Annals of Pure and Applied Logic最新文献

{"title":"First-order reasoning and efficient semi-algebraic proofs","authors":"Fedor Part ,&nbsp;Neil Thapen ,&nbsp;Iddo Tzameret","doi":"10.1016/j.apal.2024.103496","DOIUrl":"10.1016/j.apal.2024.103496","url":null,"abstract":"<div><p>Semi-algebraic proof systems such as sum-of-squares (<span><math><mi>SoS</mi></math></span>) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important <span><math><mi>NP</mi></math></span>-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with <span><math><mi>SoS</mi></math></span> proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or <span><math><mi>SoS</mi></math></span> refutations, respectively; and using a reflection principle, the converse also holds.</p><p>This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.</p><p>We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree <span><math><mi>SoS</mi></math></span>. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by <span><math><mi>SoS</mi></math></span> to PC with the radical rule.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103496"},"PeriodicalIF":0.6,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224001003/pdfft?md5=0c4bf895df8d8576c65657d12bc0c25e&pid=1-s2.0-S0168007224001003-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781080","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":"Owings-like theorems for infinitely many colours or finite monochromatic sets","authors":"David J. Fernández-Bretón,&nbsp;Eliseo Sarmiento Rosales,&nbsp;Germán Vera","doi":"10.1016/j.apal.2024.103495","DOIUrl":"10.1016/j.apal.2024.103495","url":null,"abstract":"<div><p>Inspired by Owings's problem, we investigate whether, for a given an Abelian group <em>G</em> and cardinal numbers <span><math><mi>κ</mi><mo>,</mo><mi>θ</mi></math></span>, every colouring <span><math><mi>c</mi><mo>:</mo><mi>G</mi><mo>⟶</mo><mi>θ</mi></math></span> yields a subset <span><math><mi>X</mi><mo>⊆</mo><mi>G</mi></math></span> with <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>κ</mi></math></span> such that <span><math><mi>X</mi><mo>+</mo><mi>X</mi></math></span> is monochromatic. (Owings's problem asks this for <span><math><mi>G</mi><mo>=</mo><mi>Z</mi></math></span>, <span><math><mi>θ</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>κ</mi><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>; this is known to be false for the same <em>G</em> and <em>κ</em> but <span><math><mi>θ</mi><mo>=</mo><mn>3</mn></math></span>.) We completely settle the question for <em>κ</em> and <em>θ</em> both finite (by obtaining sufficient and necessary conditions for a positive answer) and for <em>κ</em> and <em>θ</em> both infinite (with a negative answer). Also, in the case where <em>θ</em> is infinite but <em>κ</em> is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103495"},"PeriodicalIF":0.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717895","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":"Computably and punctually universal spaces","authors":"Ramil Bagaviev ,&nbsp;Ilnur I. Batyrshin ,&nbsp;Nikolay Bazhenov ,&nbsp;Dmitry Bushtets ,&nbsp;Marina Dorzhieva ,&nbsp;Heer Tern Koh ,&nbsp;Ruslan Kornev ,&nbsp;Alexander G. Melnikov ,&nbsp;Keng Meng Ng","doi":"10.1016/j.apal.2024.103491","DOIUrl":"10.1016/j.apal.2024.103491","url":null,"abstract":"<div><p>We prove that the standard computable presentation of the space <span><math><mi>C</mi><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.</p><p>We prove that the original Urysohn's construction of the universal separable Polish space <span><math><mi>U</mi></math></span> is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msup></math></span>; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103491"},"PeriodicalIF":0.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141715638","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":"Concerning Keisler measures over ultraproducts","authors":"Kyle Gannon","doi":"10.1016/j.apal.2024.103492","DOIUrl":"10.1016/j.apal.2024.103492","url":null,"abstract":"<div><p>As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if <em>μ</em> is definable and both <em>μ</em> and <em>ν</em> are pseudo-finite, then the Morley product of <em>μ</em> and <em>ν</em> agrees with the pseudo-finite product of <em>μ</em> and <em>ν</em>. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103492"},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717817","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":"Elimination of quantifiers for a theory of real closed rings","authors":"Jorge I. Guier","doi":"10.1016/j.apal.2024.103494","DOIUrl":"10.1016/j.apal.2024.103494","url":null,"abstract":"<div><p>Let <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> be the theory of lattice-ordered, convex subrings of von Neumann regular real closed rings that are divisible-projectable, sc-regular (<span><span>[12]</span></span>) and have no minimal (non zero) idempotents. In this paper, we introduce and study a local divisibility binary relation that, added to the language for lattice-ordered rings, together with the (usual) divisibility relation and the radical relation associated to the minimal prime spectrum (<span><span>[19]</span></span>) yields quantifier elimination for <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103494"},"PeriodicalIF":0.6,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141712380","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":"Model-completeness and decidability of the additive structure of integers expanded with a function for a Beatty sequence","authors":"Mohsen Khani ,&nbsp;Ali N. Valizadeh ,&nbsp;Afshin Zarei","doi":"10.1016/j.apal.2024.103493","DOIUrl":"10.1016/j.apal.2024.103493","url":null,"abstract":"<div><p>We introduce a model-complete theory which completely axiomatizes the structure <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>=</mo><mo>〈</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>f</mi><mo>〉</mo></math></span> where <span><math><mi>f</mi><mo>:</mo><mi>x</mi><mo>↦</mo><mo>⌊</mo><mi>α</mi><mi>x</mi><mo>⌋</mo></math></span> is a unary function with <em>α</em> a fixed transcendental number. Moreover, we show that decidability of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> is equivalent to computability of <em>α</em>. This result fits into the more general theme of adding traces of multiplication to integers without losing decidability.