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We present a new model of computation, described in terms of monoidal categories. It conforms the Church-Turing Thesis, and captures the same computable functions as the standard models. It provides a succinct categorical interface to most…
We introduce string diagrams as a formal mathematical, graphical language to represent, compose, program and reason about games. The language is well established in quantum physics, quantum computing and quantum linguistic with the…
Bond graph is a unified graphical approach for describing the dynamics of complex engineering and physical systems and is widely adopted in a variety of domains, such as, electrical, mechanical, medical, thermal and fluid mechanics.…
Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…
Modal logic is a paradigm for several useful and applicable formal systems in computer science. It generally retains the low complexity of classical propositional logic, but notable exceptions exist in the domains of description, temporal,…
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…
The study of Description Logics have been historically mostly focused on features that can be translated to decidable fragments of first-order logic. In this paper, we leave this restriction behind and look for useful and decidable…
Although neural sequence-to-sequence models have been successfully applied to semantic parsing, they fail at compositional generalization, i.e., they are unable to systematically generalize to unseen compositions of seen components.…
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to…
We propose a set of compositional design patterns to describe a large variety of systems that combine statistical techniques from machine learning with symbolic techniques from knowledge representation. As in other areas of computer science…
This work concerns the proof theory of (left) skew monoidal categories and their variants (e.g. closed monoidal, symmetric monoidal), continuing the line of work initiated in recent years by Uustalu et al. Skew monoidal categories are a…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
This paper introduces a new kind of propositional encoding for reasoning about partial orders. The symbols in an unspecified partial order are viewed as variables which take integer values and are interpreted as indices in the order. For a…
We suggest an enhancement to structural coding through the use of (a) causally bound codes, (b) basic constructs of graph theory and (c) statistics. As is the norm with structural coding, the codes are collected into categories. The…
Morphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal…
We present in this paper a reformulation of the usual set-theoretical semantics of the description logic $\mathcal{ALC}$ with general TBoxes by using categorical language. In this setting, $\mathcal{ALC}$ concepts are represented as…
We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof…
In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category. In particular, categorical quantum…
Inspired by recent work on the categorical semantics of dependent type theories, we investigate the following question: When is logical structure (crucially, dependent-product and subobject-classifier structure) induced from a category to…
We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe…