Pub Date : 2025-07-16DOI: 10.1016/j.apal.2025.103636
Pavel E. Alaev
We describe structures computable in polynomial time (P-computable) by a simple and short criterion. We prove that every substructure of a P-computable structure generated by a c.e. set also has a P-computable presentation. For example, we easily prove that every computable torsion-free Abelian group has a P-computable presentation.
{"title":"Description of structures computable in polynomial time","authors":"Pavel E. Alaev","doi":"10.1016/j.apal.2025.103636","DOIUrl":"10.1016/j.apal.2025.103636","url":null,"abstract":"<div><div>We describe structures computable in polynomial time (P-computable) by a simple and short criterion. We prove that every substructure of a P-computable structure generated by a c.e. set also has a P-computable presentation. For example, we easily prove that every computable torsion-free Abelian group has a P-computable presentation.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103636"},"PeriodicalIF":0.6,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703217","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}
Pub Date : 2025-07-09DOI: 10.1016/j.apal.2025.103635
Alessandra Palmigiano , Mattia Panettiere
We generalize Kracht's theory of internal describability from classical modal logic to the family of all logics canonically associated with varieties of normal lattice expansions (LE algebras). We work in the purely algebraic setting of perfect LEs; the formulas playing the role of Kracht's formulas in this generalized setting pertain to a first order language whose atoms are special inequalities between terms of perfect algebras. Via duality, formulas in this language can be equivalently translated into first order conditions in the frame correspondence languages of several types of relational semantics for LE-logics.
{"title":"Unified inverse correspondence for LE-logics","authors":"Alessandra Palmigiano , Mattia Panettiere","doi":"10.1016/j.apal.2025.103635","DOIUrl":"10.1016/j.apal.2025.103635","url":null,"abstract":"<div><div>We generalize Kracht's theory of internal describability from classical modal logic to the family of all logics canonically associated with varieties of normal lattice expansions (LE algebras). We work in the purely algebraic setting of perfect LEs; the formulas playing the role of Kracht's formulas in this generalized setting pertain to a first order language whose atoms are special inequalities between terms of perfect algebras. Via duality, formulas in this language can be equivalently translated into first order conditions in the frame correspondence languages of several types of relational semantics for LE-logics.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103635"},"PeriodicalIF":0.6,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694400","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}
Pub Date : 2025-07-08DOI: 10.1016/j.apal.2025.103634
Igor Arrieta
In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.
{"title":"The DeMorganization of a locale","authors":"Igor Arrieta","doi":"10.1016/j.apal.2025.103634","DOIUrl":"10.1016/j.apal.2025.103634","url":null,"abstract":"<div><div>In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103634"},"PeriodicalIF":0.6,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144587704","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}
Pub Date : 2025-07-07DOI: 10.1016/j.apal.2025.103633
M. Malliaris , S. Moran
This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new statistical learning model, called “Probably Eventually Correct” or PEC. We characterize Littlestone (stable) classes in terms of this model. As a corollary, Littlestone classes have frequent short definitions in a natural statistical sense. In order to obtain a characterization of Littlestone classes in terms of frequent definitions, we build an equivalence theorem highlighting what is common to many existing approximation algorithms, and to the new PEC. This is guided by an analogy to definability of types in model theory, but has its own character. Drawing on these theorems and on other recent work, we present a complete algorithmic analogue of Shelah's celebrated Unstable Formula Theorem, with algorithmic properties taking the place of the infinite.
{"title":"The unstable formula theorem revisited via algorithms","authors":"M. Malliaris , S. Moran","doi":"10.1016/j.apal.2025.103633","DOIUrl":"10.1016/j.apal.2025.103633","url":null,"abstract":"<div><div>This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new statistical learning model, called “Probably Eventually Correct” or PEC. We characterize Littlestone (stable) classes in terms of this model. As a corollary, Littlestone classes have frequent short definitions in a natural statistical sense. In order to obtain a characterization of Littlestone classes in terms of frequent definitions, we build an equivalence theorem highlighting what is common to many existing approximation algorithms, and to the new PEC. This is guided by an analogy to definability of types in model theory, but has its own character. Drawing on these theorems and on other recent work, we present a complete algorithmic analogue of Shelah's celebrated Unstable Formula Theorem, with algorithmic properties taking the place of the infinite.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103633"},"PeriodicalIF":0.6,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632237","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}
Pub Date : 2025-07-04DOI: 10.1016/j.apal.2025.103632
Bruno Dinis , Mário J. Edmundo
In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over ∅ and finitely many open intervals each a union of a ∅-definable family of group-intervals with fixed positive elements.
{"title":"On definable Skolem functions and trichotomy","authors":"Bruno Dinis , Mário J. Edmundo","doi":"10.1016/j.apal.2025.103632","DOIUrl":"10.1016/j.apal.2025.103632","url":null,"abstract":"<div><div>In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over ∅ and finitely many open intervals each a union of a ∅-definable family of group-intervals with fixed positive elements.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103632"},"PeriodicalIF":0.6,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580943","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}
Pub Date : 2025-06-30DOI: 10.1016/j.apal.2025.103631
Sakaé Fuchino , Toshimichi Usuba
The Recurrence Axiom for a class of posets and a set A of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from A is forced by a poset in , then there is a ground containing the parameters and satisfying the statement.
