This special issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Dagstuhl Seminar on Computing with Infinite Data: Topological and Logical Foundations. The workshop took place in the Schloss Dagstuhl - Leibniz Center for Informatics in the first half of October 2011.

We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is that the class of models of a generalized geometric theory is set-generated. Here, a class $\mathcal{X}$ of subsets of a set is set-generated if there exists a subset G of $\mathcal{X}$ such that for each α ∈ $\mathcal{X}$, and finitely enumerable subset τ of α there exists a subset β ∈ G such that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo–Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable in CZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.

We provide a realizability model based on infinite time Turing machines in which there is an injection from the internal Baire space, the object of infinite sequences of numbers, to the object of natural numbers.

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Selivanov in a series of papers centred on algebraic domains. And recently it has been considered by de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper, we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

We prove some results on the Wadge order on the space of sets of natural numbers endowed with Scott topology, and more generally, on omega-continuous domains. Using alternating decreasing chains we characterize the property of Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated differences of open sets). A similar characterization holds for Wadge one-to-one and finite-to-one completeness. We consider the same questions for the effectivization of the Wadge relation. We also show that for the space of sets of natural numbers endowed with the Scott topology, in each class of the Hausdorff difference hierarchy there are two strictly increasing chains of Wadge degrees of sets properly in that class. The length of these chains is the rank of the considered class, and each element in one chain is incomparable with all the elements in the other chain.

Although the language Lucid was not originally intended to support computing with infinite data structures, the notion of (infinite) sequence quickly came to the fore, together with a demand-driven computation model in which demands are propagated for the values of particular values at particular index points. This naturally generalized to sequences of multiple dimensions so that a programmer could, for example, write a program that could be understood as a (nonterminating) loop in which one of the loop variables is an infinite vector.

Programmers inevitably found use for more and more dimensions, which led to a problem which is fully solved for the first time in this paper. The problem is that the implementation's cache requires some estimate of the dimensions actually used to compute a value being fetched. This estimate can be difficult or (if dimensions are passed as parameters) impossible to obtain, and the demand-driven evaluation model for Lucid breaks down.

We outline the evolution of Lucid which gave rise to this problem, and outline the solution, as used for the implementation of TransLucid, the latest descendant of Lucid.

Ishihara's tricks have proven to be a highly useful tool in constructive mathematics, since they enable one to make decisions that seem, on first glance, impossible. They do, however, require that one deals with strongly extensional mappings on complete spaces. In this short note, we show how these assumptions can be weakened. Furthermore, we apply these generalizations to give a partial answer to the question, whether constructively we can rule out the existence of injections from Baire space into the natural numbers, to a version of Riemann's per mutation theorem and to a classification problem about cardinalities in constructive reverse mathematics.

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

In this paper, we continue the study of the geometry of Brownian motions which are encoded by Kolmogorov–Chaitin random reals (complex oscillations). We unfold Kolmogorov–Chaitin complexity in the context of Brownian motion and specifically to phenomena emerging from the random geometric patterns generated by a Brownian motion.

The first part of the paper presents a generalization of the well-known Baire category theorem. The generalization consists in replacing the dense open sets of the original formulation by dense UCO sets, where UCO means union of closed and open. This topological theorem is exactly what is needed to prove in the second part of the paper the locale-theoretic result that locales whose frame of opens has a countable presentation (countably many generators and countably many relations) are spatial. This spatiality theorem does not require choice.