An infinite binary sequence is Bennett deep if, for any computable time bound, the difference between the time-bounded prefix-free Kolmogorov complexity and the prefix-free Kolmogorov complexity of its initial segments is eventually unbounded. It is known that weakly 2-generic sets are shallow, i.e. not deep. In this paper, we show that there is a deep 1-generic set.
{"title":"There is a deep 1-generic set","authors":"Ang Li","doi":"arxiv-2409.00631","DOIUrl":"https://doi.org/arxiv-2409.00631","url":null,"abstract":"An infinite binary sequence is Bennett deep if, for any computable time\u0000bound, the difference between the time-bounded prefix-free Kolmogorov\u0000complexity and the prefix-free Kolmogorov complexity of its initial segments is\u0000eventually unbounded. It is known that weakly 2-generic sets are shallow, i.e.\u0000not deep. In this paper, we show that there is a deep 1-generic set.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187747","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}
Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj
The structure of automorphism groups of $kappa$-existentially closed groups are studied by Kaya-Kuzucuou{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^kappa$ whenever $kappa$ is an inaccessible cardinal and $G$ is the unique $kappa$-existentially closed group of cardinality $kappa$. The cardinality of the automorphism group of a $kappa$-existentially closed group of cardinality $lambda>kappa$ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case $kappa=lambda$ namely: If $G$ is a $kappa$-existentially closed group of cardinality $kappa$, then $|Aut(G)|=2^{kappa}$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $kappa$, there exist universal groups of cardinality $kappa$.
{"title":"Limit Groups and Automorphisms of $κ$-Existentially Closed Groups","authors":"Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj","doi":"arxiv-2409.00545","DOIUrl":"https://doi.org/arxiv-2409.00545","url":null,"abstract":"The structure of automorphism groups of $kappa$-existentially closed groups\u0000are studied by Kaya-Kuzucuou{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving\u0000automorphisms and $|Aut(G)|=2^kappa$ whenever $kappa$ is an inaccessible cardinal and $G$ is the unique $kappa$-existentially closed group of cardinality $kappa$. The cardinality of the automorphism group of a\u0000$kappa$-existentially closed group of cardinality $lambda>kappa$ is asked in Kourovka Notebook\u0000Question 20.40. Here we answer positively the promised case $kappa=lambda$ namely: If $G$ is a $kappa$-existentially closed group of cardinality $kappa$, then $|Aut(G)|=2^{kappa}$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $kappa$, there exist universal groups of cardinality $kappa$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187746","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}
We study Medvedev reducibility in the context of set theory -- specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li cite{HaLi}, we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary ``reasonably-definable" reducibilities, under appropriate set-theoretic hypotheses. We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and ``measure" on that class. We end by discussing some directions for future research.
{"title":"Strong reducibilities and set theory","authors":"Noah Schweber","doi":"arxiv-2408.17393","DOIUrl":"https://doi.org/arxiv-2408.17393","url":null,"abstract":"We study Medvedev reducibility in the context of set theory -- specifically,\u0000forcing and large cardinal hypotheses. Answering a question of Hamkins and Li\u0000cite{HaLi}, we show that the Medvedev degrees of countable ordinals are far\u0000from linearly ordered in multiple ways, our main result here being that there\u0000is a club of ordinals which is an antichain with respect to Medvedev\u0000reducibility. We then generalize these results to arbitrary\u0000``reasonably-definable\" reducibilities, under appropriate set-theoretic\u0000hypotheses. We then turn from ordinals to general structures. We show that some of the\u0000results above yield characterizations of counterexamples to Vaught's\u0000conjecture; another applies to all situations, assigning an ordinal to any\u0000reasonable class of structures and ``measure\" on that class. We end by\u0000discussing some directions for future research.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187749","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}
In this work we study integral residuated chains, and we solve some open problems related to the study of the amalgamation property in varieties of residuated lattices, or equivalently, about the deductive interpolation property in substructural logics. More precisely, we find a V-formation consisting of 2-potent finite commutative integral chains that does not have an amalgam, nor a one-amalgam, in residuated chains; as most relevant consequences, this entails that the following varieties do not have the amalgamation property: semilinear commutative (integral) residuated lattices, MTL-algebras, involutive and pseudocomplemented MTL-algebras, and all of their n-potent subvarieties for n strictly larger than 1. These results entail the failure of the deductive interpolation property for the corresponding substructural logics.
