The ten papers in this special issue arose from the conference CiE 2005: New Computational Paradigms, held at the University of Amsterdam in June, 2005. CiE 2005 was the first of a new series of conferences associated with the interdisciplinary network Computability in Europe focused on computability in theoretical computer science and mathematical logic, and ranging over a broad spectrum of research areas from the application of novel approaches to computation, through computability-theoretic aspects of physical systems to set-theoretic analyses of infinitary computing models.

Lipschitz continuity is used as a tool for analysing the relationship between incomputability and randomness. We present a simpler proof of one of the major results in this area – the theorem of Yu and Ding, which states that there exists no cl-complete c.e. real – and go on to consider the global theory. The existential theory of the cl degrees is decidable, but this does not follow immediately by the standard proof for classical structures, such as the Turing degrees, since the cl degrees are a structure without join. We go on to show that strictly below every random cl degree there is another random cl degree. Results regarding the phenomenon of quasi-maximality in the cl degrees are also presented.

We prove a general strong normalisation theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domain-theoretic semantics. The theorem applies to extensions of Gödel's system T, but also to various forms of barrecursion for which strong normalisation was hitherto unknown.

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.

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 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.

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 notion of ordinal computability is defined by generalising standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. The fundamental theorem on ordinal computability states that a set $x$ of ordinals is ordinal computable from ordinal parameters if and only if $x$ is an element of the constructible universe $\mathbf{L}$. In this paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo–Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.

Recent work in constructive mathematics shows that Hilbert's program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform most of the highly abstract proofs of ‘concrete’ statements into elementary proofs. Surprisingly, the arguments we produce are not only elementary but also mathematically clearer, and not necessarily longer. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem. For this we use a general method to transform some logically complex first-order formulae into a geometrical form, which may be interesting in itself.

In the paper we introduce and study the uniform regular enumerations for arbitrary recursive ordinals. As an application of the technique, we obtain a uniform generalisation of a theorem of Ash and a characterisation of a class of uniform operators on transfinite sequences of sets of natural numbers.

We investigate natural systems of fundamental sequences for ordinals below the Howard–Bachmann ordinal and study growth rates of the resulting slow growing hierarchies. We consider a specific assignment of fundamental sequences that depends on a non-negative real number $\varepsilon$. We show that the resulting slow growing hierarchy is eventually dominated by a fixed elementary recursive function if $\varepsilon$ is equal to zero. We show further that the resulting slow growing hierarchy exhausts the provably recursive functions of $\sfb{ID}_1$ if $\varepsilon$ is strictly greater than zero. Finally, we show that the resulting fast growing hierarchies exhaust the provably recursive functions of $\sfb{ID}_1$ for all non-negative values of $\varepsilon$. Our result is somewhat surprising since usually the slow growing hierarchy along the Howard–Bachmann ordinal exhausts precisely the provably recursive functions of $\sfb{PA}$. Note that the elementary functions are a very small subclass of the provably recursive functions of $\sfb{PA}$, and the provably recursive functions of $\sfb{PA}$ are a very small subclass of the provably recursive functions of $\sfb{ID}_1$. Thus the jump from $\varepsilon$ equal to zero to $\varepsilon$ greater than zero is one of the biggest jumps in growth rates for subrecursive hierarchies one might think of.