Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The Fraenkel-Mostowski permutation model of set theory with atoms (FM-sets) can serve as the semantic basis of meta-logics for specifying and reasoning about formal systems involving name binding, ff-conversion, capture avoiding substitution, and so on. We show that in FM-set theory one can express statements quantifying over `fresh' names and we use this to give a novel set-theoretic interpretation of name abstraction. Inductively defined FM-sets involving this name-abstraction set former (together with cartesian product and disjoint union) can correctly encode object-level syntax modulo ff-conversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated n...

The manipulation of objects with state which changes over time is allpervasive in computing. Perhaps the simplest example of such objects are the program variables of classical imperative languages. An important strand of work within the study of such languages, pioneered by John Reynolds, focusses on "Idealized Algol", an elegant synthesis of imperative and functional features. We present a novel semantics for Idealized Algol using games, which is quite unlike traditional denotational models of state. The model takes into account the irreversibility of changes in state, and makes explicit the difference between copying and sharing of entities. As a formal measure of the accuracy of our model, we obtain a full abstraction theorem for Idealized Algol with active expressions.

In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a built-in notion of bisimulation. We show how

We study syntax-free models for name-passing processes. For interleaving semantics, we identify the indexing structure required of an early labelled transition system to support the usual pi-calculus operations, defining Indexed Labelled Transition Systems. For noninterleaving causal semantics we define Indexed Labelled Asynchronous Transition Systems, smoothly generalizing both our interleaving model and the standard Asynchronous Transition Systems model for CCS-like calculi. In each case we relate a denotational semantics to an operational view, for bisimulation and causal bisimulation respectively. We establish completeness properties of, and adjunctions between, categories of the two models. Alternative indexing structures and possible applications are also discussed. These are first steps towards a uniform understanding of the semantics and operations of name-passing calculi.

Recent work has shown that presheaf categories provide a general model of concurrency, with an inbuilt notion of bisimulation based on open maps. Here it is shown how this approach can also handle systems where the language of actions may change dynamically as a process evolves. The example is the pi-calculus, a calculus for `mobile processes' whose communication topology varies as channels are created and discarded. A denotational semantics is described for the pi-calculus within an indexed category of profunctors; the model is fully abstract for bisimilarity, in the sense that bisimulation in the model, obtained from open maps, coincides with the usual bisimulation obtained from the operational semantics of the pi-calculus. While attention is concentrated on the `late' semantics of the pi-calculus, it is indicated how the `early' and other variants can also be captured.

This paper presents a filter model for π-calculus, and shows its full abstraction with respect to a "may" operational semantics. The model is introduced in the form of a type assignment system. Types are related by a preorder which mimics the operational behaviour of terms. A subject expansion theorem holds. Terms are interpreted as filters of types: this interpretation is compositional. The proof of full abstraction relies on a notion of realizability of types, and on the construction of terms, which test when an arbitrary term has a fixed type.

. This paper introduces a formal specification technique for mobile systems based on input/output relations on streams. We consider networks of components communicating asynchronously via unbounded directed channels. Mobility is achieved by allowing the components to communicate channel ports. We distinguish between many-to-many and two variants of point-to-point communication. The communication paradigms are semantically under-pinned by denotational models. The models are formulated in the context of timed nondeterministic data-flow networks and presented in a step-wise fashion. The emphasis is on capturing the special kind of dynamic hiding characterizing mobile systems. We demonstrate the proposed approach in a number of small examples. 1. Introduction Motivated by the need to model object-oriented programming languages and openness in distributed applications, the study of mobile systems has become a very popular research area. Most of the early theoretical research on mobility is...

We study the use of the π-calculus for semantical descriptions of languages such as Concurrent Idealised ALGOL (CIA), combining imperative, functional and concurrent features. We first present an operational semantics for CIA, given by SOS rules and a contextual form of behavioural equivalence; then a π-calculus semantics. As behavioural equivalence on π-calculus processes we choose the standard (weak early) bisimilarity. We compare the two semantics, demonstrating that there is a close operational correspondence between them and that the π-calculus semantics is sound. This allows for applying the-calculus theory in proving behavioural properties of CIA phrases. We discuss laws and examples which have served as benchmarks to various semantics, and a more complex example involving procedures of higher order.

In this paper we discuss final semantics for the -calculus, a process algebra which models systems that can dynamically change the topology of the channels. We show that the final semantics paradigm, originated by Aczel and Rutten for CCS-like languages, can be successfully applied also here. This is achieved by suitably generalizing the standard techniques so as to accommodate the mechanism of name creation and the behaviour of the binding operators peculiar to the -calculus. As a preliminary step, we give a higher order presentation of the -calculus using as metalanguage LF , a logical framework based on typed -calculus. Such a presentation highlights the nature of the binding operators and elucidates the role of free and bound channels. The final semantics is defined making use of this higher order presentation, within a category of hypersets.

We introduce a theory of behavioural types as a semantic foundation of typed ß-calculi. In this theory, a type is a set of behaviours, represented by early name passing synchronisation trees, which conform to a certain behavioural constraint. Operations on typed processes are derived from typed variants of well-known process-theoretic operations for mobile processes, and each model of typed ß-calculi in a typed universe induces a compositional theory of typed bisimilarities. The construction is simple and intuitive, yet offers a rich class of typed universes of name passing interactive behaviours, which contain, among others, models of known typed ß-calculi and universes of game semantics. As a simple but non-trivial application, we show how the sorting by Milner can be given a sound model in a basic universe of types. The soundness states not only that the interpretation is sound in the standard sense, but also that the untyped interactive behaviour of typed terms is justifiable on t...