The tightly super---Laver generic hyperhuge continuum implies the Recurrence Axiom for and . The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly -generic hyperhuge cardinal κ, and that κ in the bedrock is genuinely hyperhuge, or even super hyperhuge if κ is a tightly super---Laver generic hyperhuge definable cardinal.
The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For instance, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM.
{"title":"On Recurrence Axioms","authors":"Sakaé Fuchino , Toshimichi Usuba","doi":"10.1016/j.apal.2025.103631","DOIUrl":"10.1016/j.apal.2025.103631","url":null,"abstract":"<div><div>The Recurrence Axiom for a class <span><math><mi>P</mi></math></span> of posets and a set <em>A</em> of parameters is an axiom scheme in the language of <span>ZFC</span> asserting that if a statement with parameters from <em>A</em> is forced by a poset in <span><math><mi>P</mi></math></span>, then there is a ground containing the parameters and satisfying the statement.</div><div>The tightly super-<span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span>-<span><math><mi>P</mi></math></span>-Laver generic hyperhuge continuum implies the Recurrence Axiom for <span><math><mi>P</mi></math></span> and <span><math><mi>H</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>)</mo></math></span>. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly <span><math><mi>P</mi></math></span>-generic hyperhuge cardinal <em>κ</em>, and that <em>κ</em> in the bedrock is genuinely hyperhuge, or even super <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span> hyperhuge if <em>κ</em> is a tightly super-<span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span>-<span><math><mi>P</mi></math></span>-Laver generic hyperhuge definable cardinal.</div><div>The Laver Generic Maximum (<span>LGM</span>), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For instance, double plus version of Martin's Maximum is a consequence of <span>LGM</span> while Cichoń's Maximum is a phenomenon in many grounds of the universe under <span>LGM</span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103631"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144556730","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}
Pub Date : 2025-06-26DOI: 10.1016/j.apal.2025.103630
Julián C. Cano , Carlos A. Di Prisco
Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.
{"title":"Topological games in Ramsey spaces","authors":"Julián C. Cano , Carlos A. Di Prisco","doi":"10.1016/j.apal.2025.103630","DOIUrl":"10.1016/j.apal.2025.103630","url":null,"abstract":"<div><div>Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103630"},"PeriodicalIF":0.6,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514036","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}
Pub Date : 2025-06-17DOI: 10.1016/j.apal.2025.103629
Mark Kamsma , Jiří Rosický
We give a category-theoretic construction of simple and NSOP1-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying properties of a class of monomorphisms such that the pullback squares consisting of morphisms in form the desired independence relation. This generalizes the category-theoretic construction of stable independence relations using effective unions or cellular squares by M. Lieberman, S. Vasey and the second author to the unstable setting.
{"title":"Unstable independence from the categorical point of view","authors":"Mark Kamsma , Jiří Rosický","doi":"10.1016/j.apal.2025.103629","DOIUrl":"10.1016/j.apal.2025.103629","url":null,"abstract":"<div><div>We give a category-theoretic construction of simple and NSOP<sub>1</sub>-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying properties of a class of monomorphisms <span><math><mi>M</mi></math></span> such that the pullback squares consisting of morphisms in <span><math><mi>M</mi></math></span> form the desired independence relation. This generalizes the category-theoretic construction of stable independence relations using effective unions or cellular squares by M. Lieberman, S. Vasey and the second author to the unstable setting.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103629"},"PeriodicalIF":0.6,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322319","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}
Pub Date : 2025-06-11DOI: 10.1016/j.apal.2025.103628
Mamuka Jibladze, Evgeny Kuznetsov
An embedding of arbitrary Heyting algebra H into a reduct from the variety of Kuznetsov-Muravitsky algebras is constructed. An algebraic proof is given that this reduct belongs to the variety of Heyting algebras generated by H.
{"title":"An explicit Kuznetsov-Muravitsky enrichment","authors":"Mamuka Jibladze, Evgeny Kuznetsov","doi":"10.1016/j.apal.2025.103628","DOIUrl":"10.1016/j.apal.2025.103628","url":null,"abstract":"<div><div>An embedding of arbitrary Heyting algebra <em>H</em> into a reduct from the variety of Kuznetsov-Muravitsky algebras is constructed. An algebraic proof is given that this reduct belongs to the variety of Heyting algebras generated by <em>H</em>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103628"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364463","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}
Pub Date : 2025-06-10DOI: 10.1016/j.apal.2025.103627
Alan Dow
We prove that it is consistent with that all automorphisms of are trivial.
证明了P(ω)/fin的所有自同构都是平凡的，这与c>；
{"title":"Automorphisms of P(ω)/fin and large continuum","authors":"Alan Dow","doi":"10.1016/j.apal.2025.103627","DOIUrl":"10.1016/j.apal.2025.103627","url":null,"abstract":"<div><div>We prove that it is consistent with <span><math><mi>c</mi><mo>></mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> that all automorphisms of <span><math><mi>P</mi><mo>(</mo><mi>ω</mi><mo>)</mo><mo>/</mo><mtext>fin</mtext></math></span> are trivial.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103627"},"PeriodicalIF":0.6,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263932","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}
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