theories, it still does in several fundamental probabilistic examples.

cs.LO↗

arXiv2022

A category-theoretic proof of the ergodic decomposition theorem

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, - ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories); - the invariant $\sigma$-algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much simpler than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

math.DS↗

arXiv2022

Concrete categories and higher-order recursion (With applications including probability, differentiability, and full abstraction)

We study concrete sheaf models for a call-by-value higher-order language with recursion. Our family of sheaf models is a generalization of many examples from the literature, such as models for probabilistic and differentiable programming, and fully abstract logical relations models. We treat recursion in the spirit of synthetic domain theory. We provide a general construction of a lifting monad starting from a class of admissible monomorphisms in the site of the sheaf category. In this way, we obtain a family of models parametrized by a concrete site and a class of monomorphisms, for which we prove a general computational adequacy theorem.

cs.PL↗

arXiv2022

Probability monads with submonads of deterministic states - Extended version

Probability theory can be studied synthetically as the computational effect embodied by a commutative monad. In the recently proposed Markov categories, one works with an abstraction of the Kleisli category and then defines deterministic morphisms equationally in terms of copying and discarding. The resulting difference between 'pure' and 'deterministic' leads us to investigate the 'sober' objects for a probability monad, for which the two concepts coincide. We propose natural conditions on a probability monad which allow us to identify the sober objects and define an idempotent sobrification functor. Our framework applies to many examples of interest, including the Giry monad on measurable spaces, and allows us to sharpen a previously given version of de Finetti's theorem for Markov categories. This is an extended version of the paper accepted for the Logic In Computer Science (LICS) conference 2022. In this document we include more mathematical details, including all the proofs, of the statements and constructions given in the published version.

cs.LO↗

arXiv2021

Recursion and Sequentiality in Categories of Sheaves

We present a fully abstract model of a call-by-value language with higher-order functions, recursion and natural numbers, as an exponential ideal in a topos. Our model is inspired by the fully abstract models of O'Hearn, Riecke and Sandholm, and Marz and Streicher. In contrast with semantics based on cpo's, we treat recursion as just one feature in a model built by combining a choice of modular components.

cs.PL↗

arXiv2015

Another approach to the Kan-Quillen model structure

By careful analysis of the comparison map from a simplicial set to its image under Kan's ex-infinity functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we are able to quickly deduce some basic facts about ex-infinity and hence provide a new construction of the Kan-Quillen model structure on simplicial sets, which avoids the use of topological spaces or minimal fibrations.

math.CT↗