Annals of Pure and Applied Logic最新文献

{"title":"Random expansions of finite structures with bounded degree","authors":"Vera Koponen","doi":"10.1016/j.apal.2025.103665","DOIUrl":"10.1016/j.apal.2025.103665","url":null,"abstract":"<div><div>We consider finite relational signatures <span><math><mi>τ</mi><mo>⊆</mo><mi>σ</mi></math></span>, a sequence of finite base <em>τ</em>-structures <span><math><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></math></span> the cardinalities of which tend to infinity and such that, for some number Δ, the degree of (the Gaifman graph of) every <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is at most Δ. We let <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the set of all expansions of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to <em>σ</em> and we consider a probabilistic graphical model, a concept used in machine learning and artificial intelligence, to generate a probability distribution <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> on <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all <em>n</em>. We use a many-valued “probability logic” with truth values in the unit interval to express probabilities within probabilistic graphical models and to express queries on <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This logic uses aggregation functions (e.g. the average) instead of quantifiers and it can express all queries (on finite structures) that can be expressed with first-order logic since the aggregation functions maximum and minimum can be used to express existential and universal quantifications, respectively. The main results concern asymptotic elimination of aggregation functions (the analogue of almost sure elimination of quantifiers for two-valued logics with quantifiers) and the asymptotic distribution of truth values of formulas, the analogue of logical convergence results for two-valued logics. The structure theory that is developed for sequences <span><math><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></math></span> as above may be of independent interest.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103665"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419752","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":"On Sierpiński sets, Hurewicz spaces and Hilgers functions","authors":"Witold Marciszewski,&nbsp;Roman Pol,&nbsp;Piotr Zakrzewski","doi":"10.1016/j.apal.2025.103680","DOIUrl":"10.1016/j.apal.2025.103680","url":null,"abstract":"<div><div>The Hurewicz property is a classical generalization of <em>σ</em>-compactness. Sierpiński sets (whose existence follows from CH) are standard examples of non-<em>σ</em>-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban <span><span>[19]</span></span>, that for each Sierpiński set <em>S</em> of cardinality at least <span><math><mi>b</mi></math></span> there is a Hurewicz space <em>H</em> with <span><math><mi>S</mi><mo>×</mo><mi>H</mi></math></span> not Hurewicz.</div><div>Some other questions in the literature concerning this topic are also answered.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103680"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528621","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":"MA(I) and a failure of separation on the third level","authors":"Stefan Hoffelner","doi":"10.1016/j.apal.2025.103667","DOIUrl":"10.1016/j.apal.2025.103667","url":null,"abstract":"<div><div>We present a method which forces the failure of <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-separation, while <span><math><mrow><mi>MA</mi></mrow><mo>(</mo><mi>I</mi></math></span>) holds, for <span><math><mi>I</mi></math></span> the family of indestructible ccc forcings. This shows that, in contrast to the assumption <span><math><mi>BPFA</mi></math></span> and <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>L</mi></mrow></msubsup></math></span> which implies <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-separation, that weaker forcing axioms do not decide separation on the third projective level.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103667"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145365745","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":"Arrow algebras","authors":"Benno van den Berg,&nbsp;Marcus Briët","doi":"10.1016/j.apal.2025.103664","DOIUrl":"10.1016/j.apal.2025.103664","url":null,"abstract":"<div><div>In this paper we introduce arrow algebras, simple algebraic structures which induce elementary toposes through the tripos-to-topos construction. This includes localic toposes as well as various realizability toposes, in particular, those realizability toposes which are obtained from partial combinatory algebras. Since there are many examples of arrow algebras and arrow algebras have a number of closure properties, including a notion of subalgebra given by a nucleus, arrow algebras provide a flexible tool for constructing toposes; we illustrate this by providing some general tools for creating toposes for Kreisel's modified realizability.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103664"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145340486","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":"Some remarks on weak one-basedness","authors":"Ikuo Yoneda","doi":"10.1016/j.apal.2025.103674","DOIUrl":"10.1016/j.apal.2025.103674","url":null,"abstract":"<div><div>One-basedness is equivalent to weak one-basedness and CM-triviality in rosy theories.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103674"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419753","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":"Kurepa trees, continuous images, and perfect set properties","authors":"Chris Lambie-Hanson ,&nbsp;Šárka Stejskalová","doi":"10.1016/j.apal.2025.103663","DOIUrl":"10.1016/j.apal.2025.103663","url":null,"abstract":"<div><div>Building upon work of Lücke and Schlicht, we study (higher) Kurepa trees through the lens of higher descriptive set theory, focusing in particular on various perfect set properties and representations of sets of branches through trees as continuous images of function spaces. Answering a question of Lücke and Schlicht, we prove that it is consistent with <span><math><mi>CH</mi></math></span> that there exist <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Kurepa trees and yet, for every <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Kurepa tree <span><math><mi>T</mi><msup><mrow><mo>⊆</mo></mrow><mrow><mo>&lt;</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the set <span><math><mo>[</mo><mi>T</mi><mo>]</mo><msup><mrow><mo>⊆</mo></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of cofinal branches through <em>T</em> is not a continuous image of <span><math><mmultiscripts><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mmultiscripts></math></span>. We also produce models indicating that the existence of Kurepa trees is not necessary to produce closed subsets of <span><math><mmultiscripts><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mmultiscripts></math></span> failing to satisfy strong perfect set properties, and prove a number of consistency results regarding <em>full</em> and <em>superthin</em> trees.