在这项工作中，我们研究了积分残差链，并解决了一些与研究残差格的品种中的汞齐性质有关的开放问题，或者等价于研究子结构逻辑中的演绎插值性质的开放问题。更确切地说，我们发现了一个由 2 能有限交换积分链构成的 V 形，它在残差链中不存在混汞（amalgam），也不存在一汞（one-amalgam）；作为最相关的结果，这意味着下列品种不存在混汞属性：半线性交换（积分）残差格、MTL-代数、非累加和伪补全的 MTL-代数，以及它们的 n 严格大于 1 的所有 n 能子品种。这些结果导致相应的结构逻辑的演绎内插性失效。
{"title":"Algebraic structure theory and interpolation failures in semilinear logics","authors":"Valeria Giustarini, Sara Ugolini","doi":"arxiv-2408.17400","DOIUrl":"https://doi.org/arxiv-2408.17400","url":null,"abstract":"In this work we study integral residuated chains, and we solve some open\u0000problems related to the study of the amalgamation property in varieties of\u0000residuated lattices, or equivalently, about the deductive interpolation\u0000property in substructural logics. More precisely, we find a V-formation\u0000consisting of 2-potent finite commutative integral chains that does not have an\u0000amalgam, nor a one-amalgam, in residuated chains; as most relevant\u0000consequences, this entails that the following varieties do not have the\u0000amalgamation property: semilinear commutative (integral) residuated lattices,\u0000MTL-algebras, involutive and pseudocomplemented MTL-algebras, and all of their\u0000n-potent subvarieties for n strictly larger than 1. These results entail the\u0000failure of the deductive interpolation property for the corresponding\u0000substructural logics.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187748","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}
The variable inclusion companions of logics have lately been thoroughly studied by multiple authors. There are broadly two types of these companions: the left and the right variable inclusion companions. Another type of companions of logics induced by Hilbert-style presentations (Hilbert-style logics) were introduced in a recent paper. A sufficient condition for the restricted rules companion of a Hilbert-style logic to coincide with its left variable inclusion companion was proved there, while a necessary condition remained elusive. The present article has two parts. In the first part, we give a necessary and sufficient condition for the left variable inclusion and the restricted rules companions of a Hilbert-style logic to coincide. In the rest of the paper, we recognize that the variable inclusion restrictions used to define variable inclusion companions of a logic $langlemathcal{L},vdashrangle$ are relations from $mathcal{P}(mathcal{L})$ to $mathcal{L}$. This leads to a more general idea of a relational companion of a logical structure, a framework that we borrow from the field of universal logic. We end by showing that even Hilbert-style logics and the restricted rules companions of these can be brought under the umbrella of the general notions of logical structures and their relational companions that are discussed here.
{"title":"Relational Companions of Logics","authors":"Sankha S. Basu, Sayantan Roy","doi":"arxiv-2408.17019","DOIUrl":"https://doi.org/arxiv-2408.17019","url":null,"abstract":"The variable inclusion companions of logics have lately been thoroughly\u0000studied by multiple authors. There are broadly two types of these companions:\u0000the left and the right variable inclusion companions. Another type of\u0000companions of logics induced by Hilbert-style presentations (Hilbert-style\u0000logics) were introduced in a recent paper. A sufficient condition for the\u0000restricted rules companion of a Hilbert-style logic to coincide with its left\u0000variable inclusion companion was proved there, while a necessary condition\u0000remained elusive. The present article has two parts. In the first part, we give\u0000a necessary and sufficient condition for the left variable inclusion and the\u0000restricted rules companions of a Hilbert-style logic to coincide. In the rest\u0000of the paper, we recognize that the variable inclusion restrictions used to\u0000define variable inclusion companions of a logic\u0000$langlemathcal{L},vdashrangle$ are relations from\u0000$mathcal{P}(mathcal{L})$ to $mathcal{L}$. This leads to a more general idea\u0000of a relational companion of a logical structure, a framework that we borrow\u0000from the field of universal logic. We end by showing that even Hilbert-style\u0000logics and the restricted rules companions of these can be brought under the\u0000umbrella of the general notions of logical structures and their relational\u0000companions that are discussed here.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187751","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}
It is shown that the isomorphism relation between continuous t-norms is Borel bireducible with the relation of order isomorphism between linear orders on the set of natural numbers, and therefore, it is Borel bireducible with every Borel complete equivalence relation.
{"title":"The complexity of classifying continuous t-norms up to isomorphism","authors":"Jialiang He, Lili Shen, Yi Zhou","doi":"arxiv-2408.16456","DOIUrl":"https://doi.org/arxiv-2408.16456","url":null,"abstract":"It is shown that the isomorphism relation between continuous t-norms is Borel\u0000bireducible with the relation of order isomorphism between linear orders on the\u0000set of natural numbers, and therefore, it is Borel bireducible with every Borel\u0000complete equivalence relation.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187752","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}
In this note, we prove that the constructive and intuitionistic variants of the modal logic $mathsf{KB}$ coincide. This result contrasts with a recent result by Das and Marin, who showed that the constructive and intuitionistic variants of $mathsf{K}$ do not prove the same diamond-free formulas.
