Computability theory is a discipline in the intersection of computer science and mathematical logic where the fundamental question is: Given two mathematical objects X and Y, does X compute Y in principle? In case X and Y are real numbers, Turing’s famous ‘machine’ model provides the standard interpretation of ‘computation’ for this question. To formalise computation involving (total) abstract objects, Kleene introduced his S1–S9 computation schemes. In turn, Dag Normann and the author have introduced a version of the lambda calculus involving fixed point operators that exactly captures S1–S9 and accommodates partial objects. In this paper, we use this new model to develop the computability theory of various well-known theorems due to Baire and Volterra and related results; these theorems only require basic mathematical notions such as continuity, open sets, and density. We show that these theorems due to Baire and Volterra are computationally equivalent from the point of view of our new model, sometimes working in rather tame fragments of Gödel’s T .

We show that natural structures related to the so called homomorphism preorder (or h -preorder) on the iterated labeled forests have isomorphic copies computable in polynomial time. Moreover, the polynomials in the upper bounds are of low degree which makes the computational content of the whole theory feasible. We discuss possible applications of these results to relevant questions of automata and computability theory.

We show that, in general, there is no degree corresponding to the composition of two problems in the computable reducibility lattice. We show the same in the strong computable reducibility setting.

In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems that present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms. These systems are interesting because, when they exhibit a unique evolution, a transfinite method always exists to define such evolution as a limit of a sequence of continuous functions. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of real functions with bounded, solvable derivatives and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris–Woodin, Denjoy, and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.

We study immunity properties of the transversals of computably enumerable equivalence relations (or, briefly, ceers ), where a transversal is a set which picks at most one element from every equivalence class of the given equivalence relation. Among transversals, a particular role is played by the principal transversal , whose members are the least elements of the various equivalence classes. While hyperimmunity of the principal transversal implies hyperimmunity of every infinite transversal, we show that this fails both for immunity and hyperhyperimmunity. In both cases, counterexamples are taken from the class of interval ceers , that is, ceers whose equivalence classes are either singletons or intervals of maximal length consisting of consecutive elements of some given c.e. set. We also look into the class of hyperdark ceers, that is, those ceers with infinitely many classes, whose infinite transversals are all hyperimmune, analyzing how this property relates to other computability theoretic properties of the infinite transversals. We make some preliminary observations on the hyperhyperdark ceers, that is, those ceers with infinitely many classes, whose infinite transversals are all hyperhyperimmune.

A partial order ( P , ⩽ ) admits a jump operator if there is a map j : P → P that is strictly increasing and weakly monotone. Despite its name, the jump in the Weihrauch lattice fails to satisfy both of these properties: it is not degree-theoretic, and there are functions f such that f ≡ W f ′ . This raises the question: Is there a jump operator in the Weihrauch lattice? We answer this question positively and provide an explicit definition for an operator on partial multi-valued functions that, when lifted to the Weihrauch degrees, induces a jump operator. This new operator, called the totalizing jump , can be characterized in terms of the total continuation, a well-known operator on computational problems. The totalizing jump induces an injective endomorphism of the Weihrauch degrees. We study some algebraic properties of the totalizing jump and characterize its behavior on some pivotal problems in the Weihrauch lattice.

The infinite pigeonhole theorem asserts that if f : N → m is a function with a finite range, then there is a j < m such that the set { n ∈ N ∣ f ( n ) = j } is infinite. This article uses the techniques of reverse mathematics and Weihrauch analysis to examine the strength of a theorem that finds all the values that occur infinitely often in the range of a function.

The Erdős, Grünwald, and Weiszfeld theorem is a characterization of those infinite graphs which are Eulerian. That is, infinite graphs that admit infinite Eulerian paths. In this article, we prove an effective version of the Erdős, Grünwald, and Weiszfeld theorem for a class of graphs where vertices of infinite degree are allowed, generalizing a theorem of D. Bean. Our results are obtained from a characterization of those finite paths in a graph that can be extended to infinite Eulerian paths.

Topological models are sometimes used to prove independence results in constructive mathematics. Here we show that some of the topologies that have been used are necessary for those results.