Annals of Pure and Applied Logic最新文献

Universal proof theory: Feasible admissibility in intuitionistic modal logics 通用证明理论：直觉模态逻辑中的可行可接受性

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-10-23 DOI: 10.1016/j.apal.2024.103526

Amirhossein Akbar Tabatabai , Raheleh Jalali

{"title":"Universal proof theory: Feasible admissibility in intuitionistic modal logics","authors":"Amirhossein Akbar Tabatabai , Raheleh Jalali","doi":"10.1016/j.apal.2024.103526","DOIUrl":"10.1016/j.apal.2024.103526","url":null,"abstract":"<div><div>We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities <math><mo>{</mo><mo>□</mo><mo>,</mo><mo>◇</mo><mo>}</mo></math> and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including <math><mi>CK</mi></math>, <math><mi>IK</mi></math>, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments <math><msub><mrow><mi>CK</mi></mrow><mrow><mo>□</mo></mrow></msub></math>, propositional lax logic and <math><mi>IPC</mi></math>. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than <math><mi>IPC</mi></math> has a constructive sequent calculus.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103526"},"PeriodicalIF":0.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561348","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}

引用次数: 0

Bi-colored expansions of geometric theories 几何理论的双色展开

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-10-18 DOI: 10.1016/j.apal.2024.103525

S. Jalili , M. Pourmahdian , M. Khani

{"title":"Bi-colored expansions of geometric theories","authors":"S. Jalili , M. Pourmahdian , M. Khani","doi":"10.1016/j.apal.2024.103525","DOIUrl":"10.1016/j.apal.2024.103525","url":null,"abstract":"<div><div>This paper concerns the study of expansions of models of a geometric theory T by a color predicate p, within the framework of the Fraïssé-Hrushovski construction method. For each <math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, we define a pre-dimension function <math><msub><mrow><mi>δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math> on the class of Bi-colored models of <math><msup><mrow><mi>T</mi></mrow><mrow><mo>∀</mo></mrow></msup></math> and consider the subclass <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>α</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math> consisting of models with hereditary positive <math><msub><mrow><mi>δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math>. We impose certain natural conditions on T that enable us to introduce a complete <math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-theory <math><msub><mrow><mi>T</mi></mrow><mrow><mi>α</mi></mrow></msub></math> for the rich models in <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>α</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math>. We show how the transfer of certain model-theoretic properties, such as NIP and strong-dependence, from T to <math><msub><mrow><mi>T</mi></mrow><mrow><mi>α</mi></mrow></msub></math>, depends on whether α is rational or irrational.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103525"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553194","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}

引用次数: 0

Equiconsistency of the Minimalist Foundation with its classical version 极简主义基础与其经典版本的等价一致性

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-10-16 DOI: 10.1016/j.apal.2024.103524

Maria Emilia Maietti, Pietro Sabelli

{"title":"Equiconsistency of the Minimalist Foundation with its classical version","authors":"Maria Emilia Maietti, Pietro Sabelli","doi":"10.1016/j.apal.2024.103524","DOIUrl":"10.1016/j.apal.2024.103524","url":null,"abstract":"<div><div>The Minimalist Foundation, for short MF, was conceived by the first author with G. Sambin in 2005, and fully formalized in 2009, as a common core among the most relevant constructive and classical foundations for mathematics. To better accomplish its minimality, MF was designed as a two-level type theory, with an intensional level mTT, an extensional one emTT, and an interpretation of the latter into the first.</div><div>Here, we first show that the two levels of MF are indeed equiconsistent by interpreting mTT into emTT. Then, we show that the classical extension <math><msup><mrow><mi>emTT</mi></mrow><mrow><mi>c</mi></mrow></msup></math> is equiconsistent with emTT by suitably extending the Gödel-Gentzen double-negation translation of classical logic in the intuitionistic one. As a consequence, MF turns out to be compatible with classical predicative mathematics à la Weyl, contrary to the most relevant foundations for constructive mathematics.</div><div>Finally, we show that the chain of equiconsistency results for MF can be straightforwardly extended to its impredicative version to deduce that Coquand-Huet's Calculus of Constructions equipped with basic inductive types is equiconsistent with its extensional and classical versions too.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103524"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553193","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}

