A type of higher-order two-dimensional sketch is defined which has models in suitable 2-categories. It has as special cases the ordinary sketches of Ehresmann and certain previously defined generalizations of one-dimensional sketches. These sketches allow the specification of constructions in 2-categories such as weighted limits, as well as higher-order constructions such as exponential objects and subobject classifiers, that cannot be sketched by limits and colimits. These sketches are designed to be the basis of a category-based methodology for the description of functional programming languages, complete with rewrite rules giving the operational semantics, that is independent of the usual specification methods based on formal languages and symbolic logic. A definition of ‘path grammar’, generalizing the usual notion of grammar, is given as a step towards this goal.

Linear logic has recently been introduced by Girard as a logic of actions that seems well suited for concurrent computation. In this paper, we establish a systematic correspondence between Petri nets, linear logic theories, and linear categories. Such a correspondence sheds new light on the relationships between linear logic and concurrency, and on how both areas are related to category theory. Categories are here viewed as concurrent systems the objects of which are states, and the morphisms of which are transitions. This is an instance of the Lambek-Lawvere correspondence between logic and category theory that cannot be expressed within the more restricted framework of the Curry-Howard correspondence.

In this paper, we propose a new and elegant definition of the class of recursive functions, which is analogous to Kleene's definition but differs in the primitives taken, thus demonstrating the computational power of the concurrent programming language introduced in Walters (1991), Walters (1992) and Khalil and Walters (1993).

The definition can be immediately rephrased for any distributive graph in a countably extensive category with products, thus allowing a wide, natural generalization of computable functions.

Subtyping in the presence of recursive types for the λ-calculus was studied by Amadio and Cardelli in 1991 (Amadio and Cardelli 1991). In that paper they showed that the problem of deciding whether one recursive type is a subtype of another is decidable in exponential time. In this paper we give an 0(n2) algorithm. Our algorithm is based on a simplification of the definition of the subtype relation, which allows us to reduce the problem to the emptiness problem for a certain finite automaton with quadratically many states.

It is known that equality of recursive types and the covariant Bohm order can be decided efficiently by means of finite automata, since they are just language equality and language inclusion, respectively. Our results extend the automata-theoretic approach to handle orderings based on contravariance.

Tries, a form of string-indexed look-up structure, are generalized, in a manner first discovered by Wadsworth, to permit indexing by terms built according to an arbitrary signature. The construction is parametric with respect to the type of data to be stored as values; this is essential, because the recursion that defines tries appeals from one value type to others. ‘Trie’ (for any fixed signature) is then a functor, and the corresponding look-up function is a natural isomorphism.

The trie functor is in principle definable by the ‘initial fixed point’ semantics of Smyth and Plotkin. We simplify the construction, however, by introducing the ‘category-cpo’, a class of category within which calculations can retain some domain-theoretic flavor. Our construction of tries extends easily to many-sorted signatures.

We have found both a proofnet criterion and a sequent calculus for the multiplicative fragment with units ( ⊗1,℘,0, atoms), but without the ┴-boxes of Girard (1987), which differentiate between 1 and ┴. We have also proved that for any of our proofnets there is a corresponding sequential proof.

The notion of a recognizable set of words, trees or graphs is relative to an algebraic structure on the set of words, trees or graphs respectively. We establish that several algebraic structures yield the same notion of a recognizable set of graphs. This notion is equivalent to that of a fully cutset-regular set of graphs introduced by Fellows and Abrahamson. We also establish that the class of recognizable sets of graphs is closed under the operations considered in these various equivalent definitions. This fact is not a standard consequence of the definition of recognizability.

The concept of abstract data types (ADTs) has emerged in the last fifteen years or so as one of the major programming design tools, with the emphasis on modular construction of large-scale programe. ADTs provide the means for seperating how data objects may be used from the different ways in which they actually are, or might be, implemented. A foundation for the theory of compputation with ADTs must provide answers to such questions as: What are ADTs and how may they be specified? What does it mean to implement an ADT? How may we construct new ADTs from old ones? What does it mean to compute with ADTs? This paper is devoted to a new and rather general, constructively based approach to these questions.

