Matthew de Brecht
We show some basic results on the characterization of quasi-Polish spaces in terms of spaces of ideals, with an emphasis on the connections with computable topology.
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Suppose $p$ is a computable real so that $p \geq 1$. We show that in both the real and complex case $l^p$ is computably categorical if and only if $p \neq 2$.
Notes on degrees of relative computable categoricity
本文研究可计算结构的相对可计算范畴性度，证明其可能度可由预言机0"界定（刚性结构为0'），且该估计是精确的。
A note on computable étale spaces
本文研究可计算Étale空间，将其与可计算函数联系起来，并探讨其在可计算范畴中的等价关系。
We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable topological spaces where the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable topological spaces with particular properties from wn--families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class.
We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that although $\{ω\cdot 2, ω^\star \cdot 2\}$ is computably embeddable in $\{ω^2, {(ω^2)}^\star\}$, the class $\{ω\cdot k,ω^\star \cdot k\}$ is \emph{not} computably embeddable in $\{ω^2, {(ω^2)}^\star\}$ for any natural number $k \geq 3$.
Computable Bases
本文研究可计算基底和预基底，推广Kreitz-Weihrauch定理，探讨拓扑空间的可计算描述方法，并建立预基底与T_0空间表示之间的伽罗瓦连接。
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we u…
This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.
We define a Polish topology inspired from the Gandy-Harrington topology and show how it can be used to prove Silver's dichotomy theorem while remaining in the Polish realm. In this topology, a $Π^1_1$ equivalence relation decomposes into a "sum" of a clopen relation and a meager one. We characterize it as the largest regular toplogy with a basis included in $Σ^1_1$.
Computably discrete represented spaces
研究可计算离散表示空间的结构，揭示其与可计算可枚举等价关系的对应关系，并通过构造多个示例展示其在可计算拓扑学中的分离作用。
This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are p…
We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives a…
The $NFI$-topology, introduced in [S0], is a topology on the Stone space of a theory $T$ that depends on a reduct $T^-$ of $T$. This topology has been used in [S0] to describe the set of universal transducers for $(T,T^-)$ (invariants sets that translates forking-open sets in $T^-$ to forking-open sets in $T$). In this paper we show that in contrast to the stable case, the $NFI$-topology need not be invariant over parameters in $T^-$ but a weak version of this holds for any simple $T$. We also note that for the lovely pair expansions, of theories with the \em wnfcp\em , the topology is invaria…
New definitions in the theory of Type 1 computable topological spaces
本文提出不依赖有效基的 Type 1 可计算拓扑空间一般定义，系统化形式包含关系，并首次在不依赖有效可分性的条件下定义可计算度量诱导的可计算拓扑，同时给出 Moschovakis 定理的新版本。
In this note we give a wellfoundedness proof of a computable notation system for first-order reflection.
A note on the degree structure of primitive recursive m-reducibility
本文研究可计算集在原始递归m-可归约性下的度上半格$C^{pr}_m$，证明其一阶理论是遗传不可判定的。
Computable analysis on the space of marked groups
该研究将有限生成群的判定问题与标记群空间的可计算分析结合，用Banach-Mazur可计算性给出子群不可识别的充要条件，分类群性质的有效Borel分层，还证明标记群空间是Polish但非可计算Polish的首个自然实例。
Effective bases and notions of effective second countability in computable analysis
本文研究表示空间中的可计算拓扑基，证明可计算可枚举基下不同基概念等价，确立稳健的有效第二可数性，并刻画可嵌入可计算度量空间的空间，同时探讨开放选择问题与可分性及第二可数性的关系。
In this note we axiomatize the classes of rudimentary functions, primitive recursive functions, safe recursive set functions, and predicatively computable functions.
We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, of the natural numbers, of Cantor space and of Baire space correspond to the class of computable functions, of functions computable with finitely many mind changes, of weakly computable functions and of effectively Bore…
What can Topology tell us about Logical Complexity?
论文通过拓扑结构揭示逻辑复杂性，引入游戏化Katětov序并证明其与LT序同构，探讨了组合与计算复杂性之间的关系。
An effective version of the Stone duality
本文研究拓扑T₀空间的计算理论方面，提出可计算的c-偏序集和拓扑空间的定义，并证明Stone型对偶性的有效版本，展示了可计算拓扑空间的性质。
We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on X (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of X, and we show that the resulting hierarchies of degrees are much more complicated than the classic…
In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.
Kawamura and Cook have developed a framework for studying the computability and complexity theoretic problems over "large" topological spaces. This framework has been applied to study the complexity of the differential operator and the complexity of functionals over the space of continuous functions on the unit interval $C[0,1]$. In this paper we apply the ideas of Kawamura and Cook to the product space of the real numbers endowed with the product topology. We show that no computable norm can be defined over such topology. We investigate computability and complexity of total functions over the…
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