Abstract We study the uniform computational content of Ramsey’s theorem using both Weihrauch reducibility and a new variant of Weihrauch reducibility, where the functions in the reduction are required to be total. In the latter setting, we show that the strength of Ramsey’s theorem varies significantly depending on how one represents its solutions, for example, using characteristic functions or using enumerations. Some of our results extend beyond variants of Ramsey’s theorem. In particular, we show that RT 2 2 $\mathsf {RT}^{2}_{2}$ sans serif upper R upper T 2 squared where solutions are represented using characteristic functions is not totally Weihrauch reducible to any computational problem whose solutions are represented using enumerations. Next, we study the computational problems RT ∞ n $\mathsf {RT}^{n}_{\infty }$ sans serif upper R upper T Subscript infinity Superscript n which take as input a colouring of n -tuples which has some infinite homogeneous set and asks for any such set. The problem RT ∞ 1 $\mathsf {RT}^{1}_{\infty }$ sans serif upper R upper T Subscript infinity Superscript 1 is fairly well-studied, as it is Weihrauch equivalent to the cluster point problem on N $\mathbb {N}$ double struck upper N . Our main result shows that RT ∞ 1 $\mathsf {RT}^{1}_{\infty }$ sans serif upper R upper T Subscript infinity Superscript 1 is not Weihrauch reducible to RT N 2 $\mathsf {RT}^{2}_{\mathbb {N}}$ sans serif upper R upper T Subscript double struck upper N Superscript 2 , strengthening a result of Soldà and Valenti. We also show that the jump of RT ∞ 1 $\mathsf {RT}^{1}_{\infty }$ sans serif upper R upper T Subscript infinity Superscript 1 is Weihrauch equivalent to SRT ∞ 2 $\mathsf {SRT}^{2}_{\infty }$ sans serif upper S upper R upper T Subscript infinity Superscript 2 .
Abstract SJT reducibility between sets A , B ⊆ N $A,B \subseteq \mathbb N$ upper A comma upper B subset of or equal to double struck upper N is defined by A ≤ S J T B $A \le _{SJT} B$ upper A less than or equals Subscript upper S upper J upper T Baseline upper B if for each computable function h that is unbounded and nondecreasing, there is an h -bounded uniformly B -c.e. trace ( T n ) n ∈ N $(T_n)_{n \in \mathbb N} $ left parenthesis upper T Subscript n Baseline right parenthesis Subscript n element of double struck upper N such that for each n , the value J A ( n ) $J^A(n)$ upper J Superscript upper A Baseline left parenthesis n right parenthesis of the jump is in T n $T_n$ upper T Subscript n , if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the K -trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.
Abstract We prove the consistency of the inequality $\mathfrak {r}_{\mathsf {nwd}}<\mathfrak {irr}$ , which in turn implies the consistency of $\mathfrak {r}_{\mathsf {nwd}}<\mathfrak {i}$ and $\mathfrak {r}_{\mathsf {scatt}}<\mathfrak {irr}$ . This answers one question from [1] and one question from [2]. We also prove the consistency of the inequality $\mathfrak {r}_{\mathbb {Q}}<\mathfrak {u}_{\mathbb {Q}}$ .
Abstract For $\lambda $ inaccessible, we show that “there is a nowhere trivial automorphism of $\mathcal P(\lambda )/[\lambda ]^{<\lambda }$ ” follows (in ZFC) from $2^\lambda =\lambda ^+$ , and is consistent (via forcing) with $2^\lambda>\lambda ^+$ .
We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as an interpreted vocabulary over an infinite domain. This formalism was denoted embedded finite model theory in the past. It is clear that the expressiveness and evaluating complexity of formulas of this type depend heavily on the infinite structure. If we embed in a wild structure like the integers with additive and multiplicative arithmetic, logic is extremely expressive and formulas are impossible to evaluate. On the other hand, for some well-known decidable structures, the expressiveness and evaluating complexity are similar to the situation without the additional infrastructure. The latter phenomenon was formalized via the notion of "Restricted Quantifier Collapse": adding quantification over the infinite structure does not add expressiveness. Beyond these two extremes little was known. In this work we show that the possibilities for expressiveness and complexity are much wider. We show that we can get almost any possible complexity of evaluation while staying within a decidable structure. We also show that in some decidable structures, there is a disconnect between expressiveness of the logic and complexity, in that we cannot eliminate quantification over the structure, but this is not due to an ability to embed complex relational computation in the logic. We show failure of collapse for the theory of finite fields and the related theory of pseudo-finite fields, which will involve coding computation in the logic. As a by-product of this, we establish the first lower bounds for the complexity of decision procedures for several decidable theories of fields, including the theory of finite fields. This article includes material in the extended abstract "Embedded Finite Models: Beyond Restricted Quantifier Collapse," that appeared in the Logic in Computer Science conference (LICS '23).
We continue the investigation of spaces whose topology is definable in the sense of descriptive set theory. We prove a general combinatorial principle we call the Definable Ideal Dichotomy. This principle is then applied to classify convergence in the class of definable countable groups and to prove other results of topological and set-theoretic nature. In the last section, we build limiting examples of interesting definable groups and spaces.
We prove the consistency of the inequality & rfr;(nwd) < i & rfr;& rfr;, which in turn implies the consistency of & rfr;(nwd) < i and & rfr;(scatt) < i & rfr;& rfr;. This answers one question from [1] and one question from [2]. We also prove the consistency of the inequality & rfr;(Q) < u(Q).
We prove a Thomason-style duality for the category of instantial neighbourhood frames and instantial neighbourhood morphisms, use it to define and investigate ultrafilter extensions, and show that it restricts to Thomason duality for Kripke frames.
mu-Abstract Elementary Classes are a model theoretic framework introduced in [4] to encompass classes axiomatized by L-infinity,L-infinity. We show that the framework extends beyond these logics by showing classes axiomatized in L(aa) with just the aa quantifier are an N-1-Abstract Elementary Class.
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We show how to extend Selivanov's fine hierarchy using descriptions of Borel Wadge classes. We give a game characterisation of containment between classes. We show that every class in the extended fine hierarchy has an admissible description, and use this to calculate heights in the hierarchy.
Hyperenumeration reducibility was first introduced by Sanchis [11]. The relationship between hyperenumeration and hyperarithmetic reducibility shares many parallels with the relationship between enumeration and Turing reducibility. We ask if this relationship can be pushed to prove and analog of Selman's Theorem for hyperenumeration reducibility. By studying e-pointed trees in Baire space we are able to get a counter example. An e-pointed tree T is a tree with no dead ends and the property that every path in T enumerates T. We prove that if T is an e-pointed tree then for all X if T is $\Pi <^>1_1$ in X then $\overline {T}$ is $\Pi <^>1_1$ in X. We build an e-pointed tree T such that $\overline {T}$ is not hyperenumeration reducible to T.
