The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.

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 non-well-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.

We present a denotational continuation semantics for Prolog with cut. First a uniform language B is studied, which captures the control flow aspects of Prolog. The denotational semantics for B is proven equivalent to a transition system based operational semantics. The congruence proof relies on the representation of the operational semantics as a chain of approximations and on a convenient induction principle. Finally, we interpret the abstract language B such that we obtain equivalent denotational and operational models for Prolog itself. Section 1 Introduction In the nice textbook of Lloyd [Ll] the cut, available in all Prolog-systems, is described as a controversial control facility. The cut, added to the Horn clause logic for efficiency reasons, affects the completeness of the refutation procedure. Therefore the standard declarative semantics using Herbrand models does not adequately capture the computational aspects of the Prolog-language. In the present paper we study the Prolog...

) Christel Baier Fakultat fur Mathematik & Informatik Universitat Mannheim 68131 Mannheim, Germany Marta Kwiatkowska 1 School of Computer Science University of Birmingham Edgbaston, Birmingham B15 2TT, UK Abstract In this paper we consider Milner's calculus CCS enriched by a probabilistic choice operator. The calculus is given operational semantics based on probabilistic transition systems. We define operational notions of preorder and equivalence as probabilistic extensions of the simulation preorder and the bisimulation equivalence respectively. We extend existing category-theoretic techniques for solving domain equations to the probabilistic case and give two denotational semantics for the calculus. The first, "smooth", semantic model arises as a solution of a domain equation involving the probabilistic powerdomain and solved in the category CONT? of continuous domains. The second model also involves appropriately restricted probabilistic powerdomain, but is constructed in the c...

We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for non-deterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation. The final object in this category is the canonical abstract model for trace equivalence and can be obtained by extending the final coalgebra of the deterministic action behaviour to the Kleisli category of the non-empty powerset monad. The corresponding final coalgebra semantics is fully abstract with respect to trace equivalence.

Abstract. We describe a method for combining formal program development with a disciplined and documented way of introducing realistic compromises, for example necessitated by resource bounds. Idealistic specifications are identified with the limits of sequences of more “realistic” specifications, and such sequences can then be refined in their entirety. Compromises amount to focusing the attention on a particular element of the sequence instead of the sequence as a whole. This method addresses the problem that initial formal specifications can be abstract or complete but rarely both. Various potential application areas are sketched, some illustrated with examples. Key research issues are found in identifying metric spaces and properties that make them usable for refinement using approximations.

. This paper aims at substantiating a recently introduced categorical theory of `well-behaved' operational semantics. A variety of concrete examples of structural operational rules is modelled categorically illustrating the versatility and modularity of the theory. Further, a novel functorial notion of guardedness is introduced which allows for a general and formal treatment of guarded recursive programs. Introduction The predominant approach to operational semantics is Plotkin's SOS [13], which is based on structural rules. One finds in the literature various formats of structural rules which guarantee a good behaviour such as having adequate denotational models and behavioural equivalence (eg bisimulation) being a congruence. In [17], it is shown that the rules in the best known of these formats, namely GSOS [5], are in 1-1 correspondence with natural transformations of a suitable type, depending on specific functorial notions of syntax and behaviour. This led to studying abstract ...

Semantic models are presented for two simple imperative languages with higher order constructs. In the first language the interesting notion is that of second order assignment x := s, for x a procedure variable and s a statement. The second language extends this idea by a form of higher order communication, with statements c ! s and c ? x, for c a channel. We develop operational and denotational models for both languages, and study their relationships. Both in the definitions and the comparisons of the semantic models, convenient use is made of some tools from (metric) topology. The operational models are based on (SOS-style) transition systems; the denotational definitions use domains specified as solutions of domain equations in a category of 1-bounded complete ultrametric spaces. In establishing the connection between the two kinds of models, fruitful use is made of Rutten's processes as terms technique. Another new tool consists in the use of metric transition systems, with a metric defined on the configurations of the system. In addition to higher order programming notions, we use higher order definitional techniques, e.g., in defining the semantic mappings as fixed points of (contractive) higher order operators. By Banach's theorem, such fixed points are unique, yielding another important proof principle for our paper.

A new metric domain of processes is presented. This domain is located in between two metric process domains introduced by De Bakker and Zucker. The new process domain characterizes the collection of image finite processes. This domain has as advantages over the other process domains that no complications arise in the definitions of operators like sequential composition and parallel composition, and that image finite language constructions like random assignment can be modelled in an elementary way. As in the other domains, bisimilarity and equality coincide in this domain. The three domains are obtained as unique (up to isometry) solutions of equations in a category of 1-bounded complete metric spaces. In the case the action set is finite, the three domains are shown to be equal (up to isometry). For infinite action sets, e.g., equipollent to the set of natural or real numbers, the process domains are proved not to be isometric. AMS Subject Classification (1991): 68Q55 CR Subject Classification (1991): D.3.1, F.3.2 Keywords & Phrases: process, complete metric space, bisimulation, finitely branching, image finite, sequential composition Note: This work was partially supported by the Netherlands Nationale Faciliteit Informatica programme, project Research and Education in Concurrent Systems (REX). This paper will appear in Proceedings of the Ninth Conference on the Mathematical Foundations of Programming Semantics, New Orleans, LA, USA, April 7-10, 1993.

We present an operational model O and a continuation based denotational model D for a uniform variant of Prolog, including the cut operator. The two semantical definitions make use of higher order transformations F and Y, respectively. We prove O and D equivalent in a novel way by comparing yet another pair of higher order transformations F and Y , that yield F and Y, respectively, by application of a suitable abstraction operator. Section 1 Introduction In [BV] we presented both an operational and a denotational continuation based semantics for the core of Prolog, and we proved these two semantics equivalent. We used a two step approach, by first deriving these results for an intermediate language, obtained by stripping the logic programming aspects (substitutions, most general unifiers and all that) from Prolog. This resulted in the abstract language B in which only the control structure from Prolog remained, such as the backtrack mechanism and the cut operator. After having co...