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Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a…
Many applications, e.g., Web service composition, complex system design, team formation, etc., rely on methods for identifying collections of objects or entities satisfying some functional requirement. Among the collections that satisfy the…
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…
Gradually typed programming languages, which allow for soundly mixing static and dynamically typed programming styles, present a strong challenge for metatheorists. Even the simplest sound gradually typed languages feature at least…
The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories…
While real world challenges typically define visual categories with language words or phrases, most visual classification methods define categories with numerical indices. However, the language specification of the classes provides an…
The $\pi$-calculus is used as a model for programming languages. Its contexts exhibit arbitrary concurrency, making them very discriminating. This may prevent validating desirable behavioural equivalences in cases when more disciplined…
Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have…
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to…
This paper defines multiset-valued linear index grammar and unordered vector grammar with dominance links. The former models certain uses of multiset-valued feature structures in unification-based formalisms, while the latter is motivated…
We introduce a variation on Barthe et al.'s higher-order logic in which formulas are interpreted as predicates over open rather than closed objects. This way, concepts which have an intrinsically functional nature, like continuity,…
We consider the call-by-value lambda-calculus extended with a may-convergent non-deterministic choice and a must-convergent parallel composition. Inspired by recent works on the relational semantics of linear logic and non-idempotent…
Monads play an important role in both the syntax and semantics of modern functional programming languages. The problem of combining them has been of profound interest at least since the 90s, and different approaches have been employed to…
A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and…
In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on…
We consider the challenge of preference elicitation in systems that help users discover the most desirable item(s) within a given database. Past work on preference elicitation focused on structured models that provide a factored…
In this paper we describe how to leverage higher-order unification to type check a dependently typed language with meta-variables. The literature usually presents the unification algorithm as a standalone component, however the need to…
The humble $\dagger$ ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains).…