Term graph rewriting is concerned with the representation of functional expressions as graphs and the evaluation of these expressions by rule-based graph transformation. The advantage of computing with graphs rather than terms is that common subexpressions can be shared, improving the efficiency of computations in space and time. Sharing is ubiquitous in implementations of programming languages: many functional, logic, object-oriented and concurrent calculi are implemented using term graphs.

This second special issue devoted to ‘developments in computational models’ (the first was Volume 16 Issue 4) came out of an open call for papers following the First International Workshop on Developments in Computational Models (DCM). This took place in Lisbon, Portugal, on the 10th July 2005, and was a satellite event of ICALP 2005 focused on abstract models of computation and their associated programming paradigms.

The title Constructive analysis, types and exact real numbers covers the wide field of research dealing with ‘precise’ computationson continuous structures. The adjective ‘precise’ is used here in an informal way, referring to computations where the rounding off of the output and the approximative nature of the input are explicitly taken nto account in some way.

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing.

The first two papers deal with entanglement. The paper by R. Mosseri and P. Ribeiro presents a detailed description of the two- and three-qubit geometry in Hilbert space, dealing with the geometry of fibrations and discrete geometry. The paper by J.-G.Luque et al. is more algebraic and considers invariants of pure k-qubit states and their application to entanglement measurement.

This is the second part of a special issue in honour of Klaus Keimel – the first part appeared in Mathematical Structures in Computer Science16 (2). This second part consists of a single paper by John Longley, which could not be included in the earlier issue for reasons of size.

This special issue of Mathematical Structures in Computer Science contains several contributions to the Workshop

3 corps, classique-quantique, discret-continu.

The Workshop was held at the Ecole Normale Supérieure, Paris on 28–29 September 2004 in the framework of the Collectif Histoire Philosophie Sciences. The Workshop was organised by Giuseppe Longo and Thierry Paul with the aim of bringing together mathematicians, physicists, computer scientists and philosophers.

This paper reviews recent attempts to describe the two- and three-qubit Hilbert space geometries. In the first part, this is done with the help of Hopf fibrations of hyperspheres. It is shown that the associated Hopf map is strongly sensitive to states’ entanglement content. In the two-qubit case, a generalisation of the celebrated one-qubit Bloch sphere representation is described. In the second part, we present Hilbert space discrete versions, which are comparable to polyhedral approximations of spheres in standard geometry.

It is an empirical observation arising from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene–Kreisel continuous functionals, its effective substructure Ceff and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously.

In this paper we present some new results that go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, Ceff or HEO (as appropriate). We obtain versions of our results for both the ‘standard’ and ‘modified’ extensional collapse constructions. The proofs make essential use of a technique due to Normann.

Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the three type structures under consideration are highly canonical mathematical objects.

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.

In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.

We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

We present a discussion concerning the opposition between discreteness and the continuum in quantum mechanics. In particular, it is shown that this duality was not restricted to the early days of the theory, but remains current, and features different aspects of discretisation. In particular, the discreteness of quantum mechanics is key for quantum information and quantum computation. We propose a conclusion involving a concept of completeness linking discreteness and the continuum.

We define BACI(Boxed Ambients with Communication Interfaces), an ambient calculus with a flexible communication policy. Traditionally, typed ambient calculi have a fixed communication policy determining the kind of information that can be exchanged with a parent ambient, even though mobility changes the parent. BACI lifts that restriction, allowing different communication policies with different parents during computation. Furthermore, BACI separates communication and mobility by making the channels of communication between ambients explicit. In contrast with other typed ambient calculi where communication policies are global, each ambient in BACI is equipped with a description of the communication policies ruling its information exchange with parent and child ambients. The communication policies of ambients increase when they move: more precisely, when an ambient enters another ambient, the entering ambient and the host ambient can exchange their communication ports and agree on the kind of information to be exchanged. This information is recorded locally in both ambients.

