{"title":"2022 WINTER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC WITH THE AMS Seattle, Washington Joint Mathematics Meeting January 7–8, 2022","authors":"","doi":"10.1017/bsl.2022.25","DOIUrl":"https://doi.org/10.1017/bsl.2022.25","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"9 1","pages":"466 - 466"},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78218609","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}
{"title":"A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC","authors":"Emanuele Frittaion","doi":"10.1017/bsl.2022.23","DOIUrl":"https://doi.org/10.1017/bsl.2022.23","url":null,"abstract":"Abstract We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory \u0000$T_0$\u0000 extending \u0000$mathsf {RCA}_0$\u0000 and axiomatizable by a \u0000$Pi ^1_{k+2}$\u0000 sentence, and for any \u0000$ngeq k+1$\u0000 , \u0000$$begin{align*}T_0+ mathrm{RFN}_{varPi^1_{n+2}}(T) = T_0 + mathrm{TI}_{varPi^1_n}(varepsilon_0), end{align*}$$\u0000 \u0000$$begin{align*}T_0+ mathrm{RFN}_{varSigma^1_{n+1}}(T) = T_0+ mathrm{TI}_{varPi^1_n}(varepsilon_0)^{-}, end{align*}$$\u0000 where T is \u0000$T_0$\u0000 augmented with full induction, and \u0000$mathrm {TI}_{varPi ^1_n}(varepsilon _0)^{-}$\u0000 denotes the schema of transfinite induction up to \u0000$varepsilon _0$\u0000 for \u0000$varPi ^1_n$\u0000 formulas without set parameters.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"1 2 1","pages":"451 - 465"},"PeriodicalIF":0.0,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78419421","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}
Colloquium, A. Mickiewicz, Szymon Chlebowski, Andrzej Gajda, Marta Gawek, Patrycja Kupś, Paweł Łupkowski, Dawid Ratajczyk, Agata Tomczyk, A. Wasielewska, Joanna Golinska-Pilarek, L. Kolodziejczyk, M. Nasieniewski, J. Pogonowski, Tomasz F. Skura, K. Swirydowicz, M. Soskova, B. Monin, L. Ros
{"title":"2021 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM ’21 Adam Mickiewicz University Poznań, Poland July 19–24, 2021","authors":"Colloquium, A. Mickiewicz, Szymon Chlebowski, Andrzej Gajda, Marta Gawek, Patrycja Kupś, Paweł Łupkowski, Dawid Ratajczyk, Agata Tomczyk, A. Wasielewska, Joanna Golinska-Pilarek, L. Kolodziejczyk, M. Nasieniewski, J. Pogonowski, Tomasz F. Skura, K. Swirydowicz, M. Soskova, B. Monin, L. Ros","doi":"10.1017/bsl.2022.17","DOIUrl":"https://doi.org/10.1017/bsl.2022.17","url":null,"abstract":"of the invited 31st Annual Gödel Lecture ELISABETH BOUSCAREN, The ubiquity of configurations in model theory. CNRS—Université Paris-Saclay, Gif-sur-Yvette, France. E-mail: elisabeth.bouscaren@universite-paris-saclay.fr. Originally in Classification Theory, then in Geometric Stability, and now, beyond Stability, in Tame Model Theory, one common essential feature is the identification and study of some geometric configurations, of combinatorial and dimensional theoretic nature. They can witness the combinatorial and the model theoretic complexity of a theory or indicate the existence of specific definable algebraic structures. This enables model theory to tackle questions from very diverse subjects. We will attempt to illustrate the importance of these configurations through some examples. Abstract of invited tutorialsof invited tutorials KRZYSZTOF KRUPIŃSKI, Topological dynamics in model theory. University of Wrocław, Wrocław, Poland. E-mail: kkrup@math.uni.wroc.pl. Some fundamental notions and methods of topological dynamics were introduced to model theory by Newelski in the mid-2000s. In the first part of my tutorial, I will recall some basic notions of topological dynamics, discuss the flows which appear naturally in model theory (as various spaces of types), and give applications of basic topological dynamics to some group covering results of Newelski such as: if an א0-saturated group is covered by countably many 0-type-definable sets Xn , n ∈ , then for some finite A ⊆ G and n ∈ , G = AXnX –1 n . In the second part, I will define the Ellis semigroup and Ellis group of a flow, and focus on connections between the Ellis groups of natural flows in model theory and certain invariants of definable groups (quotients by model-theoretic connected components) or first order theories (Galois groups of first order theories as well as spaces of strong types). In particular, I will discuss the results of Pillay, Rzepecki, and myself which present certain invariants of this kind as quotients of compact (Hausdorff) groups (which are canonical Hausdorff quotients of Ellis groups). This has various consequences obtained by Pillay, Rzepecki, and myself, e.g., it leads to a general result that model-theoretic type-definability of a bounded invariant equivalence relation defined on a single complete type over ∅ is equivalent to descriptive set theoretic smoothness of this relation. 270 LOGIC COLLOQUIUM ’21 In the last part, I will discuss a definable variant of Kechris–Pestov–Todorčević (KPT) theory, developed by Lee, Moconja, and myself. KPT theory studies relationships between dynamical properties of the groups of automorphisms of Fraïssé structures and Ramseytheoretic (so combinatorial) properties of the underlying Fraïssé classes. In our research, the idea is to find interactions between dynamical properties of first order theories (i.e., properties related to the actions of the automorphism group of a sufficiently saturated model on various types spaces ove","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"83 1","pages":"268 - 318"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75972883","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}
{"title":"Uniform Properties of Ideals in Rings of Restricted Power Series","authors":"Madeline Grace Barnicle","doi":"10.1017/bsl.2020.26","DOIUrl":"https://doi.org/10.1017/bsl.2020.26","url":null,"abstract":"Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as \u0000$mathbb Q_{p}langle Xrangle $\u0000 , \u0000$mathbb Z_{p}langle Xrangle $\u0000 , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg. Abstract prepared by Madeline G. Barnicle. E-mail: barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"9 1","pages":"258 - 258"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84290436","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}
