We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\operatorname {\mathrm {\mathbf {ER}}}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$ needn’t be a strong minimal cover of a self-full degree $\mathbf {d}$. Finally, we show that the theory of the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic.

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$ . We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from  $[0,1]$ to ${\mathbb N}$ . Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$ . Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number.

Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR $_{0}$ (and so $\Pi _{1}^{1}$ -CA $_{0}$ or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity.

This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( $\Pi _{1}^{1}$ , r- $\Pi _{1}^{1}$ , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.

It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes.

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-minimal structures on $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot ,\mathbb {N})$ . Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$ . We also show that any given expansion of $(\mathbb {R}, <, +,\mathbb {N})$ by subsets of $\mathbb {N}^n$ (n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.

Descriptive set theory and computability theory are closely-related fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. In particular, we examine the perfect set property, determinacy, the Baire property, and Lebesgue measurability. We demonstrate that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals; we also show that the Baire property and Lebesgue measurability are both equivalent in complexity to the property of simply being a Borel set, for $\boldsymbol {\Sigma ^{1}_{2}}$ sets of reals.

Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$. On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.

An oracle A is low-for-speed if it is unable to speed up the computation of a set which is already computable: if a decidable language can be decided in time $t(n)$ using A as an oracle, then it can be decided without an oracle in time $p(t(n))$ for some polynomial p. The existence of a set which is low-for-speed was first shown by Bayer and Slaman who constructed a non-computable computably enumerable set which is low-for-speed. In this paper we answer a question previously raised by Bienvenu and Downey, who asked whether there is a minimal degree which is low-for-speed. The standard method of constructing a set of minimal degree via forcing is incompatible with making the set low-for-speed; but we are able to use an interesting new combination of forcing and full approximation to construct a set which is both of minimal degree and low-for-speed.

We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$ . Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$ , we give general conditions sufficient to eliminate parameters from interpretations.

Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees $\langle \mathcal {R}_{\mathrm {T}},\leq _{\mathrm {T}}\rangle $ , we do not in general know how to characterize the degrees $\mathbf {d}\in \mathcal {R}_{\mathrm {T}}$ below which L can be bounded. The important characterizations known are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf {d}$ if and only if $\mathbf {d}$ contains sets of “fickleness” $>\omega $ and $\geq \omega ^\omega $ respectively. We work towards finding a lattice that characterizes the levels above $\omega ^2$ , the first non-trivial level after $\omega $ . We introduced a lattice-theoretic property called “ $3$ -directness” to describe lattices that are no “wider” or “taller” than $L_7$ and $M_3$ . We exhaust the 3-direct lattices L, but they turn out to also characterize the $>\omega $ or $\geq \omega ^\omega $ levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides $M_3$ that also characterize the $\geq \omega ^\omega $ -levels. Our search for a $>\omega ^2$ -candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four $\geq \omega ^\omega $ -lattices as sublattices.

Abstract prepared by Liling Ko.

E-mail: ko.390@osu.edu

URL: http://sites.nd.edu/liling-ko/

The thin set theorem for n-tuples and k colors ( $\operatorname {\mathrm {\sf {TS}}}^n_k$ ) states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$ , nor the free set theorem ( $\operatorname {\mathrm {\sf {FS}}}^n$ ) imply the Erdős–Moser theorem ( $\operatorname {\mathrm {\sf {EM}}}$ ) whenever k is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $\mathsf {P}$ , a computable instance of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance (answering a question of Patey).

A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

For a ring R, Hilbert’s Tenth Problem $HTP(R)$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP(\mathbb {Z}[W^{-1}])$ , which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$ . It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP(R_W)$ . In contrast, we show that every Turing degree contains a set W for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP(\mathbb {Z}[W^{-1}])$ can succeed uniformly on a set of measure $1$ , and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$ .

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.

We ask when, for a pair of structures $\mathcal {A}_1,\mathcal {A}_2$ , there is a uniform effective procedure that, given copies of the two structures, unlabeled, always produces a copy of $\mathcal {A}_1$ . We give some conditions guaranteeing that there is such a procedure. The conditions might suggest that for the pair of orderings $\mathcal {A}_1$ of type $\omega _1^{CK}$ and $\mathcal {A}_2$ of Harrison type, there should not be any such procedure, but, in fact, there is one. We construct an example for which there is no such procedure. The construction involves forcing. On the way to constructing our example, we prove a general result on modifying Cohen generics.

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by these relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a $\boldsymbol {\Sigma }^1_1$ complete equivalence relation. We then investigate the algorithmic properties of this reduction. We obtain that elementary bi-embeddability on the class of computable graphs is $\Sigma ^1_1$ complete with respect to computable reducibility and show that the elementary bi-embeddability and bi-embeddability spectra realized by graphs are related.

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including, for example, difference fields).

We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

$\eta $ -representations are a way of coding sets in computable linear orders that were first introduced by Fellner in his thesis. Limitwise monotonic functions have been used to characterize the sets with $\eta $ -representations, and give characterizations for several variations of $\eta $ -representations. The one exception is the class of sets with strong $\eta $ -representations, the only class where the order type of the representation is unique.

We introduce the notion of a connected approximation of a set, a variation on $\Sigma ^0_2$ approximations. We use connected approximations to give a characterization of the many-one degrees of sets with strong $\eta $ -representations as well new characterizations of the variations of $\eta $ -representations with known characterizations.