Robert F. C. Walters

Introduction to extensive and distributive categories

Abstract In recent years, there has been considerable discussion as to the appropriate definition of distributive categories. Three definitions which have had some support are: (1) A category with finite sums and products such that the canonical map δ: A × B + A × C → A ×( B + C ) is an isomorphism (Walters). (2) A category with finite sums and products such that the canonical functor +: A / A × A / B → A /( A + B ) is an equivalence (Monro). (3) A category with finite sums and finite limits such that the canonical functor + of (2) is an equivalence (Lawvere and Schanuel). There has been some confusion as to which of these was the natural notion to consider. This resulted from the fact that there are actually two elementary notions being combined in the above three definitions. The first, to which we give the name distributivity , is exactly that of (1). The second notion, which we shall call extensivity , is that of a category with finite sums for which the canonical functor + of definitions (2) and (3) is an equivalence. Extensivity, although it implies the existence of certain pullbacks, is essentially a property of having well-behaved sums. It is the existence of these pullbacks which has caused the confusion. The connections between definition (1) and definitions (2) and (3) are that any extensive category with products is distributive in the first sense, and that any category satisfying (3) satisfies (1) locally. The purpose of this paper is to present some basic facts about extensive and distributive categories, and to discuss the relationships between the two notions.

Bicategories of processes

Abstract The suspension-loop construction is used to define a process in a symmetric monoidal category. The algebra of such processes is that of symmetric monoidal bicategories. Processes in categories with products and in categories with sums are studied in detail, and in both cases the resulting bicategories of processes are equipped with operations called feedback . Appropriate versions of traced monoidal properties are verified for feedback, and a normal form theorem for expressions of processes is proved. Connections with existing theories of circuit design and computation are established via structure preserving homomorphisms.

Span(Graph): A Categorial Algebra of Transition Systems

We have shown that a natural algebraic structure on Span(Graph) allows the compositional specification of concurrent systems. Hoares parallel operation appears as a derived operation in this algebra. The simpler basic operations of our algebra are possible because we do not insist on interleaving semantics: interleaving prevents consideration of the identity span, as well as other natural constants such as the diagonal. We have given some examples of transforming systems using the equations of the algebra. Associated to the algebra there is a geometry which expresses the distributed nature of a concurrent system. This relation between algebra and geometry makes precise the relation between process algebras and circuit diagrams as used, for example, in Ebergen [E87].

Feedback, trace and fixed-point semantics

We introduce a notion of category with feedback-with-delay, closely related to the notion of traced monoidal category, and show that the Circ construction of [15] is the free category with feedback on a symmetric monoidal category. Combining with the Int construction of Joyal et al. [12] we obtain a description of the free compact closed category on a symmetric monoidal category. We thus obtain a categorical analogue of the classical localization of a ring with respect to a multiplicative subset. In this context we define a notion of fixed-point semantics of a category with feedback which is seen to include a variety of classical semantics in computer science.

The Todd-Coxeter Procedure and Left Kan Extensions

We introduce a generalization of the Todd-Coxeter procedure for the enumeration of cosets. The generalized procedure relates to a construction in category theory known as the left Kan extension. It admits of a great variety of applications, including enumerating cosets, computing certain colimits in the category of Sets, and enumerating the arrows in a category given by generators and relations.We begin by defining the notion of a left Kan extension, and giving a number of illustrative examples. We then provide a full specification of the procedure, followed by its application in relation to each of the examples. Finally, we provide a formulation of the procedure in terms of graphs and presentations of actions of graphs (automata) which is more convenient for theoretical purposes.

Matrices, machines and behaviors

Sabadini, Walters and others have developed a categorical, machine based theory of concurrency in which there are four essential aspects: a distributive category of data-types; a bicategory Mach whose objects are data types, and whose arrows are input-output machines built from data types; a semantic category (or categories) Sem, suitable to contain the behaviors of machines, and a functor, “behavior”: Mach→Sem. Suitable operations on machines and semantics are found so that the behavior functor preserves these operations. Then, if each machine is decomposable into primitive machines using these operations, the behavior of a general machine is deducible from the behavior of its parts. The theory of non-deterministic finite state automata provides an example of the paradigm and also throws some light on the classical theory of finite state automata.We describe a bicategory whose objects are natural numbers, in which an arrow M: n→p is a finite state automaton with n input states, p output states, and some additional internal states; we require that no transitions begin at output states or end at input states. A machine is represented by an q+n by q+p matrix. The bicategory supports additional operations: non-deterministic choice, parallel interleaving, and feedback. Enough operations are imposed on machines to show that each machine may be obtained from some atomic ones by means of the operations.The semantic category is the (Bloom-Ésik) iteration theory Mat(X✶ whose objects are natural numbers and whose arrows from n to p are n×p matrices with entries in the semiring of languages. The behavior functor associates to a machine M: n→p a matrix |M| of languages, one language to each pair of input and output states. Behavior preserves composition, feedback, takes non-deterministic choice to union, and parallel-interleaving to shuffle. Thus, behavior gives a compositional semantics to a primitive notion of concurrent processes.

Representing Place/Transition Nets in Span(Graph)

The compact closed bicategory Span of spans of reflexive graphs is described and it is interpreted as an algebra for constructing specifications of concurrent systems. We describe a procedure for associating to any Place/Transition system Ω an expression Ψ Ω in the algebra Span. The value of this expression is a system whose behaviours are the same as those of the P/T system. Furthermore, along the lines of Penroses string diagrams, a geometry is associated to the expression Ω which is essentially the same geometry as that usually associated to the net underlying Ω.

Coinverters and categories of fractions for categories with structure

A category of fractions is a special case of acoinverter in the 2-categoryCat. We observe that, in a cartesian closed 2-category, the product of tworeflexive coinverter diagrams is another such diagram. It follows that an equational structure on a categoryA, if given by operationsAn →A forn εN along with natural transformations and equations, passes canonically to the categoryA [Σ−1] of fractions, provided that Σ is closed under the operations. We exhibit categories with such structures as algebras for a class of 2-monads onCat, to be calledstrongly finitary monads.

A note on recursive functions

In this paper, we propose a new and elegant definition of the class of recursive functions, which is analogous to Kleenes 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.

The Compositional Construction of Markov Processes

We describe a symmetric monoidal category whose arrows are automata in which the actions have probabilities. The endomorphisms of the identity for the tensor are classical finite Markov processes. The operations of the category permit the compositional description Markov processes. We illustrate by describing a Markov process with 12n states, which represents a model of the classical Dining Philosopher problem with n dining philosophers, showing how to calculate the probability of reaching deadlock in k steps. A straightforward application of the Perron-Frobenius Theorem yields that this probability tends to 1 as k tends to infinity.