We show the type-soundness of BACI, proving that it satisfies the subject reduction property, and we study its behavioural semantics by means of a labelled transition system.

In this paper we will discuss various aspects of computable/constructive analysis, namely semantics, proofs and computations. We will present some of the problems and solutions of exact real arithmetic varying from concrete implementations, representation and algorithms to various models for real computation. We then put these models in a uniform framework using realisability, which opens the door to the use of type theoretic and coalgebraic constructions both in computing and reasoning about these computations. We will indicate that it is often natural to use constructive logic to reason about these computations.

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models called classical categories that is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models called Dummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.

Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

We give an algorithm allowing the construction of bases of local unitary invariants of pure k-qubit states from a knowledge of the polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are made explicit and compared with various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

We extend the work of A. Ciaffaglione and P. di Gianantonio on the mechanical verification of algorithms for exact computation on real numbers, using infinite streams of digits implemented as a co-inductive type. Four aspects are studied. The first concerns the proof that digit streams correspond to axiomatised real numbers when they are already present in the proof system. The second re-visits the definition of an addition function, looking at techniques to let the proof search engine perform the effective construction of an algorithm that is correct by construction. The third concerns the definition of a function to compute affine formulas with positive rational coefficients. This is an example where we need to combine co-recursion and recursion. Finally, the fourth aspect concerns the definition of a function to compute series, with an application on the series that is used to compute Euler's number e. All these experiments should be reproducible in any proof system that supports co-inductive types, co-recursion and general forms of terminating recursion; we used the COQ system (Dowek et al. 1993; Bertot and Castéran 2004; Giménez 1994).

We propose a graphical implementation for (possibly recursive) processes of the π-calculus, encoding each process into a graph. Our implementation is sound and complete with respect to the structural congruence for the calculus: two processes are equivalent if and only if they are mapped into graphs with the same normal form. Most importantly, the encoding allows the use of standard graph rewriting mechanisms for modelling the reduction semantics of the calculus.

We extend the coalgebraic account of specification and refinement of objects and classes in object-oriented programming given by Reichel and Jacobs to (generalised) binary methods. These are methods that take more than one parameter of a class type. Class types include products, sums and powerset type constructors. To allow for class constructors, we model classes as bialgebras. We study and compare two solutions for modelling generalised binary methods, which use purely covariant functors.

In the first solution, which applies when we already have a class implementation, we reduce the behaviour of a generalised binary method to that of a bunch of unary methods. These are obtained by freezing the types of the extra class parameters to constant types. If all parameter types are finitary, the bisimilarity equivalence induced on objects by this model yields the greatest congruence with respect to method application.

In the second solution, we treat binary methods as graphs instead of functions, thus turning contravariant occurrences in the functor into covariant ones.

We show the existence of final coalgebras in both cases.

This paper presents a sample of the deep and multiple interplay between discrete and continuous behaviours and the corresponding modellings in physics. The aim of this overview is to show that discrete and continuous features coexist in any natural phenomenon, depending on the scales of observation. Accordingly, different models, either discrete or continuous in time, space, phase space or conjugate space can be considered. Some caveats about their limits of validity and their interrelationships (discretisation and continuous limits) are pointed out. Difficulties and gaps arising from the singular nature of continuous limits and from the information loss accompanying discretisation are discussed.

We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples.

The three body Coulomb problem is one of the oldest unsolved problems of theoretical physics, and holds a variety of puzzling problems, on both the classical and quantum levels. It is of paradigmatic importance since it bears dynamical and spectral features that establish a link between single and many-particle dynamics, with the absolute minimum of ingredients. When additionally exposed to a time-periodic external perturbation, the effective dimension of the classical phase space as well as of Hilbert space increases dramatically. This gives rise to new phenomena, which stimulate research in such remote areas as mathematical physics, semiclassics and intense-field laser–matter interactions. In the present review, we illustrate how these different areas are interconnected by three interacting particles.