Jan J. M. M. Rutten

A Coinductive Calculus of Component Connectors

Reo is a recently introduced channel-based model for coordination, wherein complex coordinators, called connectors, are compositionally built out of simpler ones. Using a more liberal notion of a channel, Reo generalises existing dataflow networks. In this paper, we present a simple and transparent semantical model for Reo, in which connectors are relations on timed data streams. Timed data streams constitute a characteristic of our model and consist of twin pairs of separate data and time streams. Furthermore, coinduction is our main reasoning principle and we use it to prove properties such as connector equivalence.

Automata and Coinduction (An Exercise in Coalgebra)

The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.

Behavioural differential equations: a coinductive calculus of streams, automata, and power series

We present a theory of streams (infinite sequences), automata and languages, and formal power series, in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This coalgebraic perspective leads to a unified theory, in which the observation that each of the aforementioned sets carries a so-called final automaton structure, plays a central role. Finality forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions take the shape of what we have called behavioural differential equations, after Brzozowskis notion of input derivative. A calculus is developed for coinductive reasoning about all of the afore mentioned structures, closely resembling calculus from classical analysis.

Bisimulation for probabilistic transition systems: a coalgebraic approach

The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendier in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation.

A coinductive calculus of streams

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Generalized metric spaces: completion, topology, and power domains via the Yoneda embedding

Abstract Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere, 1973). Combining Lawveres (1973) enriched-categorical and Smyths (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. (1) completion, 2. (2)two topologies, and 3. (3) powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. (1) chain completion and Cauchy completion; 2. (2) the Alexandroff and the Scott topology, and the e-ball topology; 3. (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.

On the Foundation of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders

Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of non-standard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of post-fixed point. They are also used here for giving a new comprehensive presentation of the (still) non-standard theory of nonwell-founded sets (as non-standard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages — concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single ‘coalgebraic’ definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.

Generalizing Determinization from Automata to Coalgebras

The powerset construction is a standard method for converting a nondeter- ministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coal- gebras with structured state spaces. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (F-coalgebras) and a notion of behavioural equivalence (�F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for non-deterministic automata it is ordinary bisimilarity. We give several examples of applications of our generalized determinization construc- tion, including partial Mealy machines, (structured) Moore automata, Rabin probabilistic automata, and, somewhat surprisingly, even pushdown automata. To further witness the generality of the approach we show how to characterize coalgebraically several equivalences which have been object of interest in the concurrency community, such as failure or ready semantics.

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

This paper, a revised version of Rutten and Turi (1993), is part of a programme aiming at formulating a mathematical theory of structural operational semantics to complement the established theory of domains and denotational semantics to form a coherent whole (Turi 1996; Turi and Plotkin 1997). The programme is based on a suitable interplay between the induction principle, which pervades modern mathematics, and a dual, non-standard ‘coinduction principle’, which underlies many of the recursive phenomena occurring in computer science.The aim of the present survey is to show that the elementary categorical notion of a final coalgebra is a suitable foundation for such a coinduction principle. The properties of coalgebraic coinduction are studied both at an abstract categorical level and in some specific categories used in semantics, namely categories of non-well-founded sets, partial orders and metric spaces.

The Failure of Failures in a Paradigm for Asynchronous Communication

We develop a general framework for a variety of concurrent languages all based on asynchronous communication, like data flow, concurrent logic, concurrent constraint languages and CSP with asynchronous channels. The main characteristic of these languages is that processes interact by reading and modifying the state of some common data structure. We abstract from the specific features of the various communication mechanisms by means of a uniform language where actions are interpreted as partially defined transformations on an abstract set of states. Suspension is modelled by an action being undefined in a state. The languages listed above can be seen as instances of our paradigm, and can be obtained by fixing a specific set of states and interpretation of the actions.