We prove the correctness of the AKS algorithm [1] within the bounded arithmetic theory $T^{\text {count}}_2$ or, equivalently, the first-order consequences of the theory $\text {VTC}^0$ expanded by the smash function, which we denote by $\text {VTC}^0_2$. Our approach initially demonstrates the correctness within the theory $S^1_2 + \mathrm {iWPHP}$ augmented by two algebraic axioms and then shows that they are provable in $\text {VTC}^0_2$. The two axioms are: a generalized version of Fermat’s Little Theorem and an axiom adding a new function symbol which injectively maps roots of polynomials over a definable finite field to numbers bounded by the degree of the given polynomial. To obtain our main result, we also give new formalizations of parts of number theory and algebra:

• • In $\mathrm {PV}_1$: We formalize Legendre’s Formula on the prime factorization of $n!$, key properties of the Combinatorial Number System and the existence of cyclotomic polynomials over the finite fields $\mathbb {Z}/p$.

• • In $S^1_2$: We prove the inequality $\text {lcm}(1,\dots , 2n) \geq 2^n$.

• • In $\text {VTC}^0$: We verify the correctness of the Kung–Sieveking algorithm for polynomial division.

Nies and Scholz formalized the notion of an infinite qubitstring and referred to it as a ‘state’. They defined ‘quantum Martin-Löf randomness’ for states. We give a notion of measurement of a state in a computable basis and introduce ‘quantum measurement randomness’, a randomness notion for states. A state is quantum measurement random if measuring it in any computable basis yields a Martin-Löf random bitstring with probability one. Our main result is that quantum Martin-Löf randomness strictly implies quantum measurement randomness. This uses the construction of a quantum measurement random state which is not quantum Martin-Löf random. We prove two general results on which this construction relies: The first concerns Martin-Löf randomness relative to computable measures and extends a result of V. Vovk. The second is a combinatorial result about Kronecker products.

A computable topological presentation of a space is given by an effective list of a countable basis of non-empty open sets so that the intersection of the basic sets is uniformly effectively enumerable. We show that every countably-based $T_0$-space has a computable topological presentation, and that, conversely, every (formal) computable topological presentation represents some Polish space. In the compact case, we give a computable uniform list of computable topological presentations such that every compact Polish space is represented by exactly one presentation from the list. Note that none of these results assume that the Polish (or $T_0$) spaces are effective. Quite surprisingly, the effectively compact topological presentations turn out to be rather well behaved. Not only do such presentations allow one to construct a $\Delta ^0_2$ (complete) metric compatible with the topology, but also, under a mild extra condition, they can be turned into a computably compact Polish presentation of the space.

In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. We show that the subTuring degrees (i.e., the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. We also show that there is a nonzero join-irreducible subTuring degree (which implies that there is a realizability subtopos of the effective topos that cannot be the meet of two larger realizability subtoposes of the effective topos).

In The Emperor’s New Mind [7], Roger Penrose proves a variant of the halting problem, and uses it to argue that humans have cognitive capacities beyond the computable. In this short note, I explicate his argument, and show how it fails, via a corollary of his result. My response to Penrose is in fact of a kind with a number of prior responses: he assumes human powers, that (as the corollary shows) no computer could have. However, as far as I am aware, no one has previously addressed this specific form of the argument, in the direct way that I will.

We examine the degree spectra of relations on ${(\omega , <)}$. Given an additional relation R on ${(\omega ,<)}$, such as the successor relation, the degree spectrum of R is the set of Turing degrees of R in computable copies of ${(\omega ,<)}$. It is known that all degree spectra of relations on ${(\omega ,<)}$ fall into one of four categories: the computable degree, all of the c.e. degrees, all of the $\Delta ^0_2$ degrees, or intermediate between the c.e. degrees and the $\Delta ^0_2$ degrees. Examples of the first three degree spectra are easy to construct and well-known, but until recently it was open whether there is a relation with intermediate degree spectrum on a cone. Bazhenov, Kalociński, and Wroclawski constructed an example of an intermediate degree spectrum, but their example is unnatural in the sense that it is constructed by diagonalization and thus not canonical, that is, which relation you obtain from their construction depends on which Gödel encoding (and hence order of enumeration) of the partial computable functions/programs you choose. In this article, we use the “on-a-cone” paradigm to restrict our attention to “natural” relations R. Our main result is a construction of a natural relation on ${(\omega ,<)}$ which has intermediate degree spectrum. This relation has intermediate degree spectrum because of structural reasons.

