We present a simple module calculus where selection and execution of a component is possible on open modules, that is, modules that still need to import some external definitions. Hence, it provides a kernel model for a computational paradigm in which standard execution (that is, execution of a single computation described by a fragment of code) can be interleaved with operations at the meta-level, which can manipulate in various ways the context in which this computation takes place. Formally, this is achieved by introducing configurations as basic terms. These are, roughly speaking, pairs consisting of an (open, mutually recursive) collection of named components and a term representing a program running in the context of these components. Configurations can be manipulated by classical module/fragment operators, hence reduction steps can be either execution steps of the program or steps that perform module operations (called reconfiguration steps).

Since configurations combine the features of lambda abstractions (first-class functions), records, environments with mutually recursive definitions and modules, the calculus extends and integrates both traditional module calculi and recursive lambda calculi. We state confluence of the calculus, and propose different ways to prevent errors arising from the lack of some required component, either by a purely static type system or by a combination of static and run-time checks. Moreover, we define a call-by-need strategy that performs module simplification only when needed and only once, leading to a generalisation of call-by-need lambda calculi that includes module features. We prove the soundness and completeness of this strategy using an approach based on information content, which also allows us to preserve confluence, even when local substitution rules are added to the calculus.

Quantum physics, together with the experimental (and slightly controversial) quantum computing, induces a twist in our vision of computation, and hence, since computing and logic are intimately linked, in our approach to logic and foundations. In this paper, we discuss the most mistreated notion of logic, truth.

We propose a mathematical semantics for event-based architectures that serves two main purposes: to characterise the modularisation properties that result from the algebraic structures induced on systems by this discipline of coordination; and to further validate and extend the categorical approach to architectural modelling that we have been building around the language CommUnity with the ‘implicit invocation’, also known as ‘publish/subscribe’ architectural style. We then use this formalisation to bring together synchronous and asynchronous interactions within the same modelling approach. We see this effort as a first step towards a form of engineering of architectural styles. Our approach adopts transition systems extended with events as a mathematical model of implicit invocation, and a family of logics that support abstract levels of modelling.

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.

First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.

We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

In this paper, we introduce the formalism of deduction graphs as a generalisation of both Gentzen–Prawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that, as with flag deductions (but not natural deduction), subproofs can be shared, but the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. We give a precise definition of deduction graphs, together with some illustrative examples. Furthermore, we analyse their computational behaviour by studying the process of cut-elimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation, we conclude that the process of cut-elimination is strongly normalising. The translation to simple type theory removes quite a lot of structure, so we also propose a translation to a context calculus with lets that faithfully captures the structure of deduction graphs. The proof nets of linear logic also offer a graph-like presentation of natural deduction, and we point out some similarities between the two formalisms.

Object-oriented (OO) programming techniques can be applied to equational specification logics by distinguishing visible data from hidden data (that is, by distinguishing the output of methods from the objects to which the methods apply), and then focusing on the behavioural equivalence of hidden data in the sense introduced by H. Reichel in 1984. Equational specification logics structured in this way are called hidden equational logics, HELs. The central problem is how to extend the specification of a given HEL to a specification of behavioural equivalence in a computationally effective way. S. Buss and G. Roşu showed in 2000 that this is not possible in general, but much work has been done on the partial specification of behavioural equivalence for a wide class of HELs. The OO connection suggests the use of coalgebraic methods, and J. Goguen and his collaborators have developed coinductive processes that depend on an appropriate choice of a cobasis, which is a special set of contexts that generates a subset of the behavioural equivalence relation. In this paper the theoretical aspects of coinduction are investigated, specifically its role as a supplement to standard equational logic for determining behavioural equivalence. Various forms of coinduction are explored. A simple characterisation is given of those HELs that are behaviourally specifiable. Those sets of conditional equations that constitute a complete, finite cobasis for a HEL are characterised in terms of the HEL's specification. Behavioural equivalence, in the form of logical equivalence, is also an important concept for single-sorted logics, for example, sentential logics such as the classical propositional logic. The paper is an application of the methods developed through the extensive work that has been done in this area on HELs, and to a broader class of logics that encompasses both sentential logics and HELs.

This paper is an introduction to the RealLib package for exact real number computations. The library provides certified accuracy, but tries to achieve this at performance close to the performance of hardware floating point for problems that do not require higher precision. The paper gives the motivation and features of the design of the library and compares it with other packages for exact real arithmetic.

Computers are used extensively to simulate continuous dynamical systems. However, different conceptual and mathematical structures underlie discrete machines and continuous dynamics, so the question arises as to the ability of the computer to simulate or, more generally, to check the properties of a continuous system.

We discuss and compare two notions of stability for a continuous dynamical system, viz. shadowing and robustness, and relate them to both the practical and theoretical computability of the system. We first discuss what we can learn from the stability of a system, using a finite-precision machine. We then show, following the work in Collins (2005), that shadowing fails but robustness succeeds in ensuring the checkability of a reachability property.

In this paper we discuss the following interesting question about accepting hybrid networks of evolutionary processors (AHNEP), which are a recently introduced bio-inspired computing model. The question is: how many processors are required in such a network to recognise a given language L? Two answers are proposed for the most general case, when L is a recursively enumerable language, and both answers improve on the previously known bounds. In the first case the network has a number of processors that is linearly bounded by the cardinality of the tape alphabet of a Turing machine recognising the given language L. In the second case we show that an AHNEP with a fixed underlying structure can accept any recursively enumerable language. The second construction has another useful property from a practical point of view as it includes a universal AHNEP as a subnetwork, and hence only a limited number of its parameters depend on the given language.