{"title":"Collapsing Constructive and Intuitionistic Modal Logics","authors":"Leonardo Pacheco","doi":"arxiv-2408.16428","DOIUrl":"https://doi.org/arxiv-2408.16428","url":null,"abstract":"In this note, we prove that the constructive and intuitionistic variants of\u0000the modal logic $mathsf{KB}$ coincide. This result contrasts with a recent\u0000result by Das and Marin, who showed that the constructive and intuitionistic\u0000variants of $mathsf{K}$ do not prove the same diamond-free formulas.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142188023","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}
We prove that every Polish group admits a non-trivial topological group automorphism. This answers a question posed by Forte Shinko. As a consequence, we prove that there are no uniquely homogeneous Polish groups.
{"title":"Every Polish group has a non-trivial topological group automorphism","authors":"Carlos Pérez Estrada, Ulises Ariet Ramos-García","doi":"arxiv-2408.16162","DOIUrl":"https://doi.org/arxiv-2408.16162","url":null,"abstract":"We prove that every Polish group admits a non-trivial topological group\u0000automorphism. This answers a question posed by Forte Shinko. As a consequence,\u0000we prove that there are no uniquely homogeneous Polish groups.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187753","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}
We study the equational theory of the Weihrauch lattice with composition and iterations, meaning the collection of equations between terms built from variables, the lattice operations $sqcup$, $sqcap$, the composition operator $star$ and its iteration $(-)^diamond$ , which are true however we substitute (slightly extended) Weihrauch degrees for the variables. We characterize them using B"uchi games on finite graphs and give a complete axiomatization that derives them. The term signature and the axiomatization are reminiscent of Kleene algebras, except that we additionally have meets and the lattice operations do not fully distributes over composition. The game characterization also implies that it is decidable whether an equation is universally valid. We give some complexity bounds; in particular, the problem is Pspace-hard in general and we conjecture that it is solvable in Pspace.
{"title":"The equational theory of the Weihrauch lattice with (iterated) composition","authors":"Cécilia Pradic","doi":"arxiv-2408.14999","DOIUrl":"https://doi.org/arxiv-2408.14999","url":null,"abstract":"We study the equational theory of the Weihrauch lattice with composition and\u0000iterations, meaning the collection of equations between terms built from\u0000variables, the lattice operations $sqcup$, $sqcap$, the composition operator\u0000$star$ and its iteration $(-)^diamond$ , which are true however we substitute\u0000(slightly extended) Weihrauch degrees for the variables. We characterize them\u0000using B\"uchi games on finite graphs and give a complete axiomatization that\u0000derives them. The term signature and the axiomatization are reminiscent of\u0000Kleene algebras, except that we additionally have meets and the lattice\u0000operations do not fully distributes over composition. The game characterization\u0000also implies that it is decidable whether an equation is universally valid. We\u0000give some complexity bounds; in particular, the problem is Pspace-hard in\u0000general and we conjecture that it is solvable in Pspace.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187776","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}
The main purpose of this paper is to present new and more uniform model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability is a (complete) first-order continuous theory with $NIP$/stability. The proof method for both theorems is based on the significant use of a particular type of models of $T^{R}$, namely simple models, and certain indiscernible arrays. Using simple models of $T^R$ gives the advantage of re-proving these theorems in a simpler and quantitative manner. We finally turn our attention to $NSOP$ in randomization. We show that based on the definition of $NSOP$ given [11], $T^R$ is stable if and only if it is $NIP$ and $NSOP$.
{"title":"Simple Models of Randomization and Preservation Theorems","authors":"Karim Khanaki, Massoud Pourmahdian","doi":"arxiv-2408.15014","DOIUrl":"https://doi.org/arxiv-2408.15014","url":null,"abstract":"The main purpose of this paper is to present new and more uniform\u0000model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The\u0000randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability\u0000is a (complete) first-order continuous theory with $NIP$/stability. The proof\u0000method for both theorems is based on the significant use of a particular type\u0000of models of $T^{R}$, namely simple models, and certain indiscernible arrays.\u0000Using simple models of $T^R$ gives the advantage of re-proving these theorems\u0000in a simpler and quantitative manner. We finally turn our attention to $NSOP$\u0000in randomization. We show that based on the definition of $NSOP$ given [11],\u0000$T^R$ is stable if and only if it is $NIP$ and $NSOP$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187754","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}
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