We propose a semantic framework for dynamic systems, which, in a sense, extends the well-known algebraic approach for modelling static data structures to the dynamic case. The framework is based on a new mathematical structure, called a d-oid, consisting of a set of instant structures and a set of dynamic operations. An instant structure is a static structure, e.g. an algebra; a dynamic operation is a transformation of instant structures with an associated point to point map, which allows us to keep track of the transformations of single objects and thus is called a tracking map. By an appropriate notion of morphism, the d-oids over a dynamic signature constitute a category.

It is shown that d-oids can model object systems and support an abstract notion of possibly unique object identity; moreover, for a d-oid satisfying an identity preserving condition, there exists an essentially equivalent d-oid where the elements of instant structures are just names.

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

We show that a set of finite graphs of tree-width at most k is recognizable (with respect to the algebra of graphs with an unbounded number of sources) if and only if it is recognizable with respect to the algebra of graphs of tree-width at most k with at most k sources.

The concept of algebraic high-level net transformation systems combines two important lines of research recently introduced in the literature: algebraic high-level nets (AHL-nets for short) and high-level replacement systems (HLR-systems for short). In both cases a categorical formulation of the corresponding theory has turned out to be highly important and is also a good basis for the integration of these concepts in this paper.

AHL-nets combine Petri nets with algebraic specifications and provide a powerful specification technique for distributed systems including data types and processes.

HLR-systems are transformation systems for high-level structures such as graphs, hypergraphs, algebraic specifications and different kinds of Petri nets. The theory of HLRsystems - formulated already in a categorical framework - is applied in this paper to AHLnets. Thus we obtain AHL-net transformation systems as an instantiation of HLR-systems to AHL-nets. This allows us to build up AHL-nets from basic components and to transform the net structure using rules or productions in the sense of graph grammars. This concept is illustrated by extending the well-known example of ‘dining philosophers’. We are able to show that AHL-net-transformation systems satisfy several important compatibility properties. On the one hand we obtain a local Church-Rosser and Parallelism Theorem, which is well-known for graph grammars and has recently been generalized to HLR-systems. This allows us to analyse concurrency in AHL-nets not only on the token level but also on the level of transformations of the net structure. On the other hand, we consider the ‘fusion’ and ‘union’ constructions for high-level structures, motivated by corresponding concepts for high-level Petri nets in the literature, and we show compatibility of these constructions with derivations of HLR-systems in general and AHL-nettransformations in particular. This means compatibility of vertical and horizontal structuring in terms of software development.

The partially ordered models of linear logic, a logical system developed by J. Y. Girard, turn out to be a class of quantales called Girard quantales. The notion of a quantaloid is a natural categorical generalization of a quantale and is one possible way of keeping track of types. In this paper, Girard quantales are generalized to Girard quantaloids. The general theory is developed and several key examples are studied. Turning our attention to the theory of categories enriched in a bicategory, it is then shown that if G is a Girard quantaloid, then the quantaloids Bim(G), Matr(G), and Mon(G), consisting of G-bimodules, G-matrices and G-monads respectively, are also Girard quantaloids.

Based on a categorical semantics for impredicative calculi of dependent types we prove several dependence and independence results. Especially we prove that there exists a model where all usual syntactical concepts can be interpreted with only one exception: in the model the strong sum of a family of propositions indexed over a proposition need not be isomorphic to a proposition again.

The method of proof consists of restricting the set of propositions in the well known PERw model due to E. Moggi. The subsets of PERw considered in this paper are inspired by the subset ExpO of PERw introduced by Freyd et al.

Finally we show that a weak and a strong notion of sub-locally-cartesian-closed-category coincide under rather mild completeness conditions.

Lambek used categories with indeterminates to capture explicit variables in simply typed λ-calculus. He observed that such categories with indeterminates can be described as Kleisli categories for suitable comonads. They account for ‘functional completeness’ for Cartesian (closed) categories.

Here we refine this analysis, by distinguishing ‘contextual’ and ‘functional’ completeness, and extend it to polymorphic λ-calculi. Since the latter are described as certain fibrations, we are lead to consider indeterminates, not only for ordinary categories, but also for fibred categories. Following a 2-categorical generalisation of Lambek's approach, such fibrations with indeterminates are presented as 'simple slices' in suitable 2-categories of fibrations; more precisely, as Kleisli objects. It allows us to establish contextual and functional completeness results for some polymorphic calculi.