arXiv - MATH - Logic最新文献

Positively closed parametrized models 正封闭参数模型

arXiv - MATH - Logic Pub Date : 2024-09-17 DOI: arxiv-2409.11231

Kristóf Kanalas

引用次数: 0

Denotational semantics driven simplicial homology? 指称语义学驱动简单同源性？

arXiv - MATH - Logic Pub Date : 2024-09-17 DOI: arxiv-2409.11566

Davide Barbarossa

{"title":"Denotational semantics driven simplicial homology?","authors":"Davide Barbarossa","doi":"arxiv-2409.11566","DOIUrl":"https://doi.org/arxiv-2409.11566","url":null,"abstract":"We look at the proofs of a fragment of Linear Logic as a whole: in fact,\u0000Linear Logic's coherent semantics interprets the proofs of a given formula $A$\u0000as faces of an abstract simplicial complex, thus allowing us to see the set of\u0000the (interpretations of the) proofs of $A$ as a geometrical space, not just a\u0000set. This point of view has never been really investigated. For a ``webbed''\u0000denotational semantics -- say the relational one --, it suffices to down-close\u0000the set of (the interpretations of the) proofs of $A$ in order to give rise to\u0000an abstract simplicial complex whose faces do correspond to proofs of $A$.\u0000Since this space comes triangulated by construction, a natural geometrical\u0000property to consider is its homology. However, we immediately stumble on a\u0000problem: if we want the homology to be invariant w.r.t. to some notion of\u0000type-isomorphism, we are naturally led to consider the homology functor acting,\u0000at the level of morphisms, on ``simplicial relations'' rather than simplicial\u0000maps as one does in topology. The task of defining the homology functor on this\u0000modified category can be achieved by considering a very simple monad, which is\u0000almost the same as the power-set monad; but, doing so, we end up considering\u0000not anymore the homology of the original space, but rather of its\u0000transformation under the action of the monad. Does this transformation keep the\u0000homology invariant ? Is this transformation meaningful from a geometrical or\u0000logical/computational point of view ?","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"96 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

AC and the Independence of WO in Second-Order Henkin Logic, Part II 二阶亨金逻辑中的 AC 和 WO 的独立性，第二部分

arXiv - MATH - Logic Pub Date : 2024-09-17 DOI: arxiv-2409.11126

Christine Gaßner

引用次数: 0

Neostability transfers in derivation-like theories 类推导理论中的新稳定性转移

arXiv - MATH - Logic Pub Date : 2024-09-17 DOI: arxiv-2409.11248

Omar Leon Sanchez, Shezad Mohamed

引用次数: 0

On a Generalization of Heyting Algebras II 论海廷代数的广义化 II

arXiv - MATH - Logic Pub Date : 2024-09-16 DOI: arxiv-2409.10642

Amirhossein Akbar Tabatabai, Majid Alizadeh, Masoud Memarzadeh

{"title":"On a Generalization of Heyting Algebras II","authors":"Amirhossein Akbar Tabatabai, Majid Alizadeh, Masoud Memarzadeh","doi":"arxiv-2409.10642","DOIUrl":"https://doi.org/arxiv-2409.10642","url":null,"abstract":"A $nabla$-algebra is a natural generalization of a Heyting algebra, unifying\u0000several algebraic structures, including bounded lattices, Heyting algebras,\u0000temporal Heyting algebras, and the algebraic representation of dynamic\u0000topological systems. In the prequel to this paper [3], we explored the\u0000algebraic properties of various varieties of $nabla$-algebras, their\u0000subdirectly-irreducible and simple elements, their closure under\u0000Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce $nabla$-spaces as a common generalization\u0000of Priestley and Esakia spaces, through which we develop a duality theory for\u0000certain categories of $nabla$-algebras. Then, we reframe these dualities in\u0000terms of spectral spaces and provide an algebraic characterization of natural\u0000families of dynamic topological systems over Priestley, Esakia, and spectral\u0000spaces. Additionally, we present a ring-theoretic representation for some\u0000families of $nabla$-algebras. Finally, we introduce several logical systems to\u0000capture different varieties of $nabla$-algebras, offering their algebraic,\u0000Kripke, topological, and ring-theoretic semantics, and establish a deductive\u0000interpolation theorem for some of these systems.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Tameness Properties in Multiplicative Valued Difference Fields with Lift and Section 带提升和分段的乘法有值差分域中的畸变特性

arXiv - MATH - Logic Pub Date : 2024-09-16 DOI: arxiv-2409.10406

Christoph Kesting

引用次数: 0

Transfer principles for forking and dividing in expansions of pure short exact sequences of Abelian groups 阿贝尔群纯短精确序列展开中分叉和分割的转移原理

arXiv - MATH - Logic Pub Date : 2024-09-16 DOI: arxiv-2409.10148

Akash Hossain

引用次数: 0

Hyperformalism for Bunched Natural Deduction Systems 束状自然演绎系统的超形式主义

arXiv - MATH - Logic Pub Date : 2024-09-16 DOI: arxiv-2409.10418

Shay Allen Logan, Blane Worley

引用次数: 0

Directed equality with dinaturality 定向平等与二元性

arXiv - MATH - Logic Pub Date : 2024-09-16 DOI: arxiv-2409.10237

Andrea Laretto, Fosco Loregian, Niccolò Veltri

{"title":"Directed equality with dinaturality","authors":"Andrea Laretto, Fosco Loregian, Niccolò Veltri","doi":"arxiv-2409.10237","DOIUrl":"https://doi.org/arxiv-2409.10237","url":null,"abstract":"We show how dinaturality plays a central role in the interpretation of\u0000directed type theory where types are interpreted as (1-)categories and directed\u0000equality is represented by $hom$-functors. We present a general elimination\u0000principle based on dinaturality for directed equality which very closely\u0000resembles the $J$-rule used in Martin-L\"of type theory, and we highlight which\u0000syntactical restrictions are needed to interpret this rule in the context of\u0000directed equality. We then use these rules to characterize directed equality as\u0000a left relative adjoint to a functor between (para)categories of dinatural\u0000transformations which contracts together two variables appearing naturally with\u0000a single dinatural one, with the relative functor imposing the syntactic\u0000restrictions needed. We then argue that the quantifiers of such a directed type\u0000theory should be interpreted as ends and coends, which dinaturality allows us\u0000to present in adjoint-like correspondences to a weakening functor. Using these\u0000rules we give a formal interpretation to Yoneda reductions and (co)end\u0000calculus, and we use logical derivations to prove the Fubini rule for\u0000quantifier exchange, the adjointness property of Kan extensions via (co)ends,\u0000exponential objects of presheaves, and the (co)Yoneda lemma. We show\u0000transitivity (composition), congruence (functoriality), and transport\u0000(coYoneda) for directed equality by closely following the same approach of\u0000Martin-L\"of type theory, with the notable exception of symmetry. We formalize\u0000our main theorems in Agda.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

AC and the Independence of WO in Second-Order Henkin Logic, Part I 二阶亨金逻辑中的 AC 和 WO 的独立性，第一部分

arXiv - MATH - Logic Pub Date : 2024-09-16 DOI: arxiv-2409.10276

Christine Gaßner

引用次数: 0