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103663"},"PeriodicalIF":0.6,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145340487","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":"The theory DCFpA exists for p > 0","authors":"Kai Ino ,&nbsp;Omar León Sánchez","doi":"10.1016/j.apal.2025.103661","DOIUrl":"10.1016/j.apal.2025.103661","url":null,"abstract":"<div><div>We prove that the (elementary) class of differential-difference fields in characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span> admits a model-companion. In the terminology of Chatzidakis-Pillay <span><span>[4]</span></span>, this says that the class of differentially closed fields of characteristic <em>p</em> equipped with a generic differential-automorphism is elementary; i.e., DCF_<em>p</em>A exists. Along the way, we provide alternative first-order axiomatisations for DCF (differentially closed fields) and also for DCF₀A.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 2","pages":"Article 103661"},"PeriodicalIF":0.6,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145242575","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":"Univalent material set theory","authors":"Håkon Robbestad Gylterud,&nbsp;Elisabeth Stenholm","doi":"10.1016/j.apal.2025.103662","DOIUrl":"10.1016/j.apal.2025.103662","url":null,"abstract":"<div><div>Homotopy type theory (HoTT) can be seen as a generalisation of structural set theory, in the sense that 0-types represent structural sets within the more general notion of types. For material set theory, we also have concrete models as 0-types in HoTT, but this does not currently have any generalisation to higher types. The aim of this paper is to give such a generalisation of material set theory to higher type levels within homotopy type theory. This is achieved by generalising the construction of the type of iterative sets to obtain an <em>n</em>-type universe of <em>n</em>-types. At level 1, this gives a connection between groupoids and multisets.</div><div>More specifically, we define the notion of an ∈-structure as a type with an extensional binary type family and generalise the axioms of constructive set theory to higher type levels. There is a tight connection between the univalence axiom and the extensionality axiom of ∈-structures. Once an ∈-structure is given, its elements can be seen as representing types in the ambient type theory. A useful property of these structures is that an ∈-structure of <em>n</em>-types is itself an <em>n</em>-type, as opposed to univalent universes, which have higher type levels than the types in the universe.</div><div>The theory has an alternative, coalgebraic formulation, in terms of coalgebras for a certain hierarchy of functors, <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, which generalises the powerset functor from sub-types to covering spaces and <em>n</em>-connected maps in general. The coalgebras which furthermore are fixed-points of their respective functors in the hierarchy are shown to model the axioms given in the first part.</div><div>As concrete examples of models for the theory developed we construct the initial algebras of the <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> functors. In addition to being an example of initial algebras of non-polynomial functors, this construction allows one to start with a univalent universe and get a hierarchy of ∈-structures which gives a stratified ∈-structure representation of that universe. These types are moreover <em>n</em>-type universes of <em>n</em>-types which contain all the usual types an type formers. The universes are cumulative both with respect to universe levels and with respect to type levels.</div><div>The results are formalised in the proof-assistant Agda.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 2","pages":"Article 103662"},"PeriodicalIF":0.6,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326840","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":"Compatibility between modal operators in distributive modal logic","authors":"Adam Přenosil","doi":"10.1016/j.apal.2025.103677","DOIUrl":"10.1016/j.apal.2025.103677","url":null,"abstract":"<div><div>Unlike in classical modal logic, in non-classical modal logics the box and diamond operators frequently fail to be interdefinable. Instead, these logics impose some compatibility conditions between the two operators which ensure that in terms of Kripke semantics they arise from the same accessibility relation. This is the case in the intuitionistic modal logic of Fischer Servi as well as the positive modal logic of Dunn. In these logics, however, such compatibility conditions also impose further restrictions on the accessibility relation. In this paper, we identify the basic compatibility conditions which ensure that modal operators arise from a single accessibility relation without imposing any restrictions on the relation. As in the distributive logic of Gehrke, Nagahashi, and Venema, we allow for negative box and diamond operators here in addition to the usual positive ones. Intuitionistic modal logic and positive modal logic, or more precisely the corresponding classes of algebras, are then obtained in a modular way by adding certain canonical axioms which we call locality conditions on top of these basic compatibility conditions.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 2","pages":"Article 103677"},"PeriodicalIF":0.6,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364462","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":"Presheaf automata","authors":"Georg Struth ,&nbsp;Krzysztof Ziemiański","doi":"10.1016/j.apal.2025.103660","DOIUrl":"10.1016/j.apal.2025.103660","url":null,"abstract":"<div><div>We introduce presheaf automata as a generalisation of different variants of higher-dimensional automata and other automata-like formalisms, including Petri nets and vector addition systems. We develop the foundations of a language theory for them based on notions of paths and track objects. We also define open maps for presheaf automata, extending the standard notions of simulation and bisimulation for transition systems. Apart from these conceptual contributions, we show that certain finite-type presheaf automata subsume all Petri nets, generalising a previous result by van Glabbeek, which applies to higher-dimensional automata and safe Petri nets. We also present a class of presheaf automata for which there is no Kleene theorem with respect to the notions of rational and regular languages introduced.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 2","pages":"Article 103660"},"PeriodicalIF":0.6,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270483","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}