Annals of Pure and Applied Logic最新文献

英文中文

From GTC to : Generating reset proof systems from cyclic proof systems 从 GTC 到：从循环证明系统生成重置证明系统

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-06-11 DOI: 10.1016/j.apal.2024.103485

Graham E. Leigh, Dominik Wehr

We consider cyclic proof systems in which derivations are graphs rather than trees. Such systems typically come with a condition that isolates which derivations are admitted as proofs, known as the soundness condition. This soundness condition frequently takes the form of either a global trace condition, a property dependent on all infinite paths in the proof-graph, or a reset condition, a ‘local’ condition depending on the simple cycles only which, as a result, is typically stable under more proof transformations.

In this article we present a general method for constructing cyclic proof systems with reset conditions from systems with global trace conditions. In contrast to previous approaches, this method of generation is entirely independent of logic's semantics, only relying on combinatorial aspects of the notion of ‘trace’ and ‘progress’. We apply this method to present reset proof systems for three cyclic proof systems from the literature: cyclic arithmetic, cyclic Gödel's T and cyclic tableaux for the modal μ-calculus.

我们考虑的循环证明系统中，导数是图而不是树。这种系统通常附带一个条件，即健全性条件，用来区分哪些推导可以作为证明。这种健全性条件通常有两种形式：一种是全局踪迹条件（一种是依赖于证明图中所有无限路径的属性），另一种是重置条件（一种是仅依赖于简单循环的 "局部 "条件）。与以往的方法不同，这种生成方法完全独立于逻辑语义，只依赖于 "轨迹 "和 "进展 "概念的组合方面。我们运用这种方法为文献中的三个循环证明系统提出了重置证明系统：循环算术、循环哥德尔T和模态μ微积分的循环表法。

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引用次数: 0

Posets of copies of countable ultrahomogeneous tournaments 可数超均质锦标赛副本的 Posets

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2024-06-06 DOI: 10.1016/j.apal.2024.103486

Miloš S. Kurilić , Stevo Todorčević

The poset of copies of a relational structure $X$ is the partial order $P (X) : = 〈 {Y \subset X : Y ≅ X}, \subset 〉$ and each similarity of such posets (e.g. isomorphism, forcing equivalence = isomorphism of Boolean completions, $B_{X} : = ro sq P (X)$ ) determines a classification of structures. Here we consider the structures from Lachlan's list of countable ultrahomogeneous tournaments: $Q$ (the rational line), $S (2)$ (the circular tournament), and $T^{\infty}$ (the countable homogeneous universal tournament); as well as the ultrahomogeneous digraphs $S (3)$ , $Q [I_{n}]$ , $S (2) [I_{n}]$ and $T^{\infty} [I_{n}]$ from Cherlin's list.

If $G_{Rado}$ (resp. $Q_{n}$ ) denotes the countable homogeneous universal graph (resp. n-labeled linear order), it turns out that $P (T^{\infty}) ≅ P (G_{Rado})$ and that $P (Q_{n})$ densely embeds in $P (S (n))$ , for $n \in {2, 3}$ .

