Received We investigate universal algebra over the category Nom of nominal sets. Using the fact that Nom is a full reflective subcategory of a monadic category, we obtain an HSP-like theorem for algebras over nominal sets. We isolate a 'uniform' fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give… (More)
Overview Motivation Functors with finitary presentations Equational logic for higher-order abstract syntax Modular coalgebraic logic Uniform completeness proofs Motivation • Logics for T-coalgebras are suitably described by endofunctors on A A 2 2 L , , X r r T p p • Functors having finitary presentation by operations and equations give rise to adequate… (More)
Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for… (More)
The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks… (More)
We develop the coalgebraic theory of nominal Kleene algebra , including an alternative language-theoretic semantics, a nominal extension of the Brzozowski derivative, and a bisimulation-based decision procedure for the equational theory.
We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.
The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach lies the existence of an adjunction of descent type… (More)
This paper studies several applications of the notion of a presentation of a functor by operations and equations. We show that the technically straightforward generalisation of this notion from the one-sorted to the many-sorted case has several interesting consequences. First, it can be applied to give equational logic for the binding algebras modelling… (More)
Gabbay and Pitts proved that lambda-terms up to alpha-equivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambda-calculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alpha-equivalence classes of finite and infinite… (More)