A criticism often levelled at functional languages is that they do not cope elegantly or efficiently with problems involving changes of state. In a recent paper [26], Wadler has proposed a new approach to these problems. His proposal involves the use of a type system based on the linear logic of Girard [7]. This allows the programmer to specify the "natural" imperative operations without at the same time sacrificing the crucial property of referential transparency. In this paper we investigate the practicality of Wadler's approach, describing the design and implementation of a variant of Lazy ML [2]. A small example program shows how imperative operations can be used in a referentially transparent way, and at the same time it highlights some of the problems with the approach. Our implementation is based on a variant of the G-machine [15, 1]. We give some benchmark figures to compare the performance of our machine with the original one. The results are disappointing: the cost of maintai...

A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCF-style languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...

Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1

We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. non-alternated, backtracking vs. repetitive, uniform vs. non-uniform.

The aim of these notes is to give an outline of the categorical structure required for a model of linear logic. That means that we only try to motivate the various conditions imposed on such models; this is by no means intended to be a full account. For intuitionistic linear logic a detailed description of the process of defining such a model can be found in Gavin Bierman’s thesis [Bie94]. There are a number of exercises in these notes, they either aim to give examples for the various notions defined or they state minor results that are used subsequently and which hopefully are not too difficult to establish. If the category theory in these notes is too advanced you may want to try [BS] which does not assume any knowledge in that area at all, but proceeds fairly rapidly to the notions we use here. Originally these notes were written for my PhD students, but since I made the first version available on my webpage a number of people have told me they found them useful. As a result I felt obliged to to add details which were rather sketchy in the original account. In particular the account now contains more proofs and so I am hopeful that the various formulations of the notion of linear exponential comonad are finally correct. I would like to thank Paola Maneggia, Peter Selinger, Robin Houston and Nicola Gambino for useful feedback on these

We generalise the notion of pre-logical predicates [HS02] to arbitrary simply typed formal systems and their categorical models. We establish the basic lemma of pre-logical predicates and composability of binary pre-logical relations in this generalised setting. This generalisation takes place in a categorical framework for typed higher-order abstract syntax and semantics [Fio02,MS03].

Abstract. In our PhD thesis ([4]), we showed that for the study of denotational semantics of linear logic ([6]), it is crucial to generalize the standard notion of monad on a category. An earlier generalization was already given by Hoofman ([9]): semimonads. But it was not suitable for our problem. This is why we introduced another generalization: weak monads. In this note, we present this new notion and give some examples.