In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken

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.

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 Alexandroff and the Scott topology, and the ffl-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.

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized ultrametric spaces. Restricted to the special cases of preorders and ordinary ultrametric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the ffl-ball topology; 3. lower, upper, and convex powerdomains, and the powerdomain of compact subsets. Interestingly, all constructions are formulated in terms of (an ultrametric version of) the Yoneda (1954) lemma.

Abstract. This article demonstrates how a powerful and expressive abstraction from concurrency theory—monads of resumptions—plays a dual rôle as a programming tool for concurrent applications. The article demonstrates how a wide variety of typical OS behaviors may be specified in terms of resumption monads known heretofore exclusively in the literature of programming language semantics. We illustrate the expressiveness of the resumption monad with the construction of an exemplary multitasking kernel in the pure functional language Haskell. 1

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We introduce a computationally adequate metric model of PCF, based on the fact that the category of non-expansive maps of complete bounded ultrametric spaces is cartesian closed. The model captures certain temporal aspects of higher-type computation and contains both extensional and intensional functions. We show that Scott’s model arises as its extensional collapse. The intensional aspects of the metric model are illustrated via a Gödel-number-free version of Kleene’s T-predicate.

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Every generalized metric space can be isometrically embedded in a complete function space by means of a metric version of the categorical Yoneda embedding. This simple fact gives naturally rise to: 1. a topology for generalized metric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary metric spaces reduces to the ffl-ball topology; 2. a topology for algebraic generalized metric spaces generalizing both the Scott topology for algebraic complete partial orders and the ffl-ball topology for metric spaces. AMS subject classification (1991): 68Q10, 68Q55 Keywords: generalized metric, preorder, metric, Alexandroff topology, Scott topology, ffl-ball topology, Yoneda embedding 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and [BV96]). Parts of their theory have been develop...

In this report a comparative semantics is given for a language L p containing probabilistic and non-deterministic choice. The effects of interpreting these operators as local or global are investigated. For three of the possible combinations an operational model and a denotational model are given and compared. First models for local probabilistic choice and local non-deterministic choice are given using a generative approach. By adjusting these models slightly models for global probability and local non-determinism are obtained. Finally models for local probability and global non-determinism are presented using a stratified approach. For use with the denotational models a construction of a complete ultra-metric space of finite multisets is given. 1 Introduction The goal of this paper is to construct comparative semantics for a language combining non-determinism and probabilistic choice. The main interest is the interplay between these two concepts. Since many of the interesting proper...

Since the work of van Glabbeek [9]... In this paper we define the notion of a labelled topological transition system, i.e., a labelled transition system where the state space is structured using a topology. Then, we define topological simulation and topological bisimulation. These notions extend the traditional ones by considering not only single steps but arbitrary long (accumulating) sequences of steps in the transfer (zig-zag) conditions. We prove that these topological notions are a preorder and an equivalence respectively and that they are stronger than the non-topological notions. We also prove that they are topological notions (invariant under the application of continuous transition morphisms)