Showing papers in "The Bulletin of Symbolic Logic in 2019"

Foundations of online structure theory

Abstract: The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

Large cardinals beyond choice

Abstract: The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

Games for Functions: Baire Classes, Weihrauch Degrees, Transfinite Computations, and Ranks

Abstract: Game characterizations of classes of functions in descriptive set theory have their origins in the seminal work of Wadge, with further developments by several others. In this thesis we study such characterizations from several perspectives. We define modifications of Semmes's game characterization of the Borel functions, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$. We also define a construction of games which transforms a game characterizing a class $\Lambda$ of functions into a game characterizing the class of functions which are piecewise $\Lambda$ on a countable partition by $\Pi^0_\alpha$ sets, for each $0 < \alpha < \omega_1$. We then define a parametrized Wadge game by using computable analysis, and show how the parameters affect the class of functions that is characterized by the game. As an application, we recast our games characterizing the Baire classes into this framework. Furthermore, we generalize our game characterizations of function classes to generalized Baire spaces, show how the notion of computability on Baire space can be transferred to generalized Baire spaces, and show that this is appropriate for computable analysis by defining a representation of Galeotti's generalized real line and analyzing the Weihrauch degree of the intermediate value theorem for that space. Finally, we show how the game characterizations of function classes discussed lead in a natural way to a stratification of each class into a hierarchy, intuitively measuring the complexity of functions in that class. This idea and the results presented open new paths for further research.

A conjectural classification of strongly dependent fields

Abstract: We survey the history of Shelah’s conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.

The Theory of the Generalised Real Numbers and Other Topics in Logic

Abstract: prepared by Andrea Vaccaro. E-mail: vaccaro@post.bgu.ac.il URL: https://etd.adm.unipi.it/theses/available/etd-04042019-113706/ Lorenzo Galeotti, The Theory of the Generalised Real Numbers and Other Topics in Logic, Universität Hamburg, Germany, 2019. Supervised by Benedikt Löwe. MSC: 03E15, 03D60, 03F30, 03C55.

Model Theory and Machine Learning

Abstract: We study a connection between model theory and machine learning by way of common combinatorial properties. In various settings of machine learning, combinatorial properties of the concept class being learned dictate whether the class is learnable, and if so, how long or how much data is required. In model theory, combinatorial properties can give rise to dividing lines, which create classes of theories in which a structure theory can be developed to varying degrees. Model theory and machine learning share several common combinatorial properties. The first known connection was between PAC learning and NIP theories by way of VC dimension, which led to subsequent interaction between the two fields. In this thesis, we describe a broad connection between stability and several forms of exact learning. The key combinatorial property is Littlestone dimension. This property has been known to both model theory and machine learning for decades (although by a different name in model theory), although it had not been previously pointed out that the connection existed. Finite Littlestone dimension classifies online learning, in which a learner classifies sequentially presented data. Finite Littlestone dimension also classifies stable theories, a model-theoretic class in which a rich structure theory has been developed. We extend this connection to query learning, in which the learner attempts to identify the target concept by guessing it exactly and receiving feedback to its guesses. We also consider compression schemes, where sets must be encoded by a bounded number of its elements and reconstructed by one of several reconstruction functions. The boundary between finite and infinite Littlestone dimension that corresponds to learnability and non-learnability has many similarities to the boundary between finite and infinite VC dimension. In both settings, one can define a counting function called the shatter function, and the relevant dimension controls the growth rate of this function. In particular, the appropriate Sauer-Shelah Lemma gives a polynomial bound if the relevant dimension is finite, while the function is exponential if the relevant dimension is infinite. We develop a framework for proving bounds in a uniform way and apply it in similar settings.

An extension of a theorem of zermelo

Abstract: We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E'). This extends Zermelo's 1930 theorem about second order ZFC.

Indirect proof and inversions of syllogisms

Abstract: Abstract By considering the new notion of the inverses of syllogisms such as Barbara and Celarent, we show how the rule of Indirect Proof, in the form (no multiple or vacuous discharges) used by Aristotle, may be dispensed with, in a system comprising four basic rules of subalternation or conversion and six basic syllogisms.

Circulating miR-221 and miR-222 as Potential Biomarkers for Screening of Breast Cancer

Abstract: Jungho Kim, Sehee Oh, Sunyoung Park, Sungwoo Ahn, Yeonim Choi, Geehyuk Kim, Seung Il Kim and Hyeyoung Lee† Department of Biomedical Laboratory Science, College of Health Sciences, Yonsei University, Wonju, Gangwon 26493, Korea Department of Microbiology, Institute of Immunology and Immunological Disease, College of Medicine, Yonsei University, Seoul 03722, Korea DNA Analysis Section, Division of Forensic Medicine, Busan Institute, National Forensic Service, Yangsan 50612, Korea Department of Biomedical Laboratory Science, Songho College, Hoengseong, Gangwon 25242, Korea Department of Surgery, College of Medicine, Yonsei University, Seoul 03722, Korea

Completeness of finite-rank differential varieties

Abstract: Differential algebraic geometry offers tantalizing similarities to the algebraic version as well as puzzling anomalies. This thesis builds on results of Kolchin, Blum, Morrison, van den Dries, and Pong to study the problem of completeness for projective differential varieties. The classical fundamental theorem of elimination theory asserts that if V is a projective algebraic variety defined over an algebraically closed field K and W is any algebraic variety defined over K, then the projection VxW -> W takes Zariski-closed sets to Zariski-closed sets. Differential varieties defined by differential polynomial equations over a differentially closed field are more complicated. We give the first example of an incomplete finite-rank differential variety, as well as new instances of complete differential varieties. We also explain how model theory yields multiple versions of Pong's valuative criterion for completeness and reduces the differential completeness problem to one involving algebraic varieties over the complex numbers.