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, has led rapidly to several new quantum algorithms. These all follow a unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about partial maps as they have a purely algebraic formulation.

In this paper we consider colimits and limits in restriction categories. As the notion of restriction category is not self-dual, we should not expect colimits and limits in restriction categories to behave in the same manner. The notion of colimit in the restriction context is quite straightforward, but limits are more delicate. The suitable notion of limit turns out to be a kind of lax limit, satisfying certain extra properties.

Of particular interest is the behaviour of the coproduct, both by itself and with respect to partial products. We explore various conditions under which the coproducts are ‘extensive’ in the sense that the total category (of the related partial map category) becomes an extensive category. When partial limits are present, they become ordinary limits in the total category. Thus, when the coproducts are extensive we obtain as the total category a lextensive category. This provides, in particular, a description of the extensive completion of a distributive category.

The relationship between continuous-time dynamics and the corresponding discrete schemes, and its generally limited validity, is an important and widely acknowledged field within numerical analysis. In this paper, we propose another, more physical, viewpoint on this topic in order to understand the possible failure of discretisation procedures and the way to fix it. Three basic examples, the logistic equation, the Lotka–Volterra predator–prey model and Newton's law for planetary motion, are worked out. They illustrate the deep difference between continuous-time evolutions and discrete-time mappings, hence shedding some light on the more general duality between continuous descriptions of natural phenomena and discrete numerical computations.

The Coq system is a Curry–Howard based proof assistant. Therefore, it contains a full functional, strongly typed programming language, which can be used to enhance the system with powerful automation tools through the implementation of reflexive tactics. We present the implementation of a cylindrical algebraic decomposition algorithm within the Coq system, whose certification leads to a proof producing decision procedure for the first-order theory of real numbers.

This paper reviews recent work related to the interplay between quantum information and computation on the one hand and classical and quantum chaos on the other.

First, we present several models of quantum chaos that can be simulated efficiently on a quantum computer. Then a discussion of information extraction shows that such models can give rise to complete algorithms including measurements that can achieve an increase in speed compared with classical computation. It is also shown that models of classical chaos can be simulated efficiently on a quantum computer, and again information can be extracted efficiently from the final wave function. The total gain can be exponential or polynomial, depending on the model chosen and the observable measured. The simulation of such systems is also economical in the number of qubits, allowing implementation on present-day quantum computers, some of these algorithms having been already experimentally implemented.

The second topic considered concerns the analysis of errors on quantum computers. It is shown that quantum chaos algorithms can be used to explore the effect of errors on quantum algorithms, such as random unitary errors or dissipative errors. Furthermore, the tools of quantum chaos allows a direct analysis of the effects of static errors on quantum computers. Finally, we consider the different resources used by quantum information, and show that quantum chaos has some precise consequences on entanglement generation, which becomes close to maximal. For another resource, interference, a proposal is presented for quantifying it, enabling a discussion on entanglement and interference generation in quantum algorithms.

The symmetric interaction combinators are a variant of Lafont's interaction combinators. They enjoy a weaker universality property with respect to interaction nets, but are equally expressive. They are a model of deterministic distributed computation and share the good properties of Turing machines (elementary reductions) and of the λ-calculus (higher-order functions and parallel execution). We introduce a denotational semantics for this system, which is inspired by the relational semantics for linear logic, and prove an injectivity and full completeness result for it. We also consider the algebraic semantics defined by Lafont, and prove that the two are strongly related.

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary functions for such proofs by using the monad properties of the completion operation on metric spaces. Bishop and Bridges's notion (Bishop and Bridges 1985) of regular sequences is generalised to what I call regular functions, which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (which are a common subclass of metric spaces) are created by lifting continuous functions on the original space. A prototype Haskell implementation has been created. I believe that this approach yields a real number library that is reasonably efficient for computation, and still simple enough to verify its correctness easily.

This paper introduces projective systems for topological and probabilistic event structures. The projective formalism is used for studying the domain of configurations of a prime event structure and its space of maximal elements. This is done from both a topological and a probabilistic viewpoint. We give probability measure extension theorems in this framework.

Perturbation theory has always been an important component of the natural sciences. From celestial mechanics to the quantum theory of fields, it has always played a central role, which this little note sets out to analyse briefly. We will show, in particular, how its epistemological position has changed from being just a ‘tool’ to being the basis of definition for objects in quantum field theory.

We give the first linear time (randomised) algorithm for the first order isomorphism problem, that is, the isomorphism of non-recursive types involving product- and function-type constructors, under the axioms of commutativity and associativity of products, currying and distributivity of functions over products. This problem can also be thought of as the problem of formal equality-testing of multi-variate expressions involving only multiplications and exponentiation. Previous work gave a deterministic O(n log2n) time and O(n) space algorithm for the problem (n being the input size). Our specific contribution includes two randomised algorithms for the problem:

1. (i) an O(n) time Monte Carlo algorithm (that is, with a small probability it may decide erroneously that the two types are isomorphic), and

2. (ii) an O(n log n) expected time and O(n) space Las Vegas algorithm (that is, with a small probability it may execute long).

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.