Annals of Pure and Applied Logic最新文献

{"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":"2025-10-08","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":"Constructing models of small ordered theories with maximal countable spectrum","authors":"B. Baizhanov ,&nbsp;T. Zambarnaya","doi":"10.1016/j.apal.2025.103659","DOIUrl":"10.1016/j.apal.2025.103659","url":null,"abstract":"<div><div>We propose a method for the construction of countable models of small theories. We then apply it to prove theorems concerning the maximal number of countable non-isomorphic models of linearly ordered theories.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 2","pages":"Article 103659"},"PeriodicalIF":0.6,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145242574","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":"Fundamental sequences based on localization","authors":"Gunnar Wilken","doi":"10.1016/j.apal.2025.103658","DOIUrl":"10.1016/j.apal.2025.103658","url":null,"abstract":"<div><div>Building on Buchholz' assignment for ordinals below Bachmann-Howard ordinal, see <span><span>[2]</span></span>, we introduce systems of fundamental sequences for two kinds of relativized <em>ϑ</em>-function-based notation systems of strength <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msub><mrow><mi>-CA</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and prove Bachmann property for these systems, which is essential for monotonicity properties of subrecursive hierarchies defined on the basis of fundamental sequences. The central notion of our construction is the notion of <em>localization</em>, which was introduced in <span><span>[12]</span></span>.</div><div>The first kind of <em>stepwise defined ϑ</em>-functions over ordinal addition as basic function fits the framework of the ordinal arithmetical toolkit developed in <span><span>[12]</span></span>, whereas the second kind of <em>ϑ</em>-functions is defined <em>simultaneously</em> and will allow for further generalization to larger proof-theoretic ordinals, see <span><span>[10]</span></span>.</div><div>The systems of fundamental sequences given here enable the investigation of fundamental sequences and independence phenomena also in the context of patterns of resemblance, an approach to ordinal notations that is both semantic and combinatorial and was first introduced by Carlson in <span><span>[4]</span></span> and further analyzed in <span><span>[11]</span></span>, <span><span>[13]</span></span>, <span><span>[14]</span></span>, <span><span>[5]</span></span>.</div><div>Our exposition is put into the context of the abstract approach to fundamental sequences developed by Buchholz, Cichon, and Weiermann in <span><span>[3]</span></span>. The results of this paper will be applied to the theory of Goodstein sequences, extending results of <span><span>[7]</span></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103658"},"PeriodicalIF":0.6,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219605","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":"Proof-theoretic investigation of λβ-reduction in the simply typed λ-calculus","authors":"William Stirton","doi":"10.1016/j.apal.2025.103657","DOIUrl":"10.1016/j.apal.2025.103657","url":null,"abstract":"<div><div>The paper defines a function <em>f</em> from simply typed <em>λ</em>-terms to natural numbers and proves that, if <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is a simply typed <em>λ</em>-term formed by contracting an arbitrary <em>λβ</em>-redex within another term <em>M</em>, then <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo><mo>&lt;</mo><mi>f</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span>. Unlike previous proofs of similar theorems, the redex contracted may be completely arbitrary, i.e. without any restriction on rule (<em>ξ</em>). The function <em>f</em> itself is related to, and no more computationally difficult than, a similar-looking function defined in Schütte's <em>Proof Theory</em> (1977).</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103657"},"PeriodicalIF":0.6,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157519","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":"Uncountable homogeneous structures","authors":"Adam Bartoš,&nbsp;Wiesław Kubiś","doi":"10.1016/j.apal.2025.103649","DOIUrl":"10.1016/j.apal.2025.103649","url":null,"abstract":"<div><div>We study the existence of uncountable first-order structures that are homogeneous with respect to their finitely generated substructures. In many classical cases this is either well-known or follows from general facts, for example, if the language is finite and relational then ultrapowers provide arbitrarily large such structures. On the other hand, there are no general results saying that uncountable homogeneous structures with a given age exist. We examine the monoid of self-embeddings of a fixed countable homogeneous structure and, using abstract Fraïssé theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103649"},"PeriodicalIF":0.6,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908766","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":"Transposition of variables is hard to axiomatize","authors":"Hajnal Andréka,&nbsp;István Németi,&nbsp;Zsolt Tuza","doi":"10.1016/j.apal.2025.103650","DOIUrl":"10.1016/j.apal.2025.103650","url":null,"abstract":"<div><div>The function <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> that interchanges two logical variables <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> in formulas is hard to describe in the following sense. Let <em>F</em> denote the Lindenbaum–Tarski formula-algebra of a finite-variable first-order logic, endowed with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> as a unary function. We prove that each equational axiom system for the equational theory of <em>F</em> has to contain, for each finite <em>n</em>, an equation that contains together with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> at least <em>n</em> algebraic variables, and each of the operations <span><math><mo>∃</mo><mo>,</mo><mo>=</mo><mo>,</mo><mo>∨</mo></math></span>. This gives an answer to a problem raised by Johnson (1969) <span><span>[30]</span></span>: the class <span><math><mi>R</mi><mi>P</mi><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of representable polyadic equality algebras of a finite dimension <span><math><mi>α</mi><mo>≥</mo><mn>3</mn></math></span> cannot be axiomatized by adding finitely many equations to the equational theory of representable cylindric algebras of dimension <em>α</em>. Consequences for proof systems of finite-variable logic and for defining equations of polyadic equality algebras are given.</div><div>The proof uses a family of nonrepresentable polyadic equality algebras <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that are more and more nearly representable as <em>n</em> increases: their <em>n</em>-generated subalgebras as well as their proper reducts are representable. The lattice of subvarieties of <span><math><mi>R</mi><mi>P</mi><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> is investigated and new open problems are asked about the interplay between the transposition operations and about generalizability of the results to infinite dimensions.