Consequently, $B_{X} ≅ ro (S ⁎ π)$ , where $S$ is the poset of perfect subsets of $R$ and π an $S$ -name such that $1_{S} 关系结构 X 副本的正集是偏序 P(X):=〈{Y\subsetX:Y≅X},\subset〉，这种正集的每一个相似性（例如同构、强制等价 = 布尔完成的同构，BX:=rosqP(X)）决定了结构的一个分类。在此，我们考虑拉克兰的可数超同调锦标赛列表中的结构：Q（有理线）、S(2)（循环锦标赛）和 T\infty（可数同质通用锦标赛）；以及谢林列表中的超同质数图 S(3)、Q[In]、S(2)[In]和 T\infty[In]。如果 GRado（或 Qn）表示可数同素万能图（或 n 标记线性阶），那么对于 n\in{2,3}，P(T\infty)≅P(GRado)和 P(Qn)密集嵌入 P(S(n))。因此，BX≅ro(S⁎π)，其中 S 是 R 的完全子集的正集，π 是一个 S 名，使得 1S⊩"π 是一个分离式、只要 X 是与 Q、Qn、S(2)、S(3)、Q[In] 或 S(2)[In]等价的可数结构，CH 下的 1S⊩"π\equivforc(P(ω)/Fin)+"（因此 1S⊩"π\equivforc(P(ω)/Fin)+"）。另外，BX≅ro(S⁎π)，其中 1S⊩"π是ω-分布强迫"，只要 X 是包含 GRado 副本的可数图，或包含 T\infty 副本的可数锦标赛，或 X=T\infty[In]。求助PDF {"title":"Posets of copies of countable ultrahomogeneous tournaments","authors":"Miloš S. Kurilić , Stevo Todorčević","doi":"10.1016/j.apal.2024.103486","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103486","url":null,"abstract":"<div>The poset of copies of a relational structure <math><mi>X</mi></math> is the partial order <math><mi>P</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>〈</mo><mo>{</mo><mi>Y</mi><mo>\subset</mo><mi>X</mi><mo>:</mo><mi>Y</mi><mo>≅</mo><mi>X</mi><mo>}</mo><mo>,</mo><mo>\subset</mo><mo>〉</mo></math> and each similarity of such posets (e.g. isomorphism, forcing equivalence = isomorphism of Boolean completions, <math><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>:</mo><mo>=</mo><mrow><mi>ro</mi></mrow><mspace></mspace><mrow><mi>sq</mi></mrow><mspace></mspace><mi>P</mi><mo>(</mo><mi>X</mi><mo>)</mo></math>) determines a classification of structures. Here we consider the structures from Lachlan's list of countable ultrahomogeneous tournaments: <math><mi>Q</mi></math> (the rational line), <math><mi>S</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> (the circular tournament), and <math><msup><mrow><mi>T</mi></mrow><mrow><mo>\infty</mo></mrow></msup></math> (the countable homogeneous universal tournament); as well as the ultrahomogeneous digraphs <math><mi>S</mi><mo>(</mo><mn>3</mn><mo>)</mo></math>, <math><mi>Q</mi><mo>[</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math>, <math><mi>S</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>[</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math> and <math><msup><mrow><mi>T</mi></mrow><mrow><mo>\infty</mo></mrow></msup><mo>[</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math> from Cherlin's list.If <math><msub><mrow><mi>G</mi></mrow><mrow><mi>Rado</mi></mrow></msub></math> (resp. <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math>) denotes the countable homogeneous universal graph (resp. n-labeled linear order), it turns out that <math><mi>P</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>\infty</mo></mrow></msup><mo>)</mo><mo>≅</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>Rado</mi></mrow></msub><mo>)</mo></math> and that <math><mi>P</mi><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> densely embeds in <math><mi>P</mi><mo>(</mo><mi>S</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math>, for <math><mi>n</mi><mo>\in</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math>.Consequently, <math><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>≅</mo><mrow><mi>ro</mi></mrow><mspace></mspace><mo>(</mo><mi>S</mi><mo>⁎</mo><mi>π</mi><mo>)</mo></math>, where <math><mi>S</mi></math> is the poset of perfect subsets of <math><mi>R</mi></math> and π an <math><mi>S</mi></math>-name such that <math><msub><mrow><mn>1</mn></mrow><mrow><mi>S</mi></mrow></msub><mo","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141325306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Around definable types in p-adically closed fields 围绕 p-adically 闭域中的可定义类型 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-06-04 DOI: 10.1016/j.apal.2024.103484 Pablo Andújar Guerrero , Will Johnson We prove some technical results on definable types in p -adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable n -type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if X is an interpretable set, then the space of global definable types on X is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye [7], [8] . Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if G is a definable group with definable f-generics (dfg), and G acts on a definable set X, then the quotient space X / G is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao [24] . Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families. 我们证明了 p-adically closed fields 中可定义类型的一些技术结果，这些结果对可定义群和可定义拓扑空间都有影响。首先，可定义 n 型的代码（在字段排序中）可以被视为实元组（在字段排序中），而不是虚元组（在几何排序中）。其次，在实排序或虚排序中，任何可定义类型都是由值组参数化的链的可数联盟生成的。第三，如果 X 是一个可解释集合，那么 X 上的全局可定义类型空间严格来说是亲可解释的，这是建立在 Cubides Kovacsics、Hils 和 Ye [7], [8] 的工作基础之上的。第四，全局可定义类型可以（以非规范的方式）沿着可解释的投射提升。第五，如果 G 是具有可定义 f 元（dfg）的可定义群，并且 G 作用于可定义集合 X，那么商空间 X/G 是可定义的，而不仅仅是可解释的。这解释了 Pillay 和 Yao [24] 观察到的一些现象。最后，我们证明可解释拓扑空间满足第一可数性和曲线选择的类似条件。由此，我们证明了可定义紧凑性的所有合理概念都与可解释拓扑空间一致，而且可定义紧凑性在族中是可定义的。求助PDF {"title":"Around definable types in p-adically closed fields","authors":"Pablo Andújar Guerrero , Will Johnson","doi":"10.1016/j.apal.2024.103484","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103484","url":null,"abstract":"<div>We prove some technical results on definable types in p-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable n-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if X is an interpretable set, then the space of global definable types on X is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye [7], [8]. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if G is a definable group with definable f-generics (dfg), and G acts on a definable set X, then the quotient space <math><mi>X</mi><mo>/</mo><mi>G</mi></math> is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao [24]. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141325305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Forcing axioms and the uniformization-property 强制公理和统一化属性 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-05-29 DOI: 10.1016/j.apal.2024.103466 Stefan Hoffelner We show that there are models of {MA}_{ω_{1}} where the Σ_{3}^{1} -uniformization property holds. Further we show that “ BPFA + ℵ_{1} is not inaccessible to reals” outright implies that the Σ_{3}^{1} -uniformization property is true. 我们证明，存在Σ31-均匀化性质成立的 MAω1 模型。此外，我们还证明 "BPFA+ ℵ1 并非不可访问的实数 "完全意味着 Σ31-uniformization 属性为真。下载PDF {"title":"Forcing axioms and the uniformization-property","authors":"Stefan Hoffelner","doi":"10.1016/j.apal.2024.103466","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103466","url":null,"abstract":"<div>We show that there are models of <math><msub><mrow><mi>MA</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></math> where the <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math>-uniformization property holds. Further we show that “<math><mi>BPFA</mi></math>+ <math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is not inaccessible to reals” outright implies that the <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math>-uniformization property is true.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000642/pdfft?md5=be051362a938ef048330838dac255f88&pid=1-s2.0-S0168007224000642-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Strong minimal pairs in the enumeration degrees 枚举度中的强最小对 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-05-28 DOI: 10.1016/j.apal.2024.103464 Josiah Jacobsen-Grocott We prove that there are strong minimal pairs in the enumeration degrees and that the degrees of the left and right sides of strong minimal pairs include Σ_{2}^{0} degrees, although it is unknown if there is a strong minimal pair in the Σ_{2}^{0} enumeration degrees. We define a stronger type of minimal pair we call a strong super minimal pair, and show that there are none of these in the enumeration degrees, answering a question of Lempp et al. [6] . We leave open the question of the existence of a super minimal pair in the enumeration degrees. 我们证明了在枚举度中存在强极小对，并且强极小对的左侧和右侧的度包括 Σ20 度，尽管在 Σ20 枚举度中是否存在强极小对还是未知数。我们定义了一种更强的极小对，称为强超极小对，并证明在枚举度中不存在强超极小对，从而回答了 Lempp 等人提出的问题[6]。我们对枚举度中是否存在超极小对这个问题保持开放态度。