This issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Workshop ‘Continuity, Computability, Constructivity: From Logic to Algorithms,’ held in Gregynog, a conference centre of the University of Wales located in the beautiful nature of Mid Wales, in the last week of June 2013. In addition, several colleagues accepted our invitation to contribute to this volume.

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

We consider computation with real numbers that arise through a process of physical measurement. We have developed a theory in which physical experiments that measure quantities can be used as oracles to algorithms and we have begun to classify the computational power of various forms of experiment using non-uniform complexity classes. Earlier, in Beggs et al. (2014 Reviews of Symbolic Logic7(4) 618–646), we observed that measurement can be viewed as a process of comparing a rational number z – a test quantity – with a real number y – an unknown quantity; each oracle call performs such a comparison. Experiments can then be classified into three categories, that correspond with being able to return test results

$$\begin{eqnarray*} z < y\text{ or }z > y\text{ or }\textit{timeout},\\ z < y\text{ or }\textit{timeout},\\ z \neq y\text{ or }\textit{timeout}. \end{eqnarray*} $$

We give a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda–calculus with pairing and case-construct. The purpose of this interpretation is to provide a foundation for the extraction of verified programs from formal proofs as an alternative to type-theoretic systems. The advantages of our approach are that (a) induction and coinduction are not restricted to the strictly positive case, (b) abstract mathematical structures and results may be imported, (c) the formalization is technically simpler than in other systems, for example, regarding the definition of realizability, which is a simple syntactical substitution, and the treatment of nested and simultaneous (co)inductive definitions.

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.

Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.

As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.

The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991. Complexity Theory of Real Functions).

This paper is a part of the ongoing program of analysing the complexity of various problems in computable analysis in terms of the complexity of the associated index sets. In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective ω–continuous domains and the natural numbers with the discrete topology satisfy this condition. We prove weak and strong analogues of Rice's theorem for computable elements. Then, we construct principal computable numberings of partial majorant-computable real-valued functions and co-effectively closed sets and calculate the complexity of index sets for important problems such as root verification and function equality. For example, we show that, for partial majorant-computable real functions, the equality problem is Π11-complete.

We give new proofs of effective versions of the Riemann mapping theorem, its extension to multiply connected domains and the uniformization on Riemann surfaces. Astonishingly, in the presented proofs, we need barely more than computational compactness and the classical results.

After works of Normann and the author on sequentiality (Normann 2006Mathematical Structures in Computer Science16 (2) 279–289; Normann and Sazonov 2012Annals of Pure and Applied Logic163 (5) 575–603; Sazonov 2007Logical Methods in Computer Science3 (3:7) 1–50), the necessity and possibility of a non-dcpo domain theory became evident. In this paper, the category of continuous dcpo domains is generalized to a category of ‘naturally’ continuous non-dcpo domains with ‘naturally’ continuous maps as arrows. A full subcategory of the latter, assuming a kind of bounded-completeness requirement of domains and presence of ⊥ in each, proves to be Cartesian closed and equivalent to a subclass of Ershov's general A-spaces (Ershov 1974Algebra and Logics12 (4) 369–416). This extends a non-dcpo generalization of Scott (algebraic) domains introduced and proved to be equivalent to Ershov's general f-spaces (Ershov 1972Algebra and Logic11 (4) 367–437) in Sazonov (2007 op. cit.; 2009 Annals of Pure and Applied Logic159 (3) 341–355).

The current approach to natural domains (v-domains) is different from f-spaces and A-spaces in that it has arisen in Sazonov (2007 op. cit.) in a different way from defining fully abstract models for some versions of the language PCF over Integers, whereas the Ershov's approach was not initially related with full abstraction, and non-dcpo version of f-spaces and A-spaces were originally considered in an abstract (mainly topological) style. In this paper devoted to naturally continuous natural domains (v-continuous v-domains), we also work in an abstract (mainly order-theoretic) style but with the hope to relate it in the future with the ideas of PCF over Reals by exploring and adapting the ideas in Escardó (1996Theoretical Computer Science162 (1) 79–115), Escardó et al. (2004Mathematical Structures in Computer Science, 14 (6), Cambridge University Press 803–814), Marcial-Romero and Escardó (2007Theoretical Computer Science379 (1-2) 120–141), Sazonov (2007 op. cit.).

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably based T0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case of k-partitions. In particular, we investigate the difference hierarchy of k-partitions and the fine hierarchy closely related to the Wadge hierarchy.

In this article we describe a bunch of probability logics with quantifiers over events, and develop primary techniques for proving computational complexity results (in terms of m-degrees) about these logics, mainly over discrete probability spaces. Also the article contains a comparison with some other probability logics and a discussion of interesting analogies with research in the metamathematics of Boolean algebras, demonstrating a number of attractive features and intuitive advantages of the present proposal.

The continuity problem, i.e., the question whether effective maps between effectively given topological spaces are effectively continuous, is reconsidered. In earlier work, it was shown that this is always the case, if the effective map also has a witness for non-inclusion. The extra condition does not have an obvious topological interpretation. As is shown in the present paper, it appears naturally where in the classical proof that sequentially continuous maps are continuous, the Axiom of Choice is used. The question is therefore whether the witness condition appears in the general continuity theorem only for this reason, i.e., whether effective operators are effectively sequentially continuous. For two large classes of spaces covering all important applications, it is shown that this is indeed the case. The general question, however, remains open.

Spaces in this investigation are in general not required to be Hausdorff. They only need to satisfy the weaker T0 separation condition

A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS. We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.