Computability and Analysis: the Legacy of Alan Turing
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Calcium: Computing in Exact Real and Complex Fields Fredrik Johansson
Calcium: computing in exact real and complex fields Fredrik Johansson To cite this version: Fredrik Johansson. Calcium: computing in exact real and complex fields. ISSAC ’21, Jul 2021, Virtual Event, Russia. 10.1145/3452143.3465513. hal-02986375v2 HAL Id: hal-02986375 https://hal.inria.fr/hal-02986375v2 Submitted on 15 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Calcium: computing in exact real and complex fields Fredrik Johansson [email protected] Inria Bordeaux and Institut Math. Bordeaux 33400 Talence, France ABSTRACT This paper presents Calcium,1 a C library for exact computa- Calcium is a C library for real and complex numbers in a form tion in R and C. Numbers are represented as elements of fields suitable for exact algebraic and symbolic computation. Numbers Q¹a1;:::; anº where the extension numbers ak are defined symbol- ically. The system constructs fields and discovers algebraic relations are represented as elements of fields Q¹a1;:::; anº where the exten- automatically, handling algebraic and transcendental number fields sion numbers ak may be algebraic or transcendental. The system combines efficient field operations with automatic discovery and in a unified way. -
Arxiv:1803.01386V4 [Math.HO] 25 Jun 2021
2009 SEKI http://wirth.bplaced.net/seki.html ISSN 1860-5931 arXiv:1803.01386v4 [math.HO] 25 Jun 2021 A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publi- Working-Paper cation, and 2 CVof Gisbert Hasenjaeger Claus-Peter Wirth Dept. of Computer Sci., Saarland Univ., 66123 Saarbrücken, Germany [email protected] SEKI Working-Paper SWP–2017–01 SEKI SEKI is published by the following institutions: German Research Center for Artificial Intelligence (DFKI GmbH), Germany • Robert Hooke Str.5, D–28359 Bremen • Trippstadter Str. 122, D–67663 Kaiserslautern • Campus D 3 2, D–66123 Saarbrücken Jacobs University Bremen, School of Engineering & Science, Campus Ring 1, D–28759 Bremen, Germany Universität des Saarlandes, FR 6.2 Informatik, Campus, D–66123 Saarbrücken, Germany SEKI Editor: Claus-Peter Wirth E-mail: [email protected] WWW: http://wirth.bplaced.net Please send surface mail exclusively to: DFKI Bremen GmbH Safe and Secure Cognitive Systems Cartesium Enrique Schmidt Str. 5 D–28359 Bremen Germany This SEKI Working-Paper was internally reviewed by: Wilfried Sieg, Carnegie Mellon Univ., Dept. of Philosophy Baker Hall 161, 5000 Forbes Avenue Pittsburgh, PA 15213 E-mail: [email protected] WWW: https://www.cmu.edu/dietrich/philosophy/people/faculty/sieg.html A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publication, and two CV of Gisbert Hasenjaeger Claus-Peter Wirth Dept. of Computer Sci., Saarland Univ., 66123 Saarbrücken, Germany [email protected] First Published: March 4, 2018 Thoroughly rev. & largely extd. (title, §§ 2, 3, and 4, CV, Bibliography, &c.): Jan. -
Addiction in Contemporary Mathematics FINAL 2020
Addiction in Contemporary Mathematics Newcomb Greenleaf Goddard College [email protected] Abstract 1. The Agony of Constructive Math 2. Reductio and Truth 3. Excluding the Middle 4. Brouwer Seen from Princeton 5. Errett Bishop 6. Brouwer Encodes Ignorance 7. Constructive Sets 8. Kicking the Classical Habit 9. Schizophrenia 10. Constructive Mathematics and Algorithms 11. The Intermediate Value Theorem 12. Real Numbers and Complexity 13. Natural Deduction 14. Mathematical Pluralism 15. Addiction in History of Math Abstract In 1973 Errett Bishop (1928-1983) gave the title Schizophrenia in Contemporary Mathematics to his Colloquium Lectures at the summer meetings of the AMS (American Mathematical Society). It marked the end of his seven-year campaign for Constructive mathematics, an attempt to introduce finer distinctions into mathematical thought. Seven years earlier Bishop had been a young math superstar, able to launch his revolution with an invited address at the International 1 Congress of Mathematicians, a quadrennial event held in Cold War Moscow in 1966. He built on the work of L. E. J. Brouwer (1881-1966), who had discovered in 1908 that Constructive mathematics requires a more subtle logic, in which truth has a positive quality stronger than the negation of falsity. By 1973 Bishop had accepted that most of his mathematical colleagues failed to understand even the simplest Constructive mathematics, and came up with a diagnosis of schizophrenia for their disability. We now know, from Andrej Bauer’s 2017 article, Five Stages of Accepting Constructive Mathematics that a much better diagnosis would have been addiction, whence our title. While habits can be kicked, schizophrenia tends to last a lifetime, which made Bishop’s misdiagnosis a very serious mistake. -
