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“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
Hubert Kennedy Eight Mathematical Biographies
Hubert Kennedy Eight Mathematical Biographies Peremptory Publications San Francisco 2002 © 2002 by Hubert Kennedy Eight Mathematical Biographies is a Peremptory Publications ebook. It may be freely distributed, but no changes may be made in it. Comments and suggestions are welcome. Please write to [email protected] . 2 Contents Introduction 4 Maria Gaetana Agnesi 5 Cesare Burali-Forti 13 Alessandro Padoa 17 Marc-Antoine Parseval des Chênes 19 Giuseppe Peano 22 Mario Pieri 32 Emil Leon Post 35 Giovanni Vailati 40 3 Introduction When a Dictionary of Scientific Biography was planned, my special research interest was Giuseppe Peano, so I volunteered to write five entries on Peano and his friends/colleagues, whose work I was investigating. (The DSB was published in 14 vol- umes in 1970–76, edited by C. C. Gillispie, New York: Charles Scribner's Sons.) I was later asked to write two more entries: for Parseval and Emil Leon Post. The entry for Post had to be done very quickly, and I could not have finished it without the generous help of one of his relatives. By the time the last of these articles was published in 1976, that for Giovanni Vailati, I had come out publicly as a homosexual and was involved in the gay liberation movement. But my article on Vailati was still discreet. If I had written it later, I would probably have included evidence of his homosexuality. The seven articles for the Dictionary of Scientific Biography have a uniform appear- ance. (The exception is the article on Burali-Forti, which I present here as I originally wrote it—with reference footnotes. -
Church's Thesis and the Conceptual Analysis of Computability
Church’s Thesis and the Conceptual Analysis of Computability Michael Rescorla Abstract: Church’s thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing’s work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church’s thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing’s work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these syntactic entities with numbers. I argue that, in providing this correlation, we must demand that the correlation itself be computable. Otherwise, the Turing machine will compute uncomputable functions. But if we presuppose the intuitive notion of a computable relation between syntactic entities and numbers, then our analysis of computability is circular.1 §1. Turing machines and number-theoretic functions A Turing machine manipulates syntactic entities: strings consisting of strokes and blanks. I restrict attention to Turing machines that possess two key properties. First, the machine eventually halts when supplied with an input of finitely many adjacent strokes. Second, when the 1 I am greatly indebted to helpful feedback from two anonymous referees from this journal, as well as from: C. Anthony Anderson, Adam Elga, Kevin Falvey, Warren Goldfarb, Richard Heck, Peter Koellner, Oystein Linnebo, Charles Parsons, Gualtiero Piccinini, and Stewart Shapiro. I received extremely helpful comments when I presented earlier versions of this paper at the UCLA Philosophy of Mathematics Workshop, especially from Joseph Almog, D. -
Enumerations of the Kolmogorov Function
Enumerations of the Kolmogorov Function Richard Beigela Harry Buhrmanb Peter Fejerc Lance Fortnowd Piotr Grabowskie Luc Longpr´ef Andrej Muchnikg Frank Stephanh Leen Torenvlieti Abstract A recursive enumerator for a function h is an algorithm f which enu- merates for an input x finitely many elements including h(x). f is a aEmail: [email protected]. Department of Computer and Information Sciences, Temple University, 1805 North Broad Street, Philadelphia PA 19122, USA. Research per- formed in part at NEC and the Institute for Advanced Study. Supported in part by a State of New Jersey grant and by the National Science Foundation under grants CCR-0049019 and CCR-9877150. bEmail: [email protected]. CWI, Kruislaan 413, 1098SJ Amsterdam, The Netherlands. Partially supported by the EU through the 5th framework program FET. cEmail: [email protected]. Department of Computer Science, University of Mas- sachusetts Boston, Boston, MA 02125, USA. dEmail: [email protected]. Department of Computer Science, University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA. Research performed in part at NEC Research Institute. eEmail: [email protected]. Institut f¨ur Informatik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany. fEmail: [email protected]. Computer Science Department, UTEP, El Paso, TX 79968, USA. gEmail: [email protected]. Institute of New Techologies, Nizhnyaya Radi- shevskaya, 10, Moscow, 109004, Russia. The work was partially supported by Russian Foundation for Basic Research (grants N 04-01-00427, N 02-01-22001) and Council on Grants for Scientific Schools. hEmail: [email protected]. School of Computing and Department of Mathe- matics, National University of Singapore, 3 Science Drive 2, Singapore 117543, Republic of Singapore. -
Theory of Computation
A Universal Program (4) Theory