Computability has its origins in Logic within the framework formed along the original path laid down by the founding fathers of the modern foundational analysis for Mathematics (Frege and Hilbert). This theoretical itinerary, which was largely focused on Logic and Arithmetic, departed in principle from the renewed relations between Geometry and Physics occurring at the time. In particular, the key issue of physical measurement, as our only access to ‘reality’, played no part in its theoretical framework. This is in stark contrast to the position in Physics, where the role of measurement has been a core theoretical and epistemological issue since Poincaré, Planck and Einstein. Furthermore, measurement is intimately related to unpredictability, (in-)determinism and the relationship with physical space–time. Computability, despite having exact access to its own discrete data type, provides a unique tool for the investigation of ‘unpredictability’ in both Physics and Biology through its fine-grained analysis of undecidability – note that unpredictability coincides with physical randomness in both classical and quantum frames. Moreover, it now turns out that an understanding of randomness in Physics and Biology is a key component of the intelligibility of Nature. In this paper, we will discuss a few results following along this line of thought.

In the distributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0,1}n are at the Hamming distanceH(x, y) = 0 or H(x, y) = $\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽ k ⩽ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that $\frac{1}{2}$n ⩽ k < (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

Full abstraction, i.e. that a function preserves equivalence from a source to a target, has been used extensively as a correctness criterion for mappings between models of computation. I here show that with fixed equivalences, fully abstract functions almost always exist. Also, with the function and one of the equivalences fixed the other equivalence can almost always be found.

In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.

The spi calculus is an extension of the pi calculus with cryptographic primitives, which was designed for the verification of cryptographic protocols. Because of this extension, the naive adaptation of labelled bisimulations from the pi calculus is too strong to be useful for the purposes of verification. Instead, as a viable alternative, several ‘environment-sensitive’ bisimulations have been proposed. In this paper, we present a formal study of the differences between these bisimulations.

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.

This paper presents a simple implementation of the λ-calculus in the interaction net paradigm. It is based on a two-fold translation. λ-terms are coded (for duplication) or decoded (for execution), and reduction is achieved by switching between these two states: decoding corresponds to head reduction and encoding to left reduction.

We show how to extend any commutative ring (or semiring) so that division by any element, including 0, is, in a sense, possible. The resulting structure is called a wheel. Wheels are similar to rings, but $0x=0$ does not hold in general; the subset $\{x|0x=0\}$ of any wheel is a commutative ring (or semiring), and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural, and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings.

In this paper we develop novel quantum algorithms based on Quantum Multi-Object Search (QMOS) for convex hulls and general object intersection reporting, with applications to computer graphics. These algorithms are developed and described using standard concepts from computer science by encapsulating the physics of quantum computation within black-box subroutines.

This article proposes a global escape mechanism which can handle unexpected or unwanted conditions changing the default execution of distributed communicational flows, preserving compatibility of the multiparty conversations. Our escape is realized by a collection of asynchronous local exceptions which can be thrown at any stage of the communication and to any subsets of participants in a multiparty session. This flexibility enables to model complex exceptions such as criss-crossing global interactions and error handling for distributed cooperating threads. Guided by multiparty session types, our semantics is proven to provide a termination algorithm for global escapes. Our type system guarantees further safety and liveness properties, such as progress within the session and atomicity of escapes with respect to the subset of involved participants.

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.

We investigate Girard's calculus Fω as a ‘Curry style’ type assignment system for pure lambda terms. First we show an example of a strongly normalizable term that is untypable in Fω. Then we prove that every partial recursive function is nonuniformly represented in Fω (even if quantification is restricted to constructor variables of level 1). It follows that the type reconstruction problem is undecidable and cannot be recursively separated from normalization.

