arXiv subjects
Publications and source records attributed to Steffen Lempp.
The Ziegler degrees were introduced to characterize definability in group theory. In [JLSta], the authors show that the first order-theory of the Ziegler degrees (as a partial order) is undecidable. We improve this result, showing that the theory of the Ziegler degrees is bi-interpretable with true second-order arithmetic.
We study the existence and the distribution of "long" chains in the Weihrauch degrees, mostly focusing on chains with uncountable cofinality. We characterize when such chains have an upper bound and prove that there are no cofinal chains (of any order type) in the Weihrauch degrees. Furthermore, we show that the existence of coinitial sequences of non-zero degrees is equivalent to $\mathrm{CH}$. Finally, we explore the extendibility of antichains, providing some necessary conditions for maximality.
We prove that every finite distributive lattice is isomorphic to a final segment of the d.c.e. Turing degrees (i.e., the degrees of differences of computably enumerable sets). As a corollary, we are able to infer the undecidability of the EAE-theory of the d.c.e. degrees in the language of partial ordering.
A partial order $(P,\le)$ admits a jump operator if there is a map $j\colon P \to P$ 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 $f\equiv_{\mathrm{W}} f'$. 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.
We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\pmb\Pi_\omega^0$-complete set of models. In particular, any sequential theory (a class of foundational theories isolated by Pudl\'ak) has a $\pmb\Pi_\omega^0$-complete set of models. We also give sharp conditions for theories to have a $\pmb\Pi^0_n$-complete set of models.
In this paper, we study the existence of minimal covers and strong minimal covers in the Weihrauch degrees. We characterize when a problem $f$ is a minimal cover or strong minimal cover of a problem $h$. We show that strong minimal covers only exist in the cone below $\mathsf{id}$ and that the Weihrauch lattice above $\mathsf{id}$ is dense. From this, we conclude that the degree of $\mathsf{id}$ is first-order definable in the Weihrauch degrees and that the first-order theory of the Weihrauch degrees is computably isomorphic to third-order arithmetic.
The tower number $\mathfrak t$ and the ultrafilter number $\mathfrak u$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of~$\omega$ and the almost inclusion relation $\subseteq^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory. We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin's characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e.\ set computes a maximal tower but no 1-generic $\Delta^0_2$-set does so. We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively.
We show that the order dimension of the partial order of all finite subsets of $κ$ under set inclusion is ${\log}_{2}({\log}_{2}(κ))$ whenever $κ$ is an infinite cardinal. We also show that the order dimension of any locally countable partial ordering $(P, <)$ of size $κ^+$, for any $κ$ of uncountable cofinality, is at most $κ$. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum.
We show that there is a strong minimal pair in the computably enumerable Turing degrees.
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truth-table degrees do not meet to the degree~0, even within the c.e. truth-table degrees, but when taking the meet over all such truth-table degrees, the infinite meet is indeed~0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truth-table degrees form a minimal pair.
We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti by showing that every computable stable $2$-coloring of pairs has an incomplete $Δ^0_2$ infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT$^2_2$ does not imply RT$^2_2$.
In Reverse Mathematics, the axiom system DNR, asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL$_0$ (weak weak König's Lemma).
We classify the computability-theoretic complexity of two index sets of classes of first-order theories: We show that the property of being an $\aleph_0$-categorical theory is $Π^0_3$-complete; and the property of being an Ehrenfeucht theory $Π^1_1$-complete. We also show that the property of having continuum many models is $Σ^1_1$-hard. Finally, as a corollary, we note that the properties of having only decidable models, and of having only computable models, are both $Π^1_1$-complete.
Recently, connections have been explored between the complexity of finite problems in graph theory and the complexity of their infinite counterparts. As is shown in our paper (and in independent work of Tirza Hirst and D. Harel from a different angle) there is no firm connection between these complexities, namely finite problems of equal complexity can have radically different complexity for the infinite versions and vice versa. Furthermore, the complexity of an infinite counterpart can depend heavily on precisely how the finite problem is rephrased in the infinite case. The finite problems we address include colorability of graphs and existence of subgraph isomorphisms. In particular, we give three infinite versions of the 3-colorability problem that vary considerably in their recursion theoretic and proof theoretic complexity. Additionally, we show that three subgraph isomorphism problems of varying finite complexity have infinite versions with identical recursion theoretic and proof theoretic content.