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With a view on applications in computing, in particular concurrency theory and higher-dimensional rewriting, we develop notions of $n$-fold monoid and comonoid objects in $n$-fold monoidal categories and bicategories. We present a series of…
Linearly distributive categories (LDC), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivities. A seminal result in monoidal…
Milner's bigraphs are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the pi-calculus and the Ambient calculus. This paper is only concerned with…
We provide a new approach to categorical graph and hypergraph theory by using categorical syntax and semantics. For each monoid $M$ and action on a set $X$, there is an associated presheaf topos of $(X,M)$-graphs where each object can be…
A major challenge in the deployment of scientific software solutions is the adaptation of research prototypes to production-grade code. While high-level languages like MATLAB are useful for rapid prototyping, they lack the resource…
The paper introduces a novel framework based on category theory to enhance the explainability of artificial intelligence systems, particularly focusing on word embeddings. Key topics include the construction of categories $\mathcal{L}_T$…
We consider the scenario in which a set of sources generate messages in a network and a receiver node demands an arbitrary linear function of these messages. We formulate an algebraic test to determine whether an arbitrary network can…
Arrays are such a rich and fundamental data type that they tend to be built into a language, either in the compiler or in a large low-level library. Defining this functionality at the user level instead provides greater flexibility for…
The idea of using unfolding as a way of computing a program semantics has been applied successfully to logic programs and has shown itself a powerful tool that provides concrete, implementable results, as its outcome is actually source…
We present an implementation in the functional programming language Haskell of the PLE decomposition of matrices over division rings. Our benchmarks indicate that it is competitive with the C-based implementation provided in Flint.…
An embedding is a function that maps entities from one algebraic structure into another while preserving certain characteristics. Embeddings are being used successfully for mapping relational data or text into vector spaces where they can…
Structured generation for LLM tool use highlights the value of compact DSL intermediate representations (IRs) that can be emitted directly and parsed deterministically. This paper introduces axial grammar: linear token sequences that…
Ordered, linear, and other substructural type systems allow us to expose deep properties of programs at the syntactic level of types. In this paper, we develop a family of unary logical relations that allow us to prove consequences of…
This paper is an exploration in a functional programming framework of {\em isomorphisms} between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs,…
Despite enormous progress in Natural Language Processing (NLP), our field is still lacking a common deep semantic representation scheme. As a result, the problem of meaning and understanding is typically sidestepped through more simple,…
Implementing a complex concept as an executable model in a strongly typed, purely functional language hits a sweet spot between mere simulation and formal specification. For research and education it is often desirable to enrich the…
We present the design of a new functional programming language, MLTS, that uses the lambda-tree syntax approach to encoding bindings appearing within data structures. In this approach, bindings never become free nor escape their scope:…
Network models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by…
Freer monads are a useful structure commonly used in various domains due to their expressiveness. However, a known issue with freer monads is that they are not amenable to static analysis. This paper explores freer arrows, a relatively…
A challenge in advancing Visual-Language Models (VLMs) is determining whether their failures on abstract reasoning tasks, such as Bongard problems, stem from flawed perception or faulty top-down reasoning. To disentangle these factors, we…