Andrea Schalk

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.

Abstract games for linear logic extended abstract

Abstract We draw attention to a number of constructions which lie behind many concrete models for linear logic; we develop an abstract context for these and describe their general theory. Using these constructions we give a model of classical linear logic based on an abstract notion of game. We derive this not from a category with built-in computational content but from the simple category of sets and relations. To demonstrate the computational content of the resulting model we make comparisons at each stage of the construction with a standard very simple notion of game. Our model provides motivation for a less familiar category of games (played on directed graphs) which is closely reflected by our notion of abstract game. We briefly indicate a number of variations on this theme and sketch how the abstract concept of game may be refined further. It will be clear that we have been influenced in a general way by colleagues who have worked on models for linear logic. Many have given readings of their models as abstract games. However, we would like to mention that a specific precursor of the work described here is a joint project with Robin Cockett on the analysis of sequentiality in the context of yet other categories of abstract games. We hope to report on this in the fullness of time.

Games on graphs and sequentially realizable functionals. Extended abstract

We present a new category of games on graphs and derive from it a model for Intuitionistic Linear Logic. Our category has the computational flavour of concrete data structures but embeds fully and faithfully in an abstract games model. It differs markedly from the usual Intuitionistic Linear Logic setting for sequential algorithms. However, we show that with a natural exponential we obtain a model for PCF essentially equivalent to the sequential algorithms model. We briefly consider a more extensional setting and the prospects for a better understanding of the Longley Conjecture.

Concrete Data Structures as Games

A result by Curien establishes that filiform concrete data structures can be viewed as games. We extend the idea to cover all stable concrete data structures. This necessitates a theory of games with an equivalence relation on positions. We present a faithful functor from the category of concrete data structures to this new category of games, allowing a game-like reading of the former. It is possible to restrict to a cartesian closed subcategory of these games, where the function space does not decompose and the product is given by the usual tensor product construction. There is a close connection between these games and graph games.

Poset-valued sets or how to build models for linear logics

We describe a method for constructing models of linear logic based on the category of sets and relations. The resulting categories are non-degenerate in general; in particular they are not compact closed nor do they have biproducts. The construction is simple, lifting the structure of a poset to the new category. The underlying poset thus controls the structure of this category, and different posets give rise to differently-flavoured models. As a result, this technique allows the construction of models for both, intuitionistic or classical linear logic as desired. A number of well-known models, for example coherence spaces and hypercoherences, are instances of this method.

A fully abstract denotational model for observational precongruence

A domain theoretic denotational model is given for a simple sublanguage of CCS extended with divergence operator. The model is derived as an abstraction on a suitable notion of normal forms for labelled transition systems. It is shown to be fully abstract with respect to observational precongruence.

A Fully Abstract Denotational Model for Observational Precongruence

A domain theoretic denotational model is given for a simple sublanguage of CCS extended with divergence operator. The model is derived as an abstraction on a suitable notion of normal forms for labelled transition systems. It is shown to be fully abstract with respect to observational precongruence.

Constructing Fully Complete Models of Multiplicative Linear Logic

The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan double glueing construction produces such categories, either with or without units, when applied to any of a large family of degenerate models. This process explains as special cases a number of such models from the literature. In order to achieve this result, we develop a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by this glueing construction adding to the structure already available from the original category.

Constructing Fully Complete Models for Multiplicative Linear Logic

We demonstrate how the Hyland-Tan double glueing construction produces a fully complete model of the unit-free multiplicative fragment of Linear Logic when applied to any of a large family of degenerative ones. This process explains as special cases a number of such models which appear in the literature. In order to achieve this result, we make use of a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by the construction adding to those already available from the original category.