We introduce the notions of...

To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is first-order acyclic sharing graphs represented by let-syntax, and others are extensions with higher-order constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced

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. Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a type-theoretic account of action calculi using the propositions-as-types paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a model-embedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi. 1 Introduction Action calculi arose directly from the ß-calculus [MPW92]. They were introduced by Milner [Mil96], to provide a uniform notation for capturing many calculi of interaction such as the ß-calculus, the -calculus, models of distribut...

We develop a notion of Kripke-like parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their category-theoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal cocompletion. As applications, we obtain full completeness results of translations between linear type theories.

Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category theory, domain theory, logic, type theory, etc. In support of this proposal I will show how such an approach leads to interesting connections between various areas of computer science and mathematics; concentrating on one such example in some detail. Specifically, I will consider the line of my research involving denotational models of the pi calculus and algebraic theories with variable-binding operators, indicating how the abstract mathematical structure underlying these models fits with that of Joyal’s combinatorial species of structures. This analysis suggests both the unification and generalisation of models, and in the latter vein I will introduce generalised species of structures and their calculus. These generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc.) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.

We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) -autonomous categories (models of Multiplicative Linear Logic); the glueing construction for -autonomous categories is a mild generalization of the double glueing construction due to Hyland and Tan. Each of the glueing techniques can be used for creating interesting models of linear logic. In particular, we use them, together with the free symmetric monoidal cocompletion, for deriving Kripke-like parameterized logical predicates (logical relations) for the fragments of linear logic. As an application, we show full completeness results for translations between linear type theories. Contents 1 Introduction 3 2 Preliminaries 4 2.1 Symmetric Monoidal Structures . . . . . . . ....

A Freyd-category is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in call-by-value programming languages, such as the computational λ-calculus, with computational effects. We develop the theory of Freyd-categories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freyd-category. We then give canonical, universal embeddings of Freyd-categories into closed Freyd-categories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the λc-calculus. Our leading examples of signatures arise from side-effects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism. Key words: Freyd-category, enriched Yoneda embedding, conical colimit completion, canonical model

We generalize Dress and Müller's main result in [5]. We observe that their result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponential principle in a symmetric monoidal category. We show that for any groupoid G, the !G of presheaves on the symmetric monoidal completion !G of G satisfies the exponential principle. The main result in [5] reduces to the case G = 1. We discuss two notions of functor between categories satisfying the exponential principle and express some well known combinatorial identities as instances of the preservation properties of these functors. Finally, we give a characterization of G as a subcategory of !G.

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of split-idempotent natural transformations on id A . In the special case that A is a presheaf category, we characterize them in terms of the domain, and prove that any bireflective subcategory of A is itself a presheaf category. We define diagonal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obtain a bireflective subcategory of [C op ; Set] and deduce results relating its finite product structure with the monoidal structure of [C op ; Set] determined by that of C. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(A), the category of relati...