This article studies the question of which classes of numberings can be generated by the direct sums of computable uniformly minimal sequences of numberings (in particular, Friedberg and positive numberings). For the class of Friedberg numberings, this question was initiated by Britta Schinzel in her 1982 paper. In this article, we show that the class of Gödel numberings is generated by the direct sums of computable uniformly positive sequences of universal numberings and that there exists a conull class of oracles computing sequences of Friedberg numberings with programmable direct sums. We further show that all computable numberings of a fairly wide class of families of total recursive functions (containing, for example, the family of all primitive recursion functions) are generated by the direct sums of computable sequences of their incomparable Friedberg numberings. On the other hand we prove that no family of partial recursive functions has a computable sequence of Friedberg numberings whose direct sum is acceptable.

In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for the appearance of non-computability. Namely, we show that additive sampling of orbits of certain skew products over expanding dynamics produces Turing non-computable reals. We apply this framework to Brjuno-type functions to demonstrate that they realize bijections between computable and lower-computable numbers, generalizing previous results of M. Braverman and the second author for the Yoccoz–Brjuno function to a wide class of examples, including Wilton’s functions and generalized Brjuno functions.

We explore the Weihrauch degree of the problems “find a bad sequence in a non-well quasi order” ($\mathsf {BS}$) and “find a descending sequence in an ill-founded linear order” ($\mathsf {DS}$). We prove that $\mathsf {DS}$ is strictly Weihrauch reducible to $\mathsf {BS}$, correcting our mistaken claim in [18]. This is done by separating their respective first-order parts. On the other hand, we show that $\mathsf {BS}$ and $\mathsf {DS}$ have the same finitary and deterministic parts, confirming that $\mathsf {BS}$ and $\mathsf {DS}$ have very similar uniform computational strength. We prove that König’s lemma $\mathsf {KL}$ and the problem $\mathsf {wList}_{{2^{\mathbb {N}}},\leq \omega }$ of enumerating a given non-empty countable closed subset of ${2^{\mathbb {N}}}$ are not Weihrauch reducible to $\mathsf {DS}$ or $\mathsf {BS}$, resolving two main open questions raised in [18]. We also answer the question, raised in [12], on the existence of a “parallel quotient” operator, and study the behavior of $\mathsf {BS}$ and $\mathsf {DS}$ under the quotient with some known problems.

By a celebrated result of Kučera and Slaman [5], the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye [1] strengthened this result by showing that, for all left-c.e. reals $\alpha $ and $\beta $ such that $\beta $ is Martin-Löf random and all left-c.e. approximations $a_0,a_1,\dots $ and $b_0,b_1,\dots $ of $\alpha $ and $\beta $, respectively, the limit

$$ \begin{align*} \lim\limits_{n\to\infty}\frac{\alpha - a_n}{\beta - b_n} \end{align*} $$

$\alpha $

$\beta $

Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.

We consider the problem of predicting the next bit in an infinite binary sequence sampled from the Cantor space with an unknown computable measure. We propose a new theoretical framework to investigate the properties of good computable predictions, focusing on such predictions’ convergence rate.

Since no computable prediction can be the best, we first define a better prediction as one that dominates the other measure. We then prove that this is equivalent to the condition that the sum of the KL divergence errors of its predictions is smaller than that of the other prediction for more computable measures. We call that such a computable prediction is more general than the other.

We further show that the sum of any sufficiently general prediction errors is a finite left-c.e. Martin-Löf random real. This means the errors converge to zero more slowly than any computable function.

A recursive set of formulas of first-order logic with finitely many predicate letters, including “=”, has a model over the integers in which the predicates are Boolean combinations of recursively enumerable sets, if it has an infinite model at all. The proof corrects a fallacious argument published by Hensel and Putnam in 1969.

Existentially closed groups are, informally, groups that contain solutions to every consistent finite system of equations and inequations. They were introduced in 1951 in an algebraic context and subsequent research elucidated deep connections with group theory and computability theory. We continue this investigation, with particular emphasis on illuminating the relationship with computability theory.