Eta-rules in Martin-löf type theory

Abstract: The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Lof type theory. They are, however, justified by the meaning explanations, and a higher order eta rule is part of that type theory. The main aim of this article is to clarify this somewhat puzzling situation. It will be argued that lower order eta rules do not, whereas the higher order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory.

Explicit computational paths in type theory

Abstract: The treatment of an equality as a type in type theory gives rise to an interesting type known as identity type. The idea is that, given terms a, b of a type A, one may form the type IdA(a, b), whose elements are proofs that a and b are equal elements of typeA. A term of this type, p : IdA(a, b), makes up for the grounds (or proof) that establishes that a is indeed equal to b. Many interesting results have been achieved using the identity type. One of these was the discovery of the Univalent Models in 2005 by Vladimir Voevodsky. A groundbreaking result has arisen from Voevodsky’s work: the connection between type theory and homotopy theory. The intuitive connection is simple: a term a : A can be considered as a point of the space A and p : IdA(a, b) is a homotopy path between points a, b ∈ A. This semantical interpretation has given rise to a whole new area of research known as Homotopy Type Theory. Inspired by the path-based approach of the homotopy interpretation, we propose that a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a computational path. The idea is that a term p : IdA(a, b) will be a computational path between terms a, b : A. With that in mind, our work has three main objectives. The first one is the proposal of computational paths as a new entity of type theory. In this proposal, we point out the fact that computational paths should be seen as the syntax counterpart of the homotopical paths between terms of a type. We also propose a formalization of the identity type using computational paths. The second objective is the proposal of a mathematical structure for a type using computational paths. We show that using categorical semantics it is possible to induce a groupoid structure for a type and also a higher groupoid structure, using computational paths and a rewrite system.We use this groupoid structure to prove that computational paths also refute the uniqueness of identity proofs. The last objective is to formulate and prove the main concepts and building blocks of homotopy type theory, now using terms which represent (explicit) computational paths. We wrap up this last objective with a proof of the isomorphism between the fundamental group of the circle and the group of the integers. c © 2019, Association for Symbolic Logic 1079-8986/19/2502-0005 DOI:10.1017/bsl.2019.2

Nominalistic ordinals, recursion on higher types, and finitism

Abstract: In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to e 0 , which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.

Realizing Realizability Results with Classical Constructions

Abstract: J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e. symmetric extensions. We also provide a new condition for preserving well-ordered, and other particular type of choice, in the general settings of symmetric extensions.

PRL-owski „sukces” Anieli Gruszeckiej

Abstract: This article discusses the life, career and work of Aniela Gruszecka. The author attempts to characterize her writing career after 1945, highlighting its successful turns. According to the author the fact that Gruszecka was married to the famous linguist Kazimierz Nitsch may have had an impact on her work. While the criticism addresses mainly the most famous work by this long forgotten writer – Przygoda w nieznanym kraju, the author of the article analyzes two unpublished novels – Wschodnie skrzydło and Geografia serdeczna.

Regular Tree Languages in the First Two Levels of the Borel Hierarchy

Abstract: The thesis focuses on a quite recent research field lying in between Descriptive Set Theory and Automata Theory (for infinite objects). In both areas, one is often concerned with subsets of the Cantor space or of its homeomorphic copies. In Descriptive Set Theory, such subsets are usually stratified in topological hierarchies, like the Borel hierarchy, the Wadge hierarchy and the difference hierarchy; in Automata Theory, such sets are studied in terms of regularity, that is, the property of being recognised by an automaton or, equivalently, of being expressible in Monadic Second-Order Logic. This double point of view leads to many interesting questions about the interplay and relationship between topological complexity and regularity. While we have a complete picture of what happens in the case of automata on words, the case of automata on trees is still a terra incognita. Some results have already been obtained for particular classes of languages, like Büchi languages, deterministic languages, and unambiguous languages. In this thesis we instead drop any restriction and prove some new results concerning arbitrary regular tree languages which belong to low levels of the Borel hierarchy and of the Wadge hierarchy. In particular we prove the following: Theorem 1. A regular tree language L is recognised by a weak-alternating automaton that uses only two priorities if and only if it is in the first level of the Borel hierarchy. Then we prove some results about slightly higher levels of the Wadge hierarchy that can be summed up by the following theorem: Theorem 2. The following holds: 1. Let Γ be a Wadge degree with finite Wadge rank. Then it is decidable if a regular tree language L belongs to Γ. 2. It is decidable if a regular tree language L is a Boolean combination of open sets. 3. It is decidable if a regular tree language L is in the Borel class Δ2. Finally, we give a complete characterisation of the second level of the Borel hierarchy: Theorem 3. It is decidable if a regular tree language L belongs to the second level of the Borel hierarchy. Moreover, a regular language L is in the second level of the Borel hierarchy if and only if it is recognised by a weak-alternating automaton that uses exactly three priorities. Abstract prepared by Filippo Cavallari. E-mail: filcavallari88@gmail.com URL: http://filippocavallari.altervista.org/wp-content/uploads/2018/07/prepared by Filippo Cavallari. E-mail: filcavallari88@gmail.com URL: http://filippocavallari.altervista.org/wp-content/uploads/2018/07/