Piergiulio Katis

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.

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.

The compact closed bicategory of left adjoints

We show that, for any braided compact closed bicategory B, the bicategory Ladj(B) of left adjoints in B also admits a braided compact closed structure.

Compositional Minimization in Span(Graph): Some Examples

We study a class of examples of minimizing automata with respect to branching bisimulation in the context of the Span(Graph) model. Compositional minimization is particularly efficient for the class which includes the classical dining philospher problem and variants. The reason for the efficiency is that finite subsets of the class generate finite submonoids of the bisimulation monoid. We indicate how this may be used in studying deadlock. In the case of the dining philosopher the critical fact is that (F · P ) = (F · P ) in the bisimulation monoid, where F is the fork and P the philosopher.

Comparing cospan-spans and tiles via a Hoare-style process calculus

The large diffusion of concurrent and distributed systems has spawned in recent years a variety of new formalisms, equipped with features for supporting an easy specification of such systems. The aim of our note is to compare two recent formalisms for the design of concurrent and distributed systems, namely the tile logic and the cospan-span model. We first present a simple, yet rather expressive Hoare-style process algebra; then, after presenting the basics of both approaches, we compare them via their modeling of the calculus.