We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1

We study syntax-free models for name-passing processes. For interleaving semantics, we identify the indexing structure required of an early labelled transition system to support the usual pi-calculus operations, defining Indexed Labelled Transition Systems. For noninterleaving causal semantics we define Indexed Labelled Asynchronous Transition Systems, smoothly generalizing both our interleaving model and the standard Asynchronous Transition Systems model for CCS-like calculi. In each case we relate a denotational semantics to an operational view, for bisimulation and causal bisimulation respectively. We establish completeness properties of, and adjunctions between, categories of the two models. Alternative indexing structures and possible applications are also discussed. These are first steps towards a uniform understanding of the semantics and operations of name-passing calculi.

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

Abstract not found

In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of suchsets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. Wewillhave a closer look into the connection of the iterated circular multisets and arbitrary trees. Key words: multiset, non-wellfounded set, Scott-universe, AFA, coalgebra, modal logic, graded modalities MSC2000 codes: 03B45, 03E65, 03E70, 18A15, 18A22, 18B05, 68Q85 1 Contents 1 Introduction 3 1.1 Multisets on a Given Domain . . . . . . . . . . . . . . . . . . . . 3 1.2 Iterated and Circular Multisets . . . . . . . . . . . . . . . . . . . 6 1.3 Organization of the Paper . . . . . . . . . . . . . . . . . . . . . . 7 2 Prerequisites 8 2.1 Coalgebras and Morphisms . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 A Prototype: Pow . . . . . . . . . . . . . . . ...

. Concurrent Constraint Programming (CCP) has been the subject of growing interest as the focus of a new paradigm for concurrent computation. Like logic programming it claims close relations to logic. In fact CCP languages are logics in a certain sense that we make precise in this paper. In recent work it was shown that the denotational semantics of determinate concurrent constraint programming languages forms a fibred categorical structure called a hyperdoctrine, which is used as the basis of the categorical formulation of first-order logic. What this shows is that the combinators of determinate CCP can be viewed as logical connectives. In this paper we extend these ideas to the operational semantics of such languages and thus make available similar analogies for a much broader variety of languages including indeterminate CCP languages and concurrent block-structured imperative languages. CR Classification: F3.1, F3.2, D1.3, D3.3 Key words: Concurrent constraint programming, simula...

An improved method to retrieve a library function via its Hindley/Milner type is described. Previous retrieval systems have identified types that are isomorphic in any Cartesian closed category (CCC), and have retrieved library functions of types that are either isomorphic to the query, or have instances that are. Sometimes it is useful to instantiate the query too, which requires unification modulo isomorphism. Although unifiability modulo CCCisomorphism is undecidable, it is decidable modulo linear isomorphism, that is, isomorphism in any symmetric monoidal closed (SMC) category. We argue that the linear isomorphism should retrieve library functions almost as well as CCC-isomorphism, and we report experiments with such retrieval from the Lazy ML library. When unification is used, the system retrieves too many functions, but sorting by the sizes of the unifiers tends to place the most relevant functions first. R'esum'e Ce papier pr'esente une nouvelle m'ethode pour la re...

The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,

Given suitable categories T; C and functor F : T ! C, if X; Y are objects of T, then we define an (X; Y )-relation in C to be a triple (R; r; ¯ r), where R is an object of C and r : R ! FX and ¯ r : R ! FY are morphisms of C. We define an algebra of relations in C, including operations of "relabeling," "sequential composition," "parallel composition," and "feedback," which correspond intuitively to ways in which processes can be composed into networks. Each of these operations is defined in terms of composition and limits in C, and we observe that any operations defined in this way are preserved under the mapping from relations in C to relations in C 0 induced by a continuous functor G : C ! C 0 . To apply the theory, we define a category Auto of concurrent automata, and we give an operational semantics of dataflow-like networks of processes with indeterminate behaviors, in which a network is modeled as a relation in Auto. We then define a category EvDom of "event domains," a (non...

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enriched-categorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the ffl-ball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.