The Proper Explanation of Intuitionistic Logic: on Brouwer's Demonstration
The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem Göran Sundholm, Mark van Atten To cite this version: Göran Sundholm, Mark van Atten. The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem. van Atten, Mark Heinzmann, Gerhard Boldini, Pascal Bourdeau, Michel. One Hundred Years of Intuitionism (1907-2007). The Cerisy Conference, Birkhäuser, pp.60- 77, 2008, 978-3-7643-8652-8. halshs-00791550 HAL Id: halshs-00791550 https://halshs.archives-ouvertes.fr/halshs-00791550 Submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem Göran Sundholm Philosophical Institute, Leiden University, P.O. Box 2315, 2300 RA Leiden, The Netherlands. [email protected] Mark van Atten SND (CNRS / Paris IV), 1 rue Victor Cousin, 75005 Paris, France. [email protected] Der … geführte Beweis scheint mir aber trotzdem . Basel: Birkhäuser, 2008, 60–77. wegen der in seinem Gedankengange enthaltenen Aussagen Interesse zu besitzen. (Brouwer 1927B, n. 7)1 Brouwer’s demonstration of his Bar Theorem gives rise to provocative ques- tions regarding the proper explanation of the logical connectives within intu- itionistic and constructivist frameworks, respectively, and, more generally, re- garding the role of logic within intuitionism. -
Computability on the Real Numbers
4. Computability on the Real Numbers Real numbers are the basic objects in analysis. For most non-mathematicians a real number is an infinite decimal fraction, for example π = 3•14159 ... Mathematicians prefer to define the real numbers axiomatically as follows: · (R, +, , 0, 1,<) is, up to isomorphism, the only Archimedean ordered field satisfying the axiom of continuity [Die60]. The set of real numbers can also be constructed in various ways, for example by means of Dedekind cuts or by completion of the (metric space of) rational numbers. We will neglect all foundational problems and assume that the real numbers form a well-defined set R with all the properties which are proved in analysis. We will denote the R real line topology, that is, the set of all open subsets of ,byτR. In Sect. 4.1 we introduce several representations of the real numbers, three of which (and the equivalent ones) will survive as useful. We introduce a rep- n n resentation ρ of R by generalizing the definition of the main representation ρ of the set R of real numbers. In Sect. 4.2 we discuss the computable real numbers. Sect. 4.3 is devoted to computable real functions. We show that many well known functions are computable, and we show that partial sum- mation of sequences is computable and that limit operator on sequences of real numbers is computable, if a modulus of convergence is given. We also prove a computability theorem for power series. Convention 4.0.1. We still assume that Σ is a fixed finite alphabet con- taining all the symbols we will need. -
On the Topological Aspects of the Theory of Represented Spaces∗
On the topological aspects of the theory of represented spaces∗ Arno Pauly Clare College University of Cambridge, United Kingdom [email protected] Represented spaces form the general setting for the study of computability derived from Turing ma- chines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by ESCARDO´ ,SCHRODER¨ , and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and – closely linked to these – properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property. 1 Introduction ∗ Just as numberings provide a tool to transfer computability from N (or f0;1g ) to all sorts of countable structures; representations provide a means to introduce computability on structures of the cardinality of the continuum based on computability on Cantor space f0;1gN. This is of course essential for com- n m k n m putable analysis [66], dealing with spaces such as R, C (R ;R ), C (R ;R ) or general Hilbert spaces [8]. Computable measure theory (e.g. [65, 58, 44, 30, 21]), or computability on the set of countable ordinals [39, 48] likewise rely on representations as foundation. Essentially, by equipping a set with a representation, we arrive at a represented space – to some extent1 the most general structure carrying a notion of computability that is derived from Turing ma- chines2. -
