We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming, we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example, we give the semantics of Selinger's language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilisers.

Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with ‘iterative algebras’, that is, algebras admitting a unique solution of all systems of flat recursive equations, a free iterative theory is obtained as the theory of free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic one. Despite this, our result is much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category $\cal{A}$.

Reportedly, a blow from the welterweight boxer Norman Selby, also known as Kid McCoy, left one victim proclaiming,

‘It's the real McCoy!’.

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.

We study the complexity of computable and $\Sigma^0_1$ inductive definitions of sets of natural numbers. For example, we show how to assign natural indices to monotone $\Sigma^0_1$-definitions and then use these to calculate the complexity of the set of all indices of monotone $\Sigma^0_1$-definitions that are computable. We also examine the complexity of a new type of inductive definition, which we call weakly finitary monotone inductive definitions. Applications are given in proof theory and in logic programming.

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.

Motivated by recent work on the derivation of labelled transitions and bisimulation congruences from unlabelled reaction rules, we show how to address this problem in the DPO (double-pushout) approach to graph rewriting. Unlike the case with previous approaches, we consider graphs as objects, rather than arrows, of the category under consideration. This allows us to present a very simple way of deriving labelled transitions (called rewriting steps with borrowed context), which integrates smoothly with the DPO approach, has a very constructive nature and requires only a minimum of category theory. The core part of this paper is the proof that the bisimilarity based on graph rewriting with borrowed contexts is a congruence relation. We will also introduce some proof techniques and compare our approach with the derivation of labelled transitions via relative pushouts.

We show that the model of quantum computation based on density matrices and superoperators can be decomposed into a pure classical (functional) part and an effectful part modelling probabilities and measurement. The effectful part can be modelled using a generalisation of monads called arrows. We express the resulting executable model of quantum computing in the Haskell programming language using its special syntax for arrow computations. However, the embedding in Haskell is not perfect: a faithful model of quantum computing requires type capabilities that are not directly expressible in Haskell.

In this note we show that quotients of countably based spaces (qcb spaces) and topological predomains, as introduced by M. Schröder and A. Simpson, are not closed under sobrification. As a consequence, replete topological predomains need not be sober, that is, in general, repletion is not given by sobrification. Our counterexample also shows that a certain tentative ‘equaliser construction’ of repletion fails for qcb spaces.

Our results also extend to the more general class of core compactly generated spaces.

The $\lambda$-calculus is destructive: its main computational mechanism, beta reduction, destroys the redex, which makes replaying the computational steps impossible. Combinatory logic is a variant of the $\lambda$-calculus that maintains irreversibility. Recently, reversible computational models have been studied mainly in the context of quantum computation, as (without measurements) quantum physics is inherently reversible. However, reversibility also fundamentally changes the semantical framework in which classical computation has to be investigated. We describe an implementation of classical combinatory logic in a reversible calculus for which we present an algebraic model based on a generalisation of the notion of a group.

Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš' Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

We present a Cartesian closed category ELOC of equilocales, which contains the category LOC of locales as a reflective full subcategory. The embedding of LOC into ELOC preserves products and all exponentials of exponentiable locales.

Recently, encodings in interaction nets of the call-by-name and call-by-value strategies of the $\lambda$-calculus have been proposed. The purpose of these encodings is to bridge the gap between interaction nets and traditional abstract machines, which are both used to provide lower-level specifications of strategies of the $\lambda$-calculus, but in radically different ways. The strength of these encodings is their simplicity, which comes from the simple idea of introducing an explicit syntactic object to represent the evaluation flow. Another benefit of this approach is that no artifact is needed to represent boxes. However, these encodings deliberately follow the implemented strategies (call-by-name and call-by-value) as closely as possible, and hence do not benefit from the ability of interaction nets to represent sharing easily. The aim of this paper is to show that better sharing (hence efficiency) can indeed be achieved without adding much structure. We thus present the call-by-need strategy following the same philosophy, which is, indeed, no more complicated than call-by-name. We also extend our approach to fully lazy reduction. This continues the task of bridging the gap between interaction nets and abstract machines, thus pushing forward a more uniform framework for implementations of the $\lambda$-calculus.

We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.

We show that the Borel hierarchy of the class of context free $\omega$-languages, or even of the class of $\omega$-languages accepted by Büchi 1-counter automata, is the same as the Borel hierarchy of the class of $\omega$-languages accepted by Turing machines with a Büchi acceptance condition. In particular, for each recursive non-null ordinal $\alpha$, there exist some ${\bf \Sigma}^0_\alpha$-complete and some ${\bf \Pi}^0_\alpha$-complete $\omega$-languages accepted by Büchi 1-counter automata. And the supremum of the set of Borel ranks of context free $\omega$-languages is an ordinal $\gamma_2^1$ that is strictly greater than the first non-recursive ordinal $\omega_1^{\mathrm{CK}}$. We then extend this result, proving that the Wadge hierarchy of context free $\omega$-languages, or even of $\omega$-languages accepted by Büchi 1-counter automata, is the same as the Wadge hierarchy of $\omega$-languages accepted by Turing machines with a Büchi or a Muller acceptance condition.

In this paper, we describe three distinct monoids over domains, each with a commutative analog, which define bag domain monoids. Our results were inspired by work by Varacca (Varacca 2003), and they lead to a constructive approach to his Hoare indexed valuations over a continuous poset $P$. We use our constructive approach to describe an analog of the probabilistic power domain, and the laws that characterise it, that forms a Scott-closed subset of Varacca's construct. We call these the Hoare random variables over$P$.

We present a formalism called addressed term rewriting systems, which can be used to model implementations of theorem proving, symbolic computation and programming languages, especially aspects of sharing, recursive computations and cyclic data structures. Addressed Term Rewriting Systems are therefore well suited to describing object-based languages, and as an example we present a language called $\lambda{\cal O}bj^{a}$, incorporating both functional and object-based features. As a case study in how reasoning about languages is supported in the ATRS formalism, we define a type system for $\lambda{\cal O}bj^{a}$ and prove a type soundness result.

We show that for any 2-computably enumerable Turing degree ${\bf l}$, any computably enumerable degree ${\bf a}$ and any Turing degree ${\bf s}$, if ${\bf l'=\boldsymbol{0}'}$, ${\bf l<a}$, ${\bf s\geq \boldsymbol{0}'}$, and ${\bf s}$ is c.e. in ${\bf a}$, then there is a 2-computably enumerable degree ${\bf x}$ with the following properties:

1. ${\bf l<x<a}$; and

2. ${\bf x'=s}$

The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.

In this paper we develop a functional programming language for quantum computers by extending the simply-typed lambda calculus with quantum types and operations. The design of this language adheres to the ‘quantum data, classical control’ paradigm, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and give a type system using affine intuitionistic linear logic. The main results of this paper are the safety properties of the language and the development of a type inference algorithm.