In particular, we show that there are existentially closed groups computable in the halting problem, and that this is optimal. Moreover, using the work of Martin Ziegler in computable group theory, we show that the previous result relativises in the enumeration degrees. We then tease apart the complexity contributed by “global” and “local” structure, showing that the complexity of finitely generated subgroups of existentially closed groups is captured by the PA degrees. Finally, we investigate the computability-theoretic complexity of omitting the non-principal quantifier-free types from a list of types, from which we obtain an upper bound on the complexity of building two existentially closed groups that are “as different as possible”.

A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal {L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal {L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e. A is low$_2$ then A has an atomless hyperhypersimple superset. In fact, if A is c.e. and low$_2$, then for any $\Sigma _3$-Boolean algebra B there is some c.e. $H\supseteq A$ such that $\mathcal {L}^*(H)\cong B$.

Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ that characterizes it among all countable structures. We can measure the complexity of a structure by the least complexity of a Scott sentence for that structure. It is known that there can be a difference between the least complexity of a Scott sentence and the least complexity of a computable Scott sentence; for example, Alvir, Knight, and McCoy showed that there is a computable structure with a $\Pi _2$ Scott sentence but no computable $\Pi _2$ Scott sentence. It is well known that a structure with a $\Pi _2$ Scott sentence must have a computable $\Pi _4$ Scott sentence. We show that this is best possible: there is a computable structure with a $\Pi _2$ Scott sentence but no computable $\Sigma _4$ Scott sentence. We also show that there is no reasonable characterization of the computable structures with a computable $\Pi _n$ Scott sentence by showing that the index set of such structures is $\Pi ^1_1$-m-complete.

The study of the word problems of groups dates back to Dehn in 1911, and has been a central topic of study in both group theory and computability theory. As most naturally occurring presentations of groups are recursive, their word problems can be thought of as a computably enumerable equivalence relation (ceer). In this article, we study the word problem of groups in the framework of ceer degrees, introducing a new metric with which to study word problems. This metric is more refined than the classical context of Turing degrees.

Classically, every Turing degree is realized as the word problem of some c.e. group, but this is not true for ceer degrees. This motivates us to look at the classical constructions and show that there is a group whose word problem is not universal, but becomes universal after taking any nontrivial free product, which we call $*$-universal. This shows that existing constructions of the Higman embedding theorem do not preserve ceer degrees. We also study the index set of various classes of groups defined by their properties as a ceer: groups whose word problems are dark (equivalently, algorithmically finite as defined by Miasnikov and Osin), universal, and $*$-universal groups.

We investigate the primitive recursive content of linear orders. We prove that the punctual degrees of rigid linear orders, the order of the integers $\mathbb {Z}$, and the order of the rationals $\mathbb {Q}$ embed the diamond (preserving supremum and infimum). In the cases of rigid orders and the order $\mathbb {Z}$, we further extend the result to embed the atomless Boolean algebra; we leave the case of $\mathbb {Q}$ as an open problem. We also show that our results for the rigid orders, in fact, work for orders having a computable infinite invariant rigid sub-order.

The family of finite subsets s of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $\omega $-large sets in Logic. We formulate and prove the generalizations of Friedman’s Free Set and Thin Set theorems and of Rainbow Ramsey’s theorem to colorings of the Schreier barrier. We analyze the strength of these theorems from the point of view of Computability Theory and Reverse Mathematics. Surprisingly, the exactly $\omega $-large counterparts of the Thin Set and Free Set theorems can code $\emptyset ^{(\omega )}$, while the exactly $\omega $-large Rainbow Ramsey theorem does not code the halting set.

We study the computational complexity of converting between different representations of irrational numbers. Typical examples of representations are Cauchy sequences, base-10 expansions, Dedekind cuts and continued fractions.

In this article, we give characterizations of Towsner’s relative leftmost path principles in terms of omega-model reflections of transfinite inductions. In particular, we show that the omega-model reflection of $\Pi ^1_{n+1}$ transfinite induction is equivalent to the $\Sigma ^0_n$ relative leftmost path principle over $\mathsf {RCA}_0$ for $n> 1$. As a consequence, we have that $\Sigma ^0_{n+1}\mathsf {LPP}$ is strictly stronger than $\Sigma ^0_{n}\mathsf {LPP}$.