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103493"},"PeriodicalIF":0.6,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141638914","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":"Club stationary reflection and other combinatorial principles at ℵω+2","authors":"","doi":"10.1016/j.apal.2024.103489","DOIUrl":"10.1016/j.apal.2024.103489","url":null,"abstract":"<div><p>In this paper we continue the study in <span><span>[11]</span></span> of compactness and incompactness principles at double successors, focusing here on the case of double successors of singulars of countable cofinality. We obtain models which satisfy the tree property and club stationary reflection at these double successors. Moreover, we can additionally obtain either approachability or its failure. We also show how to obtain our results on <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>ω</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span> by incorporating collapses; particularly relevant for these circumstances is a new indestructibility theorem of ours showing that posets satisfying certain linked assumptions preserve club stationary reflection.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103489"},"PeriodicalIF":0.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141577930","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":"Bi-intermediate logics of trees and co-trees","authors":"","doi":"10.1016/j.apal.2024.103490","DOIUrl":"10.1016/j.apal.2024.103490","url":null,"abstract":"<div><p>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 initiate the study of the lattice <span><math><mi>Λ</mi><mo>(</mo><mi>bi-GD</mi><mo>)</mo></math></span> of extensions of <span><math><mi>bi-GD</mi></math></span>.</p><p>We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of <span><math><mi>bi-GD</mi></math></span>. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of <span><math><mi>bi-GD</mi></math></span>. We introduce a sequence of co-trees, called the <em>finite combs</em>, and show that a logic in <span><math><mi>Λ</mi><mo>(</mo><mi>bi-GD</mi><mo>)</mo></math></span> is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of <span><math><mi>bi-GD</mi></math></span> and consequently, a unique pre-locally tabular extension of <span><math><mi>bi-GD</mi></math></span>. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103490"},"PeriodicalIF":0.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000940/pdfft?md5=c7604d9cf135b7d72a099447fc38fed7&pid=1-s2.0-S0168007224000940-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573355","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":"Searching problems above arithmetical transfinite recursion","authors":"Yudai Suzuki ,&nbsp;Keita Yokoyama","doi":"10.1016/j.apal.2024.103488","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103488","url":null,"abstract":"<div><p>We investigate some Weihrauch problems between <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></msub></math></span>. We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Furthermore, we introduce the <em>ω</em>-model reflection <span><math><msubsup><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>rfn</mi></mrow></msubsup></math></span> of <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and show that it is an upper bound for problems provable from the axiomatic system <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> which are of the form <span><math><mo>∀</mo><mi>X</mi><mo>(</mo><mi>θ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><mo>∃</mo><mi>Y</mi><mi>η</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>)</mo></math></span> with arithmetical formulas <span><math><mi>θ</mi><mo>,</mo><mi>η</mi></math></span>. We also show that Weihrauch degrees of relativized least fixed point theorems for monotone operators on the Cantor space form a linear hierarchy between <span><math><msubsup><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>rfn</mi></mrow></msubsup></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></msub></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103488"},"PeriodicalIF":0.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141540300","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":"Decidability bounds for Presburger arithmetic extended by sine","authors":"","doi":"10.1016/j.apal.2024.103487","DOIUrl":"10.1016/j.apal.2024.103487","url":null,"abstract":"<div><p>We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (<strong>sin-PA</strong>), and systematically study decision problems for sets of sentences in <strong>sin-PA</strong>. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in <strong>sin-PA</strong>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103487"},"PeriodicalIF":0.6,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000915/pdfft?md5=8ec66c0193137f3b2153f14d4d1e4bed&pid=1-s2.0-S0168007224000915-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509201","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}