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.

In this thesis we present three mathematical frameworks for the modelling of reac-tive probabilistic communicating processes. We first introduce generalised labelled transition systems as a model of such processes and introduce an equivalence, coarser than probabilistic bisimulation, over these systems. Two processes are identified with respect to this equivalence if, for all experiments, the probabilities of the respective processes passing a given experiment are equal. We next consider a probabilistic pro-cess calculus including external choice, internal choice, action-guarded probabilistic choice, synchronous parallel and recursion. We give operational semantics for this calculus be means of our generalised labelled transition systems and show that our equivalence is a congruence for this language. Following the methodology introduced by de Bakker & Zucker, we then give deno-tational semantics to the calculus by means of a complete metric space of probabilistic processes. The derived metric, although not an ultra-metric, satisfies the intuitive property that the distance between two processes tends to 0 if a measure of the dif-

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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...

Labelled transition systems are useful for giving semantics to programming languages. Kok and Rutten have developed some theory to prove semantic models defined by means of labelled transition systems to be equal to other semantic models. Metric labelled transition systems are labelled transition systems with the configurations and actions endowed with metrics. The additional metric structure allows us to generalize the theory developed by Kok and Rutten. Introduction The classical result due to Banach [Ban22] that a contractive function from a nonempty complete metric space to itself has a unique fixed point plays an important role in the theory of metric semantics for programming languages. Metric spaces and Banach's theorem were first employed by Nivat [Niv79] to give semantics to recursive program schemes. Inspired by the work of Nivat, De Bakker and Zucker [BZ82] gave semantics to concurrent languages by means of metric spaces. The metric spaces they used were defined as solutio...

For two parallel languages with recursion a compositional weakest precondition semantics is given using two new metric resumption domains. The underlying domains are characterized by domain equations involving functors that deliver `observable' and `safety' predicate transformers. Further a refinement relation is defined for this domains and illustrated by rules dealing with concurrent composition. It turns out, by extending the classical duality of predicate vs. state transformers, that the weakest precondition semantics for the parallel languages is isomorphic to the standard metric state transformers semantics. Moreover, the proposed refinement relation on the predicate transformer domain will correspond to the familiar notion of simulation in the state transformer domain. Contents 1 Introduction 1 2 Mathematical Preliminaries 3 3 Four Languages with Recursion 5 4 Domains for Predicate Transformers 8 5 Predicate Transformer Semantics 14 6 Refinement, Simulation and State Transforme...

ness of an Interleaving Semantics for Action Refinement J.I. den Hartog 1 , E.P. de Vink 1 and J.W. de Bakker 1;2 Abstract For an abstract programming language with action refinement both an operational and a denotational semantics are given. The operational semantics is based on an SOS-style transition system specification involving syntactical refinement sequences. The denotational semantics is an interleaving model which uses semantical refinement `environments'. It identifies those statements which are equal under all refinements. The denotational model is shown to be fully abstract with respect to the operational one. The underlying metric machinery is exploited to obtain this full abstractness result. Usually, action refinement is treated either in a model with some form of true concurrency, or, when an interleaving model is applied, by assuming that the refining statements are atomized. We argue that an interleaving model without such atomization is attractive as well. 1 I...

We re-examine the modal µ-calculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, m-saturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satisfies the Hennessy-Milner property. We conclude by showing that for descriptive modal µ-frames the standard interpretation coincides with the one we proposed.