Martin Hyland

A Term Calculus for Intuitionistic Linear Logic

In this paper we consider the problem of deriving a term assignment system for Girards Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky [1]) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these and consider their computational content.

Combining effects: sum and tensor

We seek a unified account of modularity for computational effects. We begin by reformulating Moggis monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggis exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggis side-effects monad transformer. Finally, we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.

Glueing and orthogonality for models of linear logic

We present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic: in particular we treat the exponentials in detail. We indicate simple applications of the methods and show that they cover familiar examples.

The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads

Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkins mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960s, and how the renewed interest in them in a computer science setting might develop in future.

Wellfounded Trees and Dependent Polynomial Functors

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

Pseudo-commutative monads and pseudo-closed 2-categories

Abstract Pseudo-commutative 2-monads and pseudo-closed 2-categories are defined. The former give rise to the latter: if T is pseudo-commutative, then the 2-category T-Alg, of strict T-algebras and pseudo-maps of algebras, is pseudo-closed. In particular, the 2-category of symmetric monoidal categories, is pseudo-closed. Subject to a biadjointness condition that is satisfied by T-Alg, pseudo-closed structure induces pseudo-monoidal structure on the 2-category.

Full intuitionistic linear logic (extended abstract)

Abstract In this paper we give a brief treatment of a theory of proofs for a system of Full Intuitionistic Linear Logic. This system is distinct from Classical Linear Logic, but unlike the standard Intuitionistic Linear Logic of Girard and Lafont includes the multiplicative disjunction par. This connective does have an entirely natural interpretation in a variety of (non-classical) categorical models of Intuitionistic Linear Logic. The main proof-theoretic problem arises from the observation of Schellinx that cut elimination fails outright for an intuitive formulation of Full Intuitionistic Linear Logic; the nub of the problem is the interaction between par and linear implication. We present here a term assignment system which gives an interpretation of proofs as some kind of non-deterministic function. In this way we find a system which does enjoy cut elimination. The system is a direct result of an analysis of the categorical semantics, though we make an effort to present the system as if it were purely a proof-theoretic construction.

Categorical Combinatorics for Innocent Strategies

We show how to construct the category of games and innocent strategies from a more primitive category of games. On that category we define a comonad and monad with the former distributing over the latter. Innocent strategies are the maps in the induced two-sided Kleisli category. Thus the problematic composition of innocent strategies reflects the use of the distributive law. The composition of simple strategies, and the combinatorics of pointers used to give the comonad and monad are themselves described in categorical terms. The notions of view and of legal play arise naturally in the explanation of the distributivity. The category-theoretic perspective provides a clear discipline for the necessary combinatorics.

Combining Computational Effects: commutativity & sum

We seek a unified account of modularity for computational effects, using the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi’s paradigm for modelling computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi’s side-effects monad transformer. And we give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer.

Combining algebraic effects with continuations

We consider the natural combinations of algebraic computational effects such as side-effects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubucs characterisation of strong monads in terms of enriched large Lawvere theories.