Annals of Pure and Applied Logic最新文献

{"title":"Automorphism groups of prime models, and invariant measures","authors":"Anand Pillay","doi":"10.1016/j.apal.2025.103568","DOIUrl":"10.1016/j.apal.2025.103568","url":null,"abstract":"<div><div>We adapt the notion from <span><span>[7]</span></span> and <span><span>[2]</span></span> of a (relatively) definable subset of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> when <em>M</em> is a saturated structure, to the case <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span> when <em>M</em> is atomic and strongly <em>ω</em>-homogeneous (over a set <em>A</em>). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span>. For example when <em>T</em> is stable, we have existence and uniqueness.</div><div>We also discuss the compatibility of our definability notions with definable Galois cohomology from <span><span>[12]</span></span> and differential Galois theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103568"},"PeriodicalIF":0.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547845","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}

{"title":"Comparing notions of presentability in Polish spaces and Polish groups","authors":"Sapir Ben-Shahar ,&nbsp;Heer Tern Koh","doi":"10.1016/j.apal.2025.103564","DOIUrl":"10.1016/j.apal.2025.103564","url":null,"abstract":"<div><div>A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103564"},"PeriodicalIF":0.6,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463576","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}

{"title":"Local tabularity is decidable for bi-intermediate logics of trees and of co-trees","authors":"Miguel Martins,&nbsp;Tommaso Moraschini","doi":"10.1016/j.apal.2025.103563","DOIUrl":"10.1016/j.apal.2025.103563","url":null,"abstract":"<div><div>A bi-Heyting algebra validates the Gödel-Dummett axiom <span><math><mo>(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo>)</mo><mo>∨</mo><mo>(</mo><mi>q</mi><mo>→</mo><mi>p</mi><mo>)</mo></math></span> iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called <em>bi-Gödel algebras</em> and form a variety that algebraizes the extension <span><math><mi>bi-GD</mi></math></span> of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of <span><math><mi>bi-GD</mi></math></span> is locally tabular.</div><div>Notably, if <em>L</em> is an axiomatic extension of <span><math><mi>bi-GD</mi></math></span>, then <em>L</em> is locally tabular iff <em>L</em> is not contained in <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span>, the logic of a particular family of finite co-trees, called the <em>finite combs</em>. We prove that <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span> is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103563"},"PeriodicalIF":0.6,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487726","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}

{"title":"Conditional algebras","authors":"Sergio Celani ,&nbsp;Rafał Gruszczyński ,&nbsp;Paula Menchón","doi":"10.1016/j.apal.2025.103556","DOIUrl":"10.1016/j.apal.2025.103556","url":null,"abstract":"<div><div>Drawing on the classic paper by Chellas <span><span>[8]</span></span>, we propose a general algebraic framework for studying a binary operation of <em>conditional</em> that models universal features of the “if …, then …” connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of <em>conditional algebras</em>, and we develop its duality and canonical extensions theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103556"},"PeriodicalIF":0.6,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143376529","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}

{"title":"Failure of the Blok–Esakia Theorem in the monadic setting","authors":"G. Bezhanishvili ,&nbsp;L. Carai","doi":"10.1016/j.apal.2024.103527","DOIUrl":"10.1016/j.apal.2024.103527","url":null,"abstract":"<div><div>The Blok–Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok–Esakia isomorphism <em>σ</em> does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that <em>σ</em> is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103527"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162353","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}

{"title":"Strength and limitations of Sherali-Adams and Nullstellensatz proof systems","authors":"Ilario Bonacina,&nbsp;Maria Luisa Bonet","doi":"10.1016/j.apal.2024.103538","DOIUrl":"10.1016/j.apal.2024.103538","url":null,"abstract":"<div><div>We compare the strength of the algebraic proof systems Sherali-Adams (<span><math><mi>SA</mi></math></span>) and Nullstellensatz (<span><math><mi>NS</mi></math></span>) with Frege-style proof systems. Unlike bounded-depth Frege, <span><math><mi>SA</mi></math></span> has polynomial-size proofs of the pigeonhole principle (<span>PHP</span>). A natural question is whether adding <span>PHP</span> to bounded-depth Frege is enough to simulate <span><math><mi>SA</mi></math></span>. We show that <span><math><mi>SA</mi></math></span>, with unary integer coefficients, lies strictly between tree-like depth-1 <span><math><mtext>Frege</mtext><mo>+</mo><mrow><mi>PHP</mi></mrow></math></span> and tree-like <span><math><mtext>Resolution</mtext></math></span>. We introduce a <em>levelled</em> version of <span>PHP</span> (<span><math><mi>L</mi><mrow><mi>PHP</mi></mrow></math></span>) and we show that <span><math><mi>SA</mi></math></span> with integer coefficients lies strictly between tree-like depth-1 <span><math><mtext>Frege</mtext><mo>+</mo><mi>L</mi><mrow><mi>PHP</mi></mrow></math></span> and <span><math><mtext>Resolution</mtext></math></span>. Analogous results are shown for <span><math><mi>NS</mi></math></span> using the bijective (i.e. onto and functional) pigeonhole principle and a leveled version of it.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103538"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162344","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}