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103650"},"PeriodicalIF":0.6,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144906840","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":"Classifying different criteria for learning algebraic structures","authors":"Nikolay Bazhenov ,&nbsp;Vittorio Cipriani ,&nbsp;Sanjay Jain ,&nbsp;Luca San Mauro ,&nbsp;Frank Stephan","doi":"10.1016/j.apal.2025.103648","DOIUrl":"10.1016/j.apal.2025.103648","url":null,"abstract":"<div><div>In the last years there has been a growing interest in the study of learning problems associated with algebraic structures. The framework we use models the scenario in which a learner is given larger and larger fragments of a structure from a given target family and is required to output an hypothesis about the structure's isomorphism type. So far researchers focused on <strong>Ex</strong>-learning, in which the learner is asked to eventually stabilize to the correct hypothesis, and on restrictions where the learner is allowed to change the hypothesis a fixed number of times. Yet, other learning paradigms coming from classical algorithmic learning theory remained unexplored. We study the ‘‘learning power’’ of such criteria, comparing them via descriptive-set-theoretic tools thanks to the novel notion of <em>E</em>-learnability. The main outcome of this paper is that such criteria admit natural syntactic characterizations in terms of infinitary formulas analogous to the one given for <strong>Ex</strong>-learning in <span><span>[8]</span></span>. Such characterizations give a powerful method to understand whether a family of structures is learnable with respect to the desired criterion.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103648"},"PeriodicalIF":0.6,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903024","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 the spectrum of limit models","authors":"Jeremy Beard ,&nbsp;Marcos Mazari-Armida","doi":"10.1016/j.apal.2025.103647","DOIUrl":"10.1016/j.apal.2025.103647","url":null,"abstract":"<div><div>We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all ‘long’ limit models are isomorphic, and all ‘short’ limit models are non-isomorphic. <section><p><strong>Theorem</strong></p><div><em>Let</em> <strong>K</strong> <em>be a</em> <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><em>-tame abstract elementary class stable in</em> <span><math><mi>λ</mi><mo>≥</mo><mi>LS</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span> <em>with amalgamation, joint embedding and no maximal models. Let</em> <span><math><mi>κ</mi><mo>&lt;</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> <em>be regular. Suppose</em> <figure><img></figure> <em>is an independence relation on the models of size λ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and</em> <span><math><mo>(</mo><mo>≥</mo><mi>κ</mi><mo>)</mo></math></span><em>-local character.</em></div><div><em>Suppose</em> <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> <em>with</em> <span><math><mi>cf</mi><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>&lt;</mo><mi>cf</mi><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span><em>. Then for any</em> <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>M</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> <em>where</em> <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> <em>is a</em> <span><math><mo>(</mo><mi>λ</mi><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>)</mo></math></span><em>-limit model over M for</em> <span><math><mi>l</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span><em>,</em><span><span><img></span></span></div></section></div><div>Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-tameness assumption and assuming <figure><img></figure> is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation.</div><div>We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103647"},"PeriodicalIF":0.6,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902348","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":"Equivalence of multiset-based consequence relations","authors":"Ali Madanshekaf ,&nbsp;Adam Přenosil ,&nbsp;Zeinab Khanjanzadeh Seresti ,&nbsp;Constantine Tsinakis","doi":"10.1016/j.apal.2025.103646","DOIUrl":"10.1016/j.apal.2025.103646","url":null,"abstract":"<div><div>The pioneering work of Blok and Jónsson, and its further development by Galatos and Tsinakis, initiated an abstract study of consequence relations through the lens of module theory, treating consequence relations over all types of syntactic objects on an equal footing. Despite this generality, their framework retains the assumption that premises in a consequence relation form a mere set, rather than a more structured collection. An attempt to extend this framework to account for inferentially substructural generalizations of consequence relations, where the premises have the structure of a finite multiset, was recently made by Cintula, Gil-Férez, Moraschini, and Paoli. In this paper, we propose a different substructural generalization of the Galatos–Tsinakis approach, where the premises are instead taken to form a set of finite multisets. This yields a smoother and more flexible framework that, unlike the approach of Cintula et al., subsumes the original theory of Galatos and Tsinakis as a special case.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103646"},"PeriodicalIF":0.6,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771198","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":"Countable ordered groups and Weihrauch reducibility","authors":"Ang Li","doi":"10.1016/j.apal.2025.103644","DOIUrl":"10.1016/j.apal.2025.103644","url":null,"abstract":"<div><div>This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem <span><span>[11]</span></span> on the order types of countable ordered groups. Solomon <span><span>[14]</span></span> showed that the theorem is equivalent to <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-<span><math><mi>C</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, the strongest of the big five subsystems of second order arithmetic. We show that the strength of the theorem comes from having a dense linear order without endpoints in its order type. Then, we show that for the related Weihrauch problem to be strong enough to be equivalent to <span><math><mover><mrow><mi>WF</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> (the analog problem of <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-<span><math><mi>C</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>), an order-preserving function is necessary in the output. Without the order-preserving function, the problems are very much to the side compared to analog problems of the big five.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103644"},"PeriodicalIF":0.6,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771199","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}