Showing papers in "Mathematical Structures in Computer Science in 2007"

Quantum computation and quantum information

Abstract: 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.

Decoherence in quantum walks – a review

Abstract: 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.

Free compact 2-categories

Abstract: Before one can attach a meaning to a sentence, one must distinguish different ways of parsing it. When analysing a language with pregroup grammars, we are thus led to replace the free pregroup by a free compact strict monoidal category. Since a strict monoidal category is a 2-category with one 0-cell, we investigate the free compact 2-category generated by a given category, and describe its 2-cells as labelled transition systems. In particular, we obtain a decision procedure for the equality of 2-cells in the free compact 2-category.

Restriction categories III: colimits, partial limits and extensivity

Abstract: 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.

On the size complexity of universal accepting hybrid networks of evolutionary processors

Abstract: 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.

Constructive analysis, types and exact real numbers

Abstract: 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.

Graph rewriting for the π-calculus

Abstract: 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.

Implementing the cylindrical algebraic decomposition within the Coq system

Abstract: 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.

The discrete versus continuous controversy in physics

Abstract: 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.

Affine functions and series with co-inductive real numbers

Abstract: 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 Casteran 2004; Gimenez 1994).

Unitary invariants of qubit systems

Abstract: 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.

RealLib: An efficient implementation of exact real arithmetic

Abstract: 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.

A monadic, functional implementation of real numbers

Abstract: 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.

Randomness and determinism in the interplay between the continuum and the discrete

Abstract: This paper provides a conceptual analysis of the role of the mathematical continuum versus the discrete in the understanding of randomness as a notion with a physical meaning or origin. The presentation is ‘informal’ as we will not write formulas; however, we will refer to non-obvious technical results from various scientific domains, and we will also propose a conceptual framework for understanding randomness (and predictability), which we believe is, essentially, original. As a matter of fact, unpredictability and randomness may be conveniently identified in various physico-mathematical contexts. This will allow us to explore these concepts in continuous versus discrete frameworks, with particular emphasis on the relationships and differences between classical approaches and quantum theories in Physics.

Behavioural reasoning for conditional equations

Abstract: 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. Rosu 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.

Spatiality for formal topologies

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

A rewriting calculus for cyclic higher-order term graphs

Abstract: 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).

On the ubiquity of certain total type structures

Abstract: 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 ℕ leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel 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.

Natural deduction via graphs: formal definition and computation rules

Abstract: 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.

Discrete-time and continuous-time modelling: some bridges and gaps

Abstract: 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.

On categorical models of classical logic and the Geometry of Interaction

Abstract: 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.

Dynamical systems: stability and simulability

Abstract: 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.

Proofs, denotational semantics and observational equivalences in Multiplicative Linear Logic

Abstract: We study full completeness and syntactical separability of MLL proof nets with the mix rule. The general method we use consists of first addressing these two questions in the less restrictive framework of proof structures, and then adapting the results to proof nets. At the level of proof structures, we find a semantical characterisation of their interpretations in relational semantics, and define an observational equivalence that is proved to be the equivalence induced by cut elimination. Hence, we obtain a semantical characterisation (in coherent spaces) and an observational equivalence for the proof nets with the mix rule.

On the status of perturbation theory

Abstract: 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.

Two preservation results for countable products of sequential spaces

Abstract: 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.

Entanglement and hilbert space geometry for systems with a few qubits

Abstract: 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.

A denotational semantics for the symmetric interaction combinators

Abstract: 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.

Truth, modality and intersubjectivity

Abstract: 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.

An algebraic semantics of event-based architectures

Abstract: 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.

Boxed ambients with communication interfaces

Abstract: 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.