Since its discovery, differential linear logic (DLL) inspired numerous domains. In denotational semantics, categorical models of DLL are now commune, and the simplest one is Rel, the category of sets and relations. In proof theory this naturally gave birth to differential proof nets that are full and complete for DLL. In turn, these tools can naturally be translated to their intuitionistic coun...

A block is a language construct in programming that temporarily enlarges the state space. It is typically opened by initialising some local variables, and closed via a return statement. The “scope” of these local variables is then restricted to the block in which they occur. In quantum computation such temporary extensions of the state space also play an important role. This paper axiomatises “...

A class K of coalgebras for an endofunctor T : Set→ Set is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). K may be thought of as the class of all coalgebras that satisfy some computationally significant property. In any logical system suitable for specif...

For functors L : A→ B and R : B→ A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B, MorB(L(A), B) α // MorA(A,R(B)) β oo . (L,R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category B, and there is a well-known correspondence between monads (or comonad...

Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere t...

The purpose of the present paper is to show that: Eilenberg–type correspondences = Birkhoff’s theorem for (finite) algebras + duality. We consider algebras for a monad T on a category D and we study (pseudo)varieties of T– algebras. Pseudovarieties of algebras are also known in the literature as varieties of finitealgebras. Two well–known theorems that characterize varieties and pseudovarie...

Exception handling is provided by most modern programming languages. It allows to deal with anomalous or exceptional events which require special processing. In computer algebra, exception handling is an efficient way to implement the dynamic evaluation paradigm: for instance, in linear algebra, dynamic evaluation can be used for applying programs which have been written for matrices with coeff...

In their purest formulation, monads are used in functional programming for two purposes: (1) to hygienically propagate effects, and (2) to globalize the effect scope – once an effect occurs, the purity of the surrounding computation cannot be restored. As a consequence, monadic typing does not provide very naturally for the practically important ability to handle effects, and there is a number ...

Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as “λ-models”, or as (structures arising from) any reflexive object in a cartesian closed category (ccc, for brevity). These notions are tightly linked in the sense that: given a λ-model A, one may define a ccc in which A (the carrier set) is a reflexive object; conve...

Modern computer programs are executed in a variety of different contexts: on servers, handheld devices, graphics cards, and across distributed environments, to name a few. Understanding a program’s contextual requirements is therefore vital for its correct execution. This dissertation studies contextual computations, ranging from application-level notions of context to lowerlevel notions of con...