arXiv subjects

Linus Richter

Publications and source records attributed to Linus Richter.

4 recordsLinked to original sources

arXiv2026

Chains and Antichains inside Many-One Degrees and Variants

The relations between many-one degrees and one-one degrees have been studied since the beginning of recursion theory; early results from the 1960s include that many-one degrees always have a largest one-one degree and either that one-one degree is the only one-one degree inside the many-one degree or every countable linear order is noneffectively embeddable into the structure of one-one degrees inside the given many-one degree. Furthermore, the greatest recursive many-one degree is a special case, as it allows to embed ascending infinite chains but not descending infinite chains, all other many-one degrees fall into the two cases mentioned above. It remained open whether infinite antichains can always be embedded when the many-one degree is nonrecursive and nonirreducible; Odifreddi stated in a survey 1981 and in his book Classical Recursion Theory in the year 1989 this question explicitly as an open problem. D\"egtev had already in 1976 constructed antichains of one-one degrees inside all nonrecursive and nonirreducible recursively enumerable many-one degrees and Batyrshin generalised the result to all nonrecursive and nonirreducible limit-recursive many-one degrees. In 2026, Cintioli showed that there is a measure $1$ class of sets whose many-one degrees contain infinite antichains of one-one degrees. This class contains all rigid many-one degrees. The present work generalises Batyrshin's result to all nonrecursive and nonirreducible many-one degrees and solves therefore Odifreddi's open problem. The present work also proposes to deepen the study of reducibilities between one-one and many-one in recursion theory in order to get a more complete and detailed picture for the structures inside many-one degrees. In particular it studies finite-one and bounded finite-one reducibilities where the first was introduced by Maslova in the 1970ies.

math.LO↗

arXiv2025

Languages of Words of Low Automatic Complexity Are Hard to Compute

The automatic complexity of a finite word (string) is an analogue for finite automata of Sipser's distinguishing complexity (1983) and was introduced by Shallit and Wang (2001). For a finite alphabet $\Sigma$ of at least two elements, we consider the non-deterministic automatic complexity given by exactly - yet not necessarily uniquely - accepting automata: a word $x \in \Sigma^*$ has exact non-deterministic automatic complexity $k \in \mathbb{N}$ if there exists a non-deterministic automaton of $k$ states which accepts $x$ while rejecting every other word of the same length as $x$, and no automaton of fewer states has this property. Importantly, and in contrast to the classical notion, the witnessing automaton may have multiple paths of computation accepting $x$. We denote this measure of complexity by $A_{Ne}$, and study a class of languages of low $A_{Ne}$-complexity defined as $L_q = \{ \, x \in \Sigma^* : A_{Ne}(x) < q|x| \, \}$, which is parameterised by rationals $q \in (0,1/2)$ (generalising a class of sets first studied by Kjos-Hanssen). We show that for every $q \in (0,1/2)$, this class is neither context-free nor recognisable by certain Boolean circuits. In the process, we answer an open question of Kjos-Hanssen quantifying the complexity of $L_{1/3}$ in terms of Boolean circuits, and also prove the Shannon effect for $A_{Ne}$.

cs.FL↗

arXiv2025

All Borel Group Extensions of Finite-Dimensional Real Space Are Trivial

For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability of $H$ by investigating its complexity as a Borel set. We do this by combining homological algebra and descriptive set theory, and hence study the Borel complexity of those functions inducing $H$, the abelian cocycles. We prove that, for every $N \geq 2$, there are no non-trivial Borel definable abelian cocycles coding group extensions of $(\mathbb{R}^N,+)$ by a countable abelian group $G$, and hence show that no non-trivial such group extensions exist. This completes the picture first investigated by Kanovei and Reeken in 2000, who proved the case $N = 1$, and whose techniques we adapt in this work.

math.LO↗

arXiv2023

Co-analytic Counterexamples to Marstrand's Projection Theorem

Assuming $V=L$, we construct a plane set $E$ of Hausdorff dimension $1$ whose every orthogonal projection onto straight lines through the origin has Hausdorff dimension $0$. This is a counterexample to J. M. Marstrand's seminal projection theorem. While counterexamples had already been constructed decades ago, initially by R. O. Davies, the novelty of our result lies in the fact that $E$ is co-analytic. Following Marstrand's original proof (and R. Kaufman's newer, and now standard, approach based on capacities), a counterexample to the projection theorem cannot be analytic, hence our counterexample is optimal. We then extend the result in a strong way: we show that for each $\epsilon \in (0, 1)$ there exists a co-analytic set $E_{\epsilon}$ of dimension $1 + \epsilon$, each of whose orthogonal projections onto straight lines through the origin has Hausdorff dimension $\epsilon$. The constructions of $E$ and $E_{\epsilon}$ are by induction on the countable ordinals, applying a theorem by Z. Vidnyanszky.

math.LO↗