We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

This paper presents an analysis of definitions and decidability for exponential functions on various matrix algebras. The main idea is to show that, generically, the entries of the exponential (or logarithm) of a matrix are Pfaffian functions of the entries of the matrix.

We compare two deforestation techniques: short cut fusion formalised in category theory and the syntactic composition of tree transducers. The former strongly depends on types and uses the parametricity property or free theorem, whereas the latter makes no use of types at all and allows more general compositions. We introduce the notion of a categorical transducer, which is a generalisation of a catamorphism, and show a corresponding fusion result, which is a generalisation of the ‘acid rain theorem’. We prove the following main theorems: (i) The class of all categorical transducers builds a category where composition is fusion; (ii) The semantics of categorical transducers is a functor. (iii) The subclass of top-down categorical transducers is a subcategory. (iv) Syntactic composition of top-down tree transducers is equivalent to the fusion of top-down categorical transducers.

By combining algorithmic learning, decision procedures, predicate abstraction and simple templates for quantified formulae, we present an automated technique for finding loop invariants. Theoretically, this technique can find arbitrary first-order invariants (modulo a fixed set of atomic propositions and an underlying satisfiability modulo theories solver) in the form of the given template and exploit the flexibility in invariants by a simple randomized mechanism. In our study, the proposed technique was able to find quantified invariants for loops from the Linux source and other realistic programs. Our contribution is a simpler technique than the previous works yet with a reasonable derivation power.

We show that the category of basic pairs (BP) and the category of concrete spaces (CSpa) are both small-complete and small-cocomplete in the framework of constructive Zermelo–Frankel set theory extended with the set generation axiom. We also show that CSpa is a coreflective subcategory of BP.

The choice of suitable high-level communication primitives for wide area network programming languages remains an open problem. This paper is driven by the practical consideration of providing an efficient and secure communication infrastructure for mobile agent systems. This has led us to formalise the Lime coordination middleware and propose a simplified model, which we call CoreLime, that addresses some of the main shortcomings of Lime while retaining its distinguishing feature, namely transient sharing of tuple spaces. We further discuss a prototype implementation along with security extensions. Our contribution is thus an exploration of the language design space rather than a theoretical investigation of properties of these models.

We discuss a formal development for the certification of Newton's method. We address several issues encountered in the formal study of numerical algorithms: developing the necessary libraries for our proofs, adapting paper proofs to suit the features of a proof assistant and designing new proofs based on the existing ones to deal with optimisations of the method. We start from Kantorovitch's Theorem, which gives the convergence of Newton's method in the case of a system of equations. To formalise this proof inside the proof assistant Coq, we first need to code the necessary concepts from multivariate analysis. We also prove that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input and that of the result. This proof is based on Kantorovitch's Theorem, but is an original result. An algorithm including rounding is a more accurate model for computations with Newton's method in practice.

We present a new kind of ambient calculus in which the open capability is replaced by direct mobility of generic processes. The calculus comes equipped with a labelled transition system in which types play a major role: this system allows us to show interesting algebraic laws. As usual, types express the communication, access and mobility properties of the modelled system, and inferred types express the minimal constraints required for the system to be well behaved.

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle.

Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(\mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $\phi$ of BISH a set ${\texttt{Prf}}(\phi)$ of “proofs” or witnesses of $\phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(\mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn of doctrines. Modal interior operators are characterised as particular adjoints in the 2-category Dtn. We show that they can be constructed from comonads in Dtn as well as from adjunctions in it, and we compare the two constructions. Finally we show the amount of information lost in the passage from a comonad, or from an adjunction, to the modal interior operator. The basis for the present work is provided by some seminal work of John Power.

We introduce a novel amortised resource analysis couched in a type-and-effect system. Our analysis is formulated in terms of the physicist’s method of amortised analysis and is potentialbased. The type system makes use of logarithmic potential functions and is the first such system to exhibit logarithmic amortised complexity. With our approach, we target the automated analysis of self-adjusting data structures, like splay trees, which so far have only manually been analysed in the literature. In particular, we have implemented a semi-automated prototype, which successfully analyses the zig-zig case of splaying, once the type annotations are fixed.

We show that a measurement $\mu$ on a continuous dcpo $D$ extends to a measurement $\skew3\bar{\mu}$ on the convex powerdomain ${\mathbf C} D$ iff it is a Lebesgue measurement. In particular, $\ker\mu$ must be metrisable in its relative Scott topology. Moreover, the space $\ker\skew3\bar{\mu}$ in its relative Scott topology is homeomorphic to the Vietoris hyperspace of $\ker\mu$, that is, the space of non-empty compact subsets of $\ker\mu$ in its Vietoris topology – the topology induced by any Hausdorff metric. This enables one to show that Hutchinson's theorem holds for any finite set of contractions on a domain with a Lebesgue measurement. Finally, after resolving the existence question for Lebesgue measurements on countably based domains, we uncover the following relationship between classical analysis and domain theory: for an $\omega$-continuous dcpo $D$ with $\max(D)$ regular, the Vietoris hyperspace of $\max(D)$ embeds in $\max({\mathbf C} D)$ as the kernel of a measurement on ${\mathbf C} D$.

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.

In the classical theory of regular languages, the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called “Substitution Principle.” In this paper, we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently, we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil’s Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.

A long-standing and fundamental issue in computer security is to control the flow of information, whether to prevent confidential information from being leaked, or to prevent trusted information from being tainted. While there have been many efforts aimed at preventing improper flows completely (see for example, the survey by Sabelfeld and Myers (2003)), it has long been recognized that perfection is often impossible in practice. A basic example is a login program – whenever it rejects an incorrect password, it unavoidably reveals that the secret password differs from the one that was entered. More subtly, systems may be vulnerable to side channel attacks, because observable characteristics like running time and power consumption may depend, at least partially, on sensitive information.

We describe quantum circuits generating four-qubit maximally entangled states, the amount of entanglement being quantified by using the absolute value of the Cayley hyperdeterminant as an entanglement monotone. More precisely we show that this type of four-qubit entangled states can be obtained by the action of a family of $\mathtt{CNOT}$ circuits on some special states of the LU orbit of the state $|0000\rangle$ .

We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are Dθ-completion and Ds2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its Dθ-completion is a domain; (2) the Dθ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

We investigate a new computational interpretation for an intuitionistic focused sequent calculus which is compatible with a resource aware semantics. For that, we associate to Herbelin's syntax a type system based on non-idempotent intersection types, together with a set of reduction rules – inspired from the substitution at a distance paradigm – that preserves (and decreases the size of) typing derivations. The non-idempotent approach allows us to use very simple combinatorial arguments, only based on this measure decreasingness, to characterize linear-head and strongly normalizing terms by means of typability. For the sake of completeness, we also study typability (and the corresponding strong normalization characterization) in the calculus obtained from the former one by projecting the explicit cuts.

Indexed monoidal algebras are introduced as an equivalent structure for self-dual compact closed categories, and a coherence theorem is proved for the category of such algebras. Turing automata and Turing graph machines are defined by generalising the classical Turing machine concept so that the collection of such machines becomes an indexed monoidal algebra. On the analogy of the von Neumann data-flow computer architecture, Turing graph machines are proposed as potentially reversible low-level universal computational devices, and a truly reversible molecular size hardware model is presented as an example.