Arxiv:2007.07560V4 [Math.LO] 8 Oct 2020
ON THE UNCOUNTABILITY OF R DAG NORMANN AND SAM SANDERS Abstract. Cantor’s first set theory paper (1874) establishes the uncountabil- ity of R based on the following: for any sequence of reals, there is another real different from all reals in the sequence. The latter (in)famous result is well-studied and has been implemented as an efficient computer program. By contrast, the status of the uncountability of R is not as well-studied, and we therefore investigate the logical and computational properties of NIN (resp. NBI) the statement there is no injection (resp. bijection) from [0, 1] to N. While intuitively weak, NIN (and similar for NBI) is classified as rather strong on the ‘normal’ scale, both in terms of which comprehension axioms prove NIN and which discontinuous functionals compute (Kleene S1-S9) the real numbers from NIN from the data. Indeed, full second-order arithmetic is essential in each case. To obtain a classification in which NIN and NBI are weak, we explore the ‘non-normal’ scale based on (classically valid) continuity axioms and non- normal functionals, going back to Brouwer. In doing so, we derive NIN and NBI from basic theorems, like Arzel`a’s convergence theorem for the Riemann inte- gral (1885) and central theorems from Reverse Mathematics formulated with the standard definition of ‘countable set’ involving injections or bijections to N. Thus, the uncountability of R is a corollary to basic mainstream mathemat- ics; NIN and NBI are even (among) the weakest principles on the non-normal scale, which serendipitously reproves many of our previous results. -
E.W. Beth Als Logicus
E.W. Beth als logicus ILLC Dissertation Series 2000-04 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ E.W. Beth als logicus Verbeterde electronische versie (2001) van: Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof.dr. J.J.M. Franse ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op dinsdag 26 september 2000, te 10.00 uur door Paul van Ulsen geboren te Diemen Promotores: prof.dr. A.S. Troelstra prof.dr. J.F.A.K. van Benthem Faculteit Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Copyright c 2000 by P. van Ulsen Printed and bound by Print Partners Ipskamp. ISBN: 90{5776{052{5 Ter nagedachtenis aan mijn vader v Inhoudsopgave Dankwoord xi 1 Inleiding 1 2 Levensloop 11 2.1 Beginperiode . 12 2.1.1 Leerjaren . 12 2.1.2 Rijpingsproces . 14 2.2 Universitaire carriere . 21 2.2.1 Benoeming . 21 2.2.2 Geleerde genootschappen . 23 2.2.3 Redacteurschappen . 31 2.2.4 Beth naar Berkeley . 32 2.3 Beth op het hoogtepunt van zijn werk . 33 2.3.1 Instituut voor Grondslagenonderzoek . 33 2.3.2 Oprichting van de Centrale Interfaculteit . 37 2.3.3 Logici en historici aan Beths leiband . -
Intuitionistic Completeness of First-Order Logic with Respect to Uniform Evidence Semantics
Intuitionistic Completeness of First-Order Logic Robert Constable and Mark Bickford October 15, 2018 Abstract We establish completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable if and only if its embedding into minimal logic, mFOL, is uniformly valid under the Brouwer Heyting Kolmogorov (BHK) semantics, the intended semantics of iFOL and mFOL. Our proof is intuitionistic and provides an effective procedure Prf that converts uniform minimal evidence into a formal first-order proof. We have implemented Prf. Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuitionistically valid, but not conversely. Our strongest result requires the Fan Theorem; it can also be proved classically by showing that Prf terminates using K¨onig’s Theorem. The fundamental idea behind our completeness theorem is that a single evidence term evd wit- nesses the uniform validity of a minimal logic formula F. Finding even one uniform realizer guarantees validity because Prf(F,evd) builds a first-order proof of F, establishing its uniform validity and providing a purely logical normalized realizer. We establish completeness for iFOL as follows. Friedman showed that iFOL can be embedded in minimal logic (mFOL) by his A-transformation, mapping formula F to F A. If F is uniformly valid, then so is F A, and by our completeness theorem, we can find a proof of F A in minimal logic. Then we intuitionistically prove F from F F alse, i.e. by taking False for A and for ⊥ of mFOL. -
From Constructive Mathematics to Computable Analysis Via the Realizability Interpretation
