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.

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.

Argumentation is a proof theoretic paradigm for reasoning under uncertainty. Whereas a ‘proof’ establishes its conclusion outright, an ‘argument’ can only lend a measure of support. Thus, the process of argumentation consists of identifying all the arguments for a particular hypothesis φ, and then calculating the support for φ from the weight attached to these individual arguments. Argumentation has been incorporated as the inference mechanism of a large scale medical expert system, the ‘Oxford System of Medicine’ (OSM), and it is therefore important to demonstrate that the approach is theoretically justified. This paper provides a formal semantics for the notion of argument embodied in the OSM. We present a categorical account in which arguments are the arrows of a semilattice enriched category. The axioms of a cartesian closed category are modified to give the notion of an ‘evidential closed category’, and we show that this provides the correct enriched setting in which to model the connectives of conjunction (&) and implication (⇒).

Finally, we develop a theory of ‘confidence measures’ over such categories, and relate this to the Dempster-Shafer theory of evidence.

It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.