引用次数: 0

Some properties of precompletely and positively numbered sets 预完全正数集的一些性质

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-10-09 DOI: 10.1016/j.apal.2024.103523

Marat Faizrahmanov

{"title":"Some properties of precompletely and positively numbered sets","authors":"Marat Faizrahmanov","doi":"10.1016/j.apal.2024.103523","DOIUrl":"10.1016/j.apal.2024.103523","url":null,"abstract":"<div><div>In this paper, we prove a joint generalization of Arslanov's completeness criterion and Visser's ADN theorem for precomplete numberings which, for the Gödel numbering <math><mi>x</mi><mo>↦</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi></mrow></msub></math>, has been proved by Terwijn (2018). The question of whether this joint generalization takes place in each precomplete numbering has been raised in his joint paper with Barendregt in 2019. Then we consider the properties of completeness and precompleteness of numberings in the context of the positivity property. We show that no completion of a positive numbering is a minimal cover of that numbering, and that the Turing completeness of any set A is equivalent to the existence of a positive precomplete A-computable numbering of any infinite family with positive A-computable numbering. In addition, we prove that each <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math>-computable numbering (<math><mi>n</mi><mo>⩾</mo><mn>2</mn></math>) of a <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math>-computable non-principal family has a <math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math>-computable minimal cover ν such that for every computable function f there exists an integer n with <math><mi>ν</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>ν</mi><mo>(</mo><mi>n</mi><mo>)</mo></math>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103523"},"PeriodicalIF":0.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420439","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}

引用次数: 0

Strong reducibilities and set theory 强还原性与集合论

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-10-01 DOI: 10.1016/j.apal.2024.103522

Noah Schweber

引用次数: 0

Dividing and forking in random hypergraphs 随机超图中的分割和分叉

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-09-24 DOI: 10.1016/j.apal.2024.103521

Hirotaka Kikyo , Akito Tsuboi

引用次数: 0

Saturation properties for compositional truth with propositional correctness 具有命题正确性的组合真理的饱和特性

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-09-03 DOI: 10.1016/j.apal.2024.103512

Bartosz Wcisło

引用次数: 0

Foundations of iterated star maps and their use in combinatorics 迭代星图的基础及其在组合学中的应用

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-08-30 DOI: 10.1016/j.apal.2024.103511

Mauro Di Nasso , Renling Jin

引用次数: 0

Theories of Frege structure equivalent to Feferman's system T0 弗雷格理论结构等同于费弗曼体系 T0

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-08-22 DOI: 10.1016/j.apal.2024.103510

Daichi Hayashi

引用次数: 0

Universal proof theory: Semi-analytic rules and Craig interpolation 通用证明理论：半解析规则和克雷格插值法

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-08-20 DOI: 10.1016/j.apal.2024.103509

Amirhossein Akbar Tabatabai , Raheleh Jalali

{"title":"Universal proof theory: Semi-analytic rules and Craig interpolation","authors":"Amirhossein Akbar Tabatabai , Raheleh Jalali","doi":"10.1016/j.apal.2024.103509","DOIUrl":"10.1016/j.apal.2024.103509","url":null,"abstract":"<div>We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including <math><mi>U</mi><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo></mrow></msup></math>, <math><mi>MTL</mi></math>, <math><mi>R</mi></math>, Łn (for <math><mi>n</mi><mo>⩾</mo><mn>3</mn></math>), Gn (for <math><mi>n</mi><mo>⩾</mo><mn>4</mn></math>), and almost all extensions of <math><mi>IMTL</mi></math>, <math><mi>Ł</mi></math>, <math><mi>BL</mi></math>, <math><mi>R</mi><msup><mrow><mi>M</mi></mrow><mrow><mi>e</mi></mrow></msup></math>, <math><mi>IPC</mi></math>, <math><mi>S4</mi></math>, and <math><mi>Grz</mi></math> (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103509"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087766","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}

引用次数: 0