求助PDF {"title":"Strong minimal pairs in the enumeration degrees","authors":"Josiah Jacobsen-Grocott","doi":"10.1016/j.apal.2024.103464","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103464","url":null,"abstract":"<div>We prove that there are strong minimal pairs in the enumeration degrees and that the degrees of the left and right sides of strong minimal pairs include <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math> degrees, although it is unknown if there is a strong minimal pair in the <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math> enumeration degrees. We define a stronger type of minimal pair we call a strong super minimal pair, and show that there are none of these in the enumeration degrees, answering a question of Lempp et al. [6]. We leave open the question of the existence of a super minimal pair in the enumeration degrees.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Finite undecidability in PAC and PRC fields PAC 和 PRC 领域中的有限不可判定性 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-05-20 DOI: 10.1016/j.apal.2024.103465 Brian Tyrrell A field K in a ring language L is finitely undecidable if Cons (Σ) is undecidable for every nonempty finite Σ \subseteq Th (K; L) . We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields) and Haran (for PRC fields) to prove all PAC and PRC fields are finitely undecidable. We describe the difficulties that arise in adapting the proof to P p C fields, and show no bounded P p C field is finitely axiomatisable. This work is drawn from the author's PhD thesis [44, Chapter 4] . 如果对于每个非空有限 Σ\subseteqTh(K;L)，环语言 L 中的场 K 是有限不可判定的，那么 Cons(Σ) 就是不可判定的。我们改编了起源于谢林-范登德里斯-麦金泰尔/埃尔绍夫（针对 PAC 场）和哈兰（针对 PRC 场）的论证，证明所有 PAC 场和 PRC 场都是有限不可判定的。我们描述了将这一证明应用于 PpC 场时遇到的困难，并证明了没有一个有界 PpC 场是有限公理可证的。这项工作来自作者的博士论文[44，第 4 章]。下载PDF {"title":"Finite undecidability in PAC and PRC fields","authors":"Brian Tyrrell","doi":"10.1016/j.apal.2024.103465","DOIUrl":"10.1016/j.apal.2024.103465","url":null,"abstract":"<div>A field K in a ring language <math><mi>L</mi></math> is finitely undecidable if <math><mtext>Cons</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math> is undecidable for every nonempty finite <math><mi>Σ</mi><mo>\subseteq</mo><mtext>Th</mtext><mo>(</mo><mi>K</mi><mo>;</mo><mi>L</mi><mo>)</mo></math>. We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields) and Haran (for PRC fields) to prove all PAC and PRC fields are finitely undecidable. We describe the difficulties that arise in adapting the proof to PpC fields, and show no bounded PpC field is finitely axiomatisable. This work is drawn from the author's PhD thesis [44, Chapter 4].</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000630/pdfft?md5=4a33b42fff6d541d26261561103e7ddd&pid=1-s2.0-S0168007224000630-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Two-cardinal ideal operators and indescribability 双心理想算子与不可描述性 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-05-13 DOI: 10.1016/j.apal.2024.103463 Brent Cody , Philip White A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner, Feng and the first author, from the cardinal setting to the two-cardinal setting, we study hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability. 关于超紧密性超滤波器的罗伯顿定理的一个著名版本自然而然地引出了双心形拉姆齐性的概念和本文介绍的相应法理想。我们将鲍姆加特纳、冯和第一作者的结果从心形环境推广到双心形环境，研究了与特定版本的双心形拉姆齐性和强版本的双心形不可描述性相关的层次结构，以及这些层次结构与无穷双心形不可描述性的自然概念之间的关系。下载PDF {"title":"Two-cardinal ideal operators and indescribability","authors":"Brent Cody , Philip White","doi":"10.1016/j.apal.2024.103463","DOIUrl":"10.1016/j.apal.2024.103463","url":null,"abstract":"<div>A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner, Feng and the first author, from the cardinal setting to the two-cardinal setting, we study hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000617/pdfft?md5=48f3bb6ab30a9410a6aa9d07709c915f&pid=1-s2.0-S0168007224000617-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal 巴赫曼-霍华德序数的基本序列和快速增长层次结构 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-05-13 DOI: 10.1016/j.apal.2024.103455 David Fernández-Duque, Andreas Weiermann Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties. In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along ϑ (ε_{Ω + 1}) . 哈代函数是通过无穷递归定义的，它为各种形式理论中可证明的总可计算性函数的增长率提供了上限，使其成为许多独立性证明的重要组成部分。它们的定义取决于基本序列的选择，基本序列以 "典型 "的方式逼近极限。在本文中，我们证明了布霍尔茨的ϑ函数基本序列系统具有这些条件，包括巴赫曼性质。我们将这些结果部分扩展到ϑ函数的变体，包括可数序数的无加法版本。我们的结论是，基于这些符号系统的哈代函数享有自然单调性，并使所有沿 ϑ(εΩ+1) 原始递归定义的函数大数化。下载PDF {"title":"Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal","authors":"David Fernández-Duque, Andreas Weiermann","doi":"10.1016/j.apal.2024.103455","DOIUrl":"10.1016/j.apal.2024.103455","url":null,"abstract":"<div>Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along <math><mi>ϑ</mi><mo>(</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>Ω</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math>.