{"title":"Proof-theoretic methods in quantifier-free definability","authors":"Zoltan A. Kocsis","doi":"10.1016/j.apal.2025.103555","DOIUrl":"10.1016/j.apal.2025.103555","url":null,"abstract":"<div><div>We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's “realizability disjunction” connective does not admit a quantifier-free definition, and use it to obtain new results and more nuanced information about the nondefinability of Kreisel's and Połacik's unary connectives. The finitary and combinatorial nature of our method makes it resilient to changes in metatheory, and suitable for settings with axioms that are explicitly incompatible with classical logic. Furthermore, the problem-specific subproofs arising from this approach can be readily transcribed into univalent type theory and verified using the Agda proof assistant.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103555"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162322","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}

{"title":"More about the cofinality and the covering of the ideal of strong measure zero sets","authors":"Miguel A. Cardona ,&nbsp;Diego A. Mejía","doi":"10.1016/j.apal.2024.103537","DOIUrl":"10.1016/j.apal.2024.103537","url":null,"abstract":"<div><div>We improve the previous work of Yorioka and the first author about the combinatorics of the ideal <span><math><mi>SN</mi></math></span> of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle <span><math><mrow><mi>DS</mi></mrow><mo>(</mo><mi>δ</mi><mo>)</mo></math></span> that helps to find simple conditions to deduce <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>≤</mo><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> (where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span> is the dominating number on <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>). In addition, we find a new upper bound of <span><math><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> by using products of relational systems and cardinal characteristics associated with Yorioka ideals.</div><div>In addition, we dissect and generalize results from Pawlikowski to force upper bounds of the covering of <span><math><mi>SN</mi></math></span>, particularly for finite support iterations of precaliber posets.</div><div>Finally, as applications of our main theorems, we prove consistency results about the cardinal characteristics associated with <span><math><mi>SN</mi></math></span> and the principle <span><math><mrow><mi>DS</mi></mrow><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>. For example, we show that <span><math><mrow><mi>cov</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>&lt;</mo><mrow><mi>non</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>=</mo><mi>c</mi><mo>&lt;</mo><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> holds in Cohen model, and we refine a result (and the proof) of the first author about the consistency of <span><math><mrow><mi>cov</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>&lt;</mo><mrow><mi>non</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>&lt;</mo><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span>, with <span><math><mi>c</mi></math></span> in any desired position with respect to <span><math><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span>, and the improvement that <span><math><mrow><mi>non</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> can be singular here.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103537"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162352","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}

{"title":"Generic multiplicative endomorphism of a field","authors":"Christian d'Elbée","doi":"10.1016/j.apal.2025.103554","DOIUrl":"10.1016/j.apal.2025.103554","url":null,"abstract":"<div><div>We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative endomorphism, which we call ACFH. Among others, we prove that this theory is NSOP₁ and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103554"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162054","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}

{"title":"Peano arithmetic, games and descent recursion","authors":"Emanuele Frittaion","doi":"10.1016/j.apal.2024.103550","DOIUrl":"10.1016/j.apal.2024.103550","url":null,"abstract":"<div><div>We analyze Coquand's game-theoretic interpretation of Peano Arithmetic <span><span>[6]</span></span> through the lens of elementary descent recursion <span><span>[8]</span></span>. In Coquand's game semantics, winning strategies correspond to infinitary cut-free proofs and cut elimination corresponds to <em>debates</em> between these winning strategies. The proof of cut elimination, i.e., the proof that such debates eventually terminate, is by transfinite induction on certain <em>interaction</em> sequences of ordinals. In this paper, we provide a direct implementation of Coquand's proof, one that allows us to describe winning strategies by descent recursive functions. As a byproduct, we obtain yet another proof of well-known results about provably recursive functions and functionals.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103550"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162345","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}