{"title":"A journey through computability, topology and analysis","authors":"Manlio Valenti","doi":"10.1017/bsl.2022.13","DOIUrl":"https://doi.org/10.1017/bsl.2022.13","url":null,"abstract":"Abstract This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and (effective) descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy. We first analyze the strength of the open and clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain \u0000$mathbb {N}$\u0000 or is single-valued. These notions proved to be extremely useful when exploring the Weihrauch degree of the problem \u0000$mathsf {DS}$\u0000 of computing descending sequences in ill-founded linear orders. They allow us to show that \u0000$mathsf {DS}$\u0000 , and the Weihrauch equivalent problem \u0000$mathsf {BS}$\u0000 of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize \u0000$mathsf {DS}$\u0000 and \u0000$mathsf {BS}$\u0000 by considering \u0000$boldsymbol {Gamma }$\u0000 -presented orders, where \u0000$boldsymbol {Gamma }$\u0000 is a Borel pointclass or \u0000$boldsymbol {Delta }^1_1$\u0000 , \u0000$boldsymbol {Sigma }^1_1$\u0000 , \u0000$boldsymbol {Pi }^1_1$\u0000 . We study the obtained \u0000$mathsf {DS}$\u0000 -hierarchy and \u0000$mathsf {BS}$\u0000 -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace \u0000$mathbf {K}([0,1])$\u0000 of compact subsets of \u0000$[0,1]$\u0000 and show that the closed Salem sets form a \u0000$boldsymbol {Pi }^0_3$\u0000 -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in \u0000$mathbf {K}([0,1]^d)$\u0000 . We also generalize the results by relaxing the compactness of the ambient space and sho","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"42 1","pages":"266 - 267"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80848571","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}
{"title":"Paraconsistent Logic Programming in Three and Four-Valued Logics","authors":"Kleidson Êglicio Carvalho da Silva Oliveira","doi":"10.1017/bsl.2021.34","DOIUrl":"https://doi.org/10.1017/bsl.2021.34","url":null,"abstract":"Abstract From the interaction among areas such as Computer Science, Formal Logic, and Automated Deduction arises an important new subject called Logic Programming. This has been used continuously in the theoretical study and practical applications in various fields of Artificial Intelligence. After the emergence of a wide variety of non-classical logics and the understanding of the limitations presented by first-order classical logic, it became necessary to consider logic programming based on other types of reasoning in addition to classical reasoning. A type of reasoning that has been well studied is the paraconsistent, that is, the reasoning that tolerates contradictions. However, although there are many paraconsistent logics with different types of semantics, their application to logic programming is more delicate than it first appears, requiring an in-depth study of what can or cannot be transferred directly from classical first-order logic to other types of logic. Based on studies of Tarcisio Rodrigues on the foundations of Paraconsistent Logic Programming (2010) for some Logics of Formal Inconsistency (LFIs), this thesis intends to resume the research of Rodrigues and place it in the specific context of LFIs with three- and four-valued semantics. This kind of logics are interesting from the computational point of view, as presented by Luiz Silvestrini in his Ph.D. thesis entitled “A new approach to the concept of quase-truth” (2011), and by Marcelo Coniglio and Martín Figallo in the article “Hilbert-style presentations of two logics associated to tetravalent modal algebras” [Studia Logica (2012)]. Based on original techniques, this study aims to define well-founded systems of paraconsistent logic programming based on well-known logics, in contrast to the ad hoc approaches to this question found in the literature. Abstract prepared by Kleidson Êglicio Carvalho da Silva Oliveira. E-mail: kecso10@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322632","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"86 1","pages":"260 - 260"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84846068","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}
{"title":"Non-Deterministic Matrices: Theory and Applications to Algebraic Semantics","authors":"A. C. Golzio","doi":"10.1017/bsl.2021.35","DOIUrl":"https://doi.org/10.1017/bsl.2021.35","url":null,"abstract":"Abstract We call multioperation any operation that return for even argument a set of values instead of a single value. Through multioperations we can define an algebraic structure equipped with at least one multioperation. This kind of structure is called multialgebra. The study of them began in 1934 with the publication of a paper of Marty. In the realm of Logic, multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) and used as semantics tool for characterizing some logics which cannot be characterized by a single finite matrix. Carnielli and Coniglio introduced the semantics of swap structures for LFIs (Logics of Formal Inconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron’s semantics. In this thesis, we will introduce a new method of algebraization of logics based on multialgebras and swap structures that is similar to classical algebraization method of Lindenbaum-Tarski, but more extensive because it can be applied to systems such that some operators are non-congruential. In particular, this method will be applied to a family of non-normal modal logics and to some LFIs that are not algebraizable by the very general techniques introduced by Blok and Pigozzi. We also will obtain representation theorems for some LFIs and we will prove that, within out approach, the classes of swap structures for some axiomatic extensions of mbC are a subclass of the class of swap structures for the logic mbC. Abstract prepared by Ana Claudia de Jesus Golzio. E-mail: anaclaudiagolzio@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322436","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"53 1","pages":"260 - 261"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87018721","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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