From Constructive Mathematics to Computable Analysis via the Realizability Interpretation Vom Fachbereich Mathematik der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Math. Peter Lietz aus Mainz Referent: Prof. Dr. Thomas Streicher Korreferent: Dr. Alex Simpson Tag der Einreichung: 22. Januar 2004 Tag der m¨undlichen Pr¨ufung: 11. Februar 2004 Darmstadt 2004 D17 Hiermit versichere ich, dass ich diese Dissertation selbst¨andig verfasst und nur die angegebenen Hilfsmittel verwendet habe. Peter Lietz Abstract Constructive mathematics is mathematics without the use of the principle of the excluded middle. There exists a wide array of models of constructive logic. One particular interpretation of constructive mathematics is the realizability interpreta- tion. It is utilized as a metamathematical tool in order to derive admissible rules of deduction for systems of constructive logic or to demonstrate the equiconsistency of extensions of constructive logic. In this thesis, we employ various realizability mod- els in order to logically separate several statements about continuity in constructive mathematics. A trademark of some constructive formalisms is predicativity. Predicative logic does not allow the definition of a set by quantifying over a collection of sets that the set to be defined is a member of. Starting from realizability models over a typed version of partial combinatory algebras we are able to show that the ensuing models provide the features necessary in order to interpret impredicative logics and type theories if and only if the underlying typed partial combinatory algebra is equivalent to an untyped pca. It is an ongoing theme in this thesis to switch between the worlds of classical and constructive mathematics and to try and use constructive logic as a method in order to obtain results of interest also for the classically minded mathematician. -
On the Computational Complexity of Algebraic Numbers: the Hartmanis–Stearns Problem Revisited
On the computational complexity of algebraic numbers : the Hartmanis-Stearns problem revisited Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec To cite this version: Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec. On the computational complexity of alge- braic numbers : the Hartmanis-Stearns problem revisited. Transactions of the American Mathematical Society, American Mathematical Society, 2020, pp.3085-3115. 10.1090/tran/7915. hal-01254293 HAL Id: hal-01254293 https://hal.archives-ouvertes.fr/hal-01254293 Submitted on 12 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON THE COMPUTATIONAL COMPLEXITY OF ALGEBRAIC NUMBERS: THE HARTMANIS–STEARNS PROBLEM REVISITED by Boris Adamczewski, Julien Cassaigne & Marion Le Gonidec Abstract. — We consider the complexity of integer base expansions of alge- braic irrational numbers from a computational point of view. We show that the Hartmanis–Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-b expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one. -
Abraham Robinson 1918–1974
NATIONAL ACADEMY OF SCIENCES ABRAHAM ROBINSON 1918–1974 A Biographical Memoir by JOSEPH W. DAUBEN Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoirs, VOLUME 82 PUBLISHED 2003 BY THE NATIONAL ACADEMY PRESS WASHINGTON, D.C. Courtesy of Yale University News Bureau ABRAHAM ROBINSON October 6, 1918–April 11, 1974 BY JOSEPH W. DAUBEN Playfulness is an important element in the makeup of a good mathematician. —Abraham Robinson BRAHAM ROBINSON WAS BORN on October 6, 1918, in the A Prussian mining town of Waldenburg (now Walbrzych), Poland.1 His father, Abraham Robinsohn (1878-1918), af- ter a traditional Jewish Talmudic education as a boy went on to study philosophy and literature in Switzerland, where he earned his Ph.D. from the University of Bern in 1909. Following an early career as a journalist and with growing Zionist sympathies, Robinsohn accepted a position in 1912 as secretary to David Wolfson, former president and a lead- ing figure of the World Zionist Organization. When Wolfson died in 1915, Robinsohn became responsible for both the Herzl and Wolfson archives. He also had become increas- ingly involved with the affairs of the Jewish National Fund. In 1916 he married Hedwig Charlotte (Lotte) Bähr (1888- 1949), daughter of a Jewish teacher and herself a teacher. 1Born Abraham Robinsohn, he later changed the spelling of his name to Robinson shortly after his arrival in London at the beginning of World War II. This spelling of his name is used throughout to distinguish Abby Robinson the mathematician from his father of the same name, the senior Robinsohn.