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000538/pdfft?md5=a9318d0df651509a7116d53069683110&pid=1-s2.0-S0168007224000538-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用The ghosts of forgotten things: A study on size after forgetting 被遗忘事物的幽灵遗忘后的大小研究 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-05-08 DOI: 10.1016/j.apal.2024.103456 Paolo Liberatore Forgetting is removing variables from a logical formula while preserving the constraints on the other variables. In spite of reducing information, it does not always decrease the size of the formula and may sometimes increase it. This article discusses the implications of such an increase and analyzes the computational properties of the phenomenon. Given a propositional Horn formula, a set of variables and a maximum allowed size, deciding whether forgetting the variables from the formula can be expressed in that size is D^{p} -hard in Σ_{2}^{p} . The same problem for unrestricted CNF propositional formulae is D_{2}^{p} -hard in Σ_{3}^{p} . 遗忘是从逻辑公式中删除变量，同时保留对其他变量的限制。尽管遗忘减少了信息量，但它并不总是减少公式的大小，有时可能会增加公式的大小。本文讨论了这种增加的影响，并分析了这种现象的计算特性。给定一个命题霍恩公式、一组变量和一个最大允许大小，在.NET中，决定从公式中遗忘变量是否可以用该大小表达是很难的。对于无限制的 CNF 命题式，同样的问题在 . 下载PDF {"title":"The ghosts of forgotten things: A study on size after forgetting","authors":"Paolo Liberatore","doi":"10.1016/j.apal.2024.103456","DOIUrl":"10.1016/j.apal.2024.103456","url":null,"abstract":"<div>Forgetting is removing variables from a logical formula while preserving the constraints on the other variables. In spite of reducing information, it does not always decrease the size of the formula and may sometimes increase it. This article discusses the implications of such an increase and analyzes the computational properties of the phenomenon. Given a propositional Horn formula, a set of variables and a maximum allowed size, deciding whether forgetting the variables from the formula can be expressed in that size is <math><msup><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msup></math>-hard in <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></msubsup></math>. The same problem for unrestricted CNF propositional formulae is <math><msubsup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></msubsup></math>-hard in <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>p</mi></mrow></msubsup></math>.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016800722400054X/pdfft?md5=664e39c8c02a1bed0bfc7e9414e88499&pid=1-s2.0-S016800722400054X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Continuum many different things: Localisation, anti-localisation and Yorioka ideals 延续许多不同的东西：本地化、反本地化和赖冈理想 IF 0.8 2区数学 Q2 Mathematics Annals of Pure and Applied Logic Pub Date : 2024-04-15 DOI: 10.1016/j.apal.2024.103453 Miguel A. Cardona , Lukas Daniel Klausner , Diego A. Mejía Combining creature forcing approaches from [16] and [8], we show that, under ch, there is a proper ω^{ω} -bounding poset with ℵ 2 -cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from [17] . 结合来自和的生物强迫方法，我们证明，在和下，存在一个具有 ℵ-cc 的适当 - 边界正集，它能强迫以下六种类型中的每一种类型都具有连续多对不同的红心特征（以有数为参数）：尤里卡理想的均匀性和覆盖数，以及两种本地化和反本地化红心。这回答了.NET 的几个悬而未决的问题。求助PDF {"title":"Continuum many different things: Localisation, anti-localisation and Yorioka ideals","authors":"Miguel A. Cardona , Lukas Daniel Klausner , Diego A. Mejía","doi":"10.1016/j.apal.2024.103453","DOIUrl":"10.1016/j.apal.2024.103453","url":null,"abstract":"<div>Combining creature forcing approaches from [16] and [8], we show that, under ch, there is a proper <math><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math>-bounding poset with ℵ2-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from [17].</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140611495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用首页上一页 1234567 ... 10 下一页尾页类型全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程数据库全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley 期刊 Annals of Pure and Applied Logic 全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. 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