{"title":"A classification of low c.e. sets and the Ershov hierarchy","authors":"Marat Faizrahmanov","doi":"10.1002/malq.202300020","DOIUrl":"10.1002/malq.202300020","url":null,"abstract":"<p>In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ-levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ-level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Sigma ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>- and <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Delta ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>-bound for every infinite computable ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Sigma ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>- or <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Delta ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>-bound of a c.e. set <i>A</i> is equivalent to the fact that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>A</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <mo>∈</mo>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$A^{prime }in Delta ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>. Finally, we consider the generalized truth-table reducibilities <math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>⩽</mo>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 <mi>t</mi>\u0000 <mo>(</mo>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximate isomorphism of metric structures","authors":"James E. Hanson","doi":"10.1002/malq.202200076","DOIUrl":"10.1002/malq.202200076","url":null,"abstract":"<p>We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isomorphism. As an application, we show that the theory of any <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8].</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200076","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84346103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pregeometry over locally o-minimal structures and dimension","authors":"Masato Fujita","doi":"10.1002/malq.202200069","DOIUrl":"10.1002/malq.202200069","url":null,"abstract":"<p>We define a discrete closure operator for definably complete locally o-minimal structures <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math>. The pair of the underlying set of <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it <math>\u0000 <semantics>\u0000 <mo>discl</mo>\u0000 <annotation>$operatorname{discl}$</annotation>\u0000 </semantics></math>-dimension. A definable set <i>X</i> is of dimension equal to the <math>\u0000 <semantics>\u0000 <mo>discl</mo>\u0000 <annotation>$operatorname{discl}$</annotation>\u0000 </semantics></math>-dimension of <i>X</i>. The structure <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> also coincides with its dimension.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76113023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bisimulations and bisimulation games between Verbrugge models","authors":"Sebastijan Horvat, Tin Perkov, Mladen Vuković","doi":"10.1002/malq.202200042","DOIUrl":"https://doi.org/10.1002/malq.202200042","url":null,"abstract":"<p>Interpretability logic is a modal formalization of relative interpretability between first-order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call w-bisimulation. We show that the new notion, while keeping the basic property that bisimilarity implies modal equivalence, is weak enough to allow the converse to hold in the finitary case. To do this, we develop and use an appropriate notion of bisimulation games between Verbrugge models.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50126665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The permutations with n non-fixed points and the subsets with n elements of a set","authors":"Supakun Panasawatwong, Pimpen Vejjajiva","doi":"10.1002/malq.202300005","DOIUrl":"https://doi.org/10.1002/malq.202300005","url":null,"abstract":"<p>We write <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {S}_n(mathfrak {a})$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>a</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$[mathfrak {a}]^n$</annotation>\u0000 </semantics></math> for the cardinalities of the set of permutations with <i>n</i> non-fixed points and the set of subsets with <i>n</i> elements, respectively, of a set which is of cardinality <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, where <i>n</i> is a natural number greater than 1. With the Axiom of Choice, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {S}_n(mathfrak {a})$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>a</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$[mathfrak {a}]^n$</annotation>\u0000 </semantics></math> are equal for all infinite cardinals <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>. We show, in <span>ZF</span>, that if <math>\u0000 <semantics>\u0000 <msub>\u0000 <mtext>AC</mtext>\u0000 <mrow>\u0000 <mo>≤</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mbox{textsf {AC}}_{le n}$</annotation>\u0000 </semantics></math> is assumed, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>a</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>≤</mo>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50126677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Borel complexity and Ramsey largeness of sets of oracles separating complexity classes","authors":"Alex Creiner, Stephen Jackson","doi":"10.1002/malq.202200068","DOIUrl":"https://doi.org/10.1002/malq.202200068","url":null,"abstract":"<p>We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>≠</mo>\u0000 <mi>NP</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {P}ne mathbf {NP}$</annotation>\u0000 </semantics></math> with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>IP</mi>\u0000 <mo>=</mo>\u0000 <mi>PSPACE</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {IP} = mathbf {PSPACE}$</annotation>\u0000 </semantics></math>, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “large” using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating <math>\u0000 <semantics>\u0000 <mi>NP</mi>\u0000 <annotation>$mathbf {NP}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>co</mi>\u0000 <mi>-</mi>\u0000 <mi>NP</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {co}text{-}mathbf {NP}$</annotation>\u0000 </semantics></math> is not small, and obtain similar results for the separation of <math>\u0000 <semantics>\u0000 <mi>PSPACE</mi>\u0000 <annotation>$mathbf {PSPACE}$</annotation>\u0000 </semantics></math> from <math>\u0000 <semantics>\u0000 <mi>PH</mi>\u0000 <annotation>$mathbf {PH}$</annotation>\u0000 </semantics></math> along with the separation of <math>\u0000 <semantics>\u0000 <mi>NP</mi>\u0000 <annotation>$mathbf {NP}$</annotation>\u0000 </semantics></math> from <math>\u0000 <semantics>\u0000 <mi>BQP</mi>\u0000 <annotation>$mathbf {BQP}$</annotation>\u0000 </semantics></math>. We also show that the set of oracles <i>equating</i> <math>\u0000 <semantics>\u0000 <mi>IP</mi>\u0000 <annotation>$mathbf {IP}$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mi>PSPACE</mi>\u0000 <annotation>$mathbf {PSPACE}$</annotation>\u0000 </semantics></math> is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50128497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On self-distributive weak Heyting algebras","authors":"Mohsen Nourany, Shokoofeh Ghorbani, Arsham Borumand Saeid","doi":"10.1002/malq.202200073","DOIUrl":"https://doi.org/10.1002/malq.202200073","url":null,"abstract":"<p>We use the left self-distributive axiom to introduce and study a special class of weak Heyting algebras, called self-distributive weak Heyting algebras (SDWH-algebras). We present some useful properties of SDWH-algebras and obtain some equivalent conditions of them. A characteristic of SDWH-algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH-algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH-algebras (RWH-algebras), and the variety of transitive WH-algebras (TWH-algebras).</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50117486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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