Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enriched-categorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion 2. the Alexandro and the Scott topology, a n d the-ball topology 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.

A coherent description of enterprise architecture provides insight, enables communication among stakeholders and guides complicated change processes. Unfortunately, so far no enterprise architecture description language exists that fully enables integrated enterprise modeling, because for each architectural domain, architects use their own modeling techniques and concepts, tool support, visualization techniques, etc. In this paper, we outline such an integrated language and we identify and study concepts that relate architectural domains. In our language, concepts for describing the relationships between architecture descriptions at the business, application, and technology levels play a central role, related to the ubiquitous problem of business-ICT alignment, whereas for each architectural domain we conform to existing languages or standards such as UML. In particular, usage of services offered by one layer to another plays an important role in relating the behaviour aspects of the layers. The structural aspects of the layers are linked through the interface concept, and the information aspects through realization relations.

Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages. © 2012 Elsevier Inc. All rights reserved

Abstract. The powerset construction is a standard method for converting a nondeterministic 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 coalgebras 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 construction, 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. ACM CCS

Coalgebras provide a uniform framework for studying dynamical systems, including several types of automata. In this article, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalized powerset construction that determinizes coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor FT , where T is a monad describing the branching of the systems (e.g., non-determinism, weights, probability, etc.), has as a quotient the rational fixpoint of the determinized type functor F , a lifting of F to the category of T -algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain nondeterministic automata in which we recover Rabinovich’s sound and complete calculus for language equivalence.

We give a new presentation of Brzozowski's algorithm to minimize finite automata using elementary facts from universal algebra and coalgebra and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal language equivalent automata from Moore nondeterministic and weighted automata.

Abstract. Recent approaches to component-based software engineering employ coordinating connectors to compose components into software systems. For maximum flexibility and reuse, such connectors can themselves be composed, resulting in an expressive calculus of connectors whose semantics encompasses complex combinations of synchronisation, mutual exclusion, non-deterministic choice and state-dependent behaviour. A more expressive notion of connector includes also contextdependent behaviour, namely, whenever the choices the connector can take change non-monotonically as the context, given by the pending activity on its ports, changes. Context dependency can express notions of priority and inhibition. Capturing context-dependent behaviour in formal models is non-trivial, as it is unclear how to propagate context information through composition. In this paper we present an intuitive automata-based formal model of context-dependent connectors, and argue that it is superior to previous attempts at such a model for the coordination language Reo.

Abstract. We present a general framework for logics of transition systems based on Stone duality. Transition systems are modelled as coalgebras for a functor T on a category X . The propositional logic used to reason about state spaces from X is modelled by the Stone dual A of X (e.g. if X is Stone spaces then A is Boolean algebras and the propositional logic is the classical one). In order to obtain a modal logic for transition systems (i.e. for T -coalgebras) we consider the functor L on A that is dual to T . An adequate modal logic for T -coalgebras is then obtained from the category of L-algebras which is, by construction, dual to the category of T -coalgebras. The logical meaning of the duality is that the logic is sound and complete and expressive (or fully abstract) in the sense that non-bisimilar states are distinguished by some formula. We apply the framework to Vietoris coalgebras on topological spaces, using the duality between spaces and observation frames, to obtain adequate logics for transition systems on posets, sets, spectral spaces and Stone spaces.