There are two main approaches to obtaining ‘topological’ cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed – for example, the category of sequential spaces. Under the other, one generalises the notion of space – for example, to Scott's notion of equilogical space. In this paper, we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. In fact, this category turns out to be equivalent to the category of all quotient spaces of countably based topological spaces. We show that the category is bicartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the category of equilogical spaces. We also show that the category of countably based equilogical spaces has a larger full subcategory that can be simultaneously viewed as a full subcategory of limit spaces. This full subcategory is locally cartesian closed and the embeddings into limit spaces and countably based equilogical spaces preserve this structure. We observe that it seems essential to go beyond the realm of topological spaces to achieve this result.

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.

A quantum algorithm to determine approximations of linear structures of Boolean functions is presented and analysed. Similar results have already been published (see Simon's algorithm) but only for some promise versions of the problem, and it has been shown that no exponential quantum speedup can be obtained for the general (no promise) version of the problem. In this paper, no additional promise assumptions are made. The approach presented is based on the method used in the Bernstein–Vazirani algorithm to identify linear Boolean functions and on ideas from Simon's period finding algorithm. A proper combination of these two approaches results here to a polynomial-time approximation to the linear structures set. Specifically, we show how the accuracy of the approximation with high probability changes according to the running time of the algorithm. Moreover, we show that the time required for the linear structure determine problem with high success probability is related to so called relative differential uniformity δf of a Boolean function f. Smaller differential uniformity is, shorter time is needed.

This paper focusses on the determination of the dihomotopy classes of dipaths in cubical complexes, representing the essentially different computations in a given concurrent program due to different schedules. Several new notions have to be developed, for example, the domains of dependence (Definition 2.4), which are borrowed from relativity theory (Penrose 1972). However, it turns out that the algorithm determining deadlocks and unsafe regions described in Fajstrup et al. (1998a; 1998b) can be modified and applied to do the essential calculational work.

Amalgamation is a well-known concept for graph transformations that is used to model synchronised parallelism of rules with shared subrules and corresponding transformations. This concept is especially important for an adequate formalisation of the operational semantics of statecharts and other visual modelling languages, where typed attributed graphs are used for multiple rules with nested application conditions. However, the theory of amalgamation for the double-pushout approach has so far only been developed on a set-theoretical basis for pairs of standard graph rules without any application conditions.

For this reason, in the current paper we present the theory of amalgamation for $\mathcal{M}$-adhesive categories, which form a slightly more general framework than (weak) adhesive HLR categories, for a bundle of rules with (nested) application conditions. The two main results are the Complement Rule Theorem, which shows how to construct a minimal complement rule for each subrule, and the Multi-Amalgamation Theorem, which generalises the well-known Parallelism and Amalgamation Theorems to the case of multiple synchronised parallelism. In order to apply the largest amalgamated rule, we use maximal matchings, which are computed according to the actual instance graph. The constructions are illustrated by a small but meaningful running example, while a more complex case study concerning the firing semantics of Petri nets is presented as an introductory example and to provide motivation.

The purpose of this short paper is to sketch the development of a few basic topics in the history of realizability. The number of topics is quite limited and reflects very much my own personal taste, prejudices and area of competence.

The notion of a recognizable set of words, trees or graphs is relative to an algebraic structure on the set of words, trees or graphs respectively. We establish that several algebraic structures yield the same notion of a recognizable set of graphs. This notion is equivalent to that of a fully cutset-regular set of graphs introduced by Fellows and Abrahamson. We also establish that the class of recognizable sets of graphs is closed under the operations considered in these various equivalent definitions. This fact is not a standard consequence of the definition of recognizability.

We propose compactly generated monotone convergence spaces as a well-behaved topological generalisation of directed-complete partial orders (dcpos). The category of such spaces enjoys the usual properties of categories of ‘predomains’ in denotational semantics. Moreover, such properties are retained if one restricts to spaces with a countable pseudobase in the sense of E. Michael, a fact that permits connections to be made with computability theory, realizability semantics and recent work on the closure properties of topological quotients of countably based spaces (qcb spaces). We compare the standard domain-theoretic constructions of products and function spaces on dcpos with their compactly generated counterparts, showing that these agree in important cases, though not in general.