Andre Nies
The 2015 Logic Blog contains a large variety of results connected to logic, some of them unlikely to be submitted to a journal. For the first time there is a group theory part. There are results in higher randomness, and in computable ergodic theory.
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This year's logic blog has focussed on: 1. Demuth randomness 2. traceability 3. The connection of computable analysis and randomness 4. $K$-triviality in metric spaces.
The 2012 logic blog has focussed on the following: Randomness and computable analysis/ergodic theory; Systematizing algorithmic randomness notions; Traceability; Higher randomness; Calibrating the complexity of equivalence relations from computability theory and algebra.
Logic Blog 2023-2024
该文合并2023至2024年的逻辑博客，汇集特定群类与逻辑关系、遍历理论、数学基础及AI定理证明等条目。
The blog is somewhat shorter than in previous years, It contains new insights in a variety of areas, including computability, quantum algorithmic version of the SMB theorem, descriptions of groups (both discrete and profinite), metric spaces. There are also lots of open questions in computability, randomness, and computable model theory.
Some notions from algorithmic randomness are extended to measures and to quantum states. There is a lot on group theory and its relation to logic. This includes some new results on oligomorphic groups. There's also metric spaces and Scott rank, and interpretability.
The 2014 Logic Blog starts with open questions from the May IMS program in Singapore. It contains results on randomness, including answers to some open questions in higher randomness. There are structural results on equivalence relations, and metric spaces. The last 20 pages contain a tutorial [arXiv:1407.4259] by Bienvenu and Shen on the coincidence of K-trivial and low for K, and related results. They formulate the golden run in game theoretic language.
This year's logic blog contains a variety of results, some of them available only here. Highlights include the resolution of the Gamma question by Monin, and a number of entries on topological group theory and its connection to logic. There's also a new proof by Yu of Harrington's 1978 result that a Pi-11 set of hyperdegrees with a perfect subset contains an upper cone. Also there's quite a bit of reverse mathematics, including its connection to randomness, and variants of Hindman's theorem in this setting.
This year's blog has focused on the connections of group theory with logic and algorithms. The first post is on automata presentable groups. Then there are several posts related to topological groups, for instance Ivanov and Majcher showing that extreme amenability of closed subgroups of $S_\infty$ is a Borel property. One post due to Harrison-Trainor and Nies reviews notes by Segal on pseudofinite groups, and attempts an effective version. About 25 percent is on computability and randomness, in particular equivalence of reducibilities weaker than Turing on the K-trivials by Greenberg, Nies an…
The blog has several entries on group theory interacting with computability and wider logic, several open questions, and an entry on undecidability in physics.
The blog focusses on algorithmic randomness and its connections to quantum information theory, group theory and its connections to logic, and computability analogs of cardinal characteristics.
The 2013 logic blog has focussed on the following: 1. Higher randomness. Among others, the Borel complexity of $Π^1_1$ randomness and higher weak 2 randomness is determined. 2. Reverse mathematics and its relationship to randomness. For instance, what is the strength of Jordan's theorem in analysis? (His theorem states that each function of bounded variation is the difference of two nondecreasing functions.) 3. Randomness and computable analysis. This focusses on the connection of randomness of a real $z$ and Lebesgue density of effectively closed sets at $z$. 4. Exploring similarity relations…
The 2022 logic blog has concentrated on the connections of group theory and logic. It discusses Gardam's 2021 refutation of the Higman/ Kaplansky unit conjecture, and its connections to logic and to computation. The rest is about topological groups of various kinds, in particular a computational theory of tdlc groups, and a duality between locally Roelcke precompact groups and certain countable structures called meet groupoids.
It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra X+ over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q+, as well as the logic of a subalgebra of C+. This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R+. In th…
This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We show they are elementary structures axiomatised by Horn sentences and we use our completeness result to state a version of algebraisability for inquisitive and dependence logics. We then foc…
There are two known general results on the finite model property (fmp) of commutators [L,L'] (bimodal logics with commuting and confluent modalities). If L is finitely axiomatisable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so have the fmp. On the negative side, if both L and L' are determined by transitive frames and have frames of arbitrarily large depth, then [L,L'] does not have the fmp. In this paper we show that commutators with a `weakly connected' component often lack the…
In [Mon11] are studied, for the axiomatic extensions of the monoidal t-norm based logic ([EG01]), the properties of single chain completeness. On the other side, in [GJKO07, Chapter 5] are studied many logical and algebraic properties (like Halldén completeness, variable separation properties, amalgamation property etc.), in the context of substructural logics. The aim of this paper is twofold: first of all we will specialize the properties studied in [GJKO07, Chapter 5] from the case of substructural logics to the one of extensions of MTL, by obtaining some general characterization. Moreover …
We present the logic BL_Chang, an axiomatic extension of BL (see P. Hájek - Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BL_Chang-algebras will be strictly connected to the one generated by Chang's MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new results concerning these las…
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