A process $M$ terminates if it cannot produce an infinite sequence of reductions $M \mathop{\rightarrow}^{\tau} M_1\mathop{\rightarrow}^{\tau} M_2 \ldots$. Termination is a useful property in concurrency. For instance, a terminating applet, when loaded on a machine, will not run for ever, possibly absorbing all computing resources (a ‘denial of service’ attack). Similarly, termination guarantees that queries to a given service originate only finite computations.

We ensure termination of a non-trivial subset of the $\pi$-calculus by a combination of conditions on types and on the syntax. The proof of termination is in two parts. The first uses the technique of logical relations – a well-know technique of $\lambda$-calculi – on a small set of non-deterministic ‘functional’ processes. The second part of the proof uses techniques of process calculi, in particular, techniques of behavioural preorders.

This paper gives the first proof that the subtyping relation of a higher-order lambda calculus, ${\cal F}^{\omega}_{\leq}$, is anti-symmetric, establishing in the process that the subtyping relation is a partial order – reflexive, transitive, and anti-symmetric up to $\beta$-equality. While a subtyping relation is reflexive and transitive by definition, anti-symmetry is a derived property. The result, which may seem obvious to the non-expert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping, and the logical relation used to prove its equivalence with the declarative presentation of ${\cal F}^{\omega}_{\leq}$, offers a powerful new technology to solve the problem: of particular importance is our extended rule for the well-formedness of types with head variables. The paper also gives a presentation of ${\cal F}^{\omega}_{\leq}$ without a relation for $\beta$-equality, which is apparently the first such, and shows its equivalence with the traditional presentation.

We study the relationship between classical phase semantics for classical linear logic (LL) and intuitionistic phase semantics for intuitionistic linear logic (ILL). We prove that (i) every intuitionistic phase space is a subspace of a classical phase space, and (ii) every intuitionistic phase space is phase isomorphic to an ‘almost classical’ phase space. Here, by an ‘almost classical’ phase space we mean an intuitionistic phase space having a double-negation-like closure operator. Based on these semantic considerations, we give a syntactic embedding of propositional ILL into LL.

We study the combination of probability and non-determinism from a categorical point of view. In category theory, non-determinism and probability are represented by suitable monads. However, these two monads do not combine well as they are. To overcome this problem, we introduce the notion of indexed valuations. This notion is used to define a new monad that can be combined with the usual non-deterministic monad via a categorical distributive law. We give an equational characterisation of our construction. We discuss the computational meaning of indexed valuations, and we show how they can be used by giving a denotational semantics of a simple imperative language.

An approach to point-free geometry based on the notion of a quasi-metric is proposed in which the primitives are the regions and a non-symmetric distance between regions. The intended models are the bounded regular closed subsets of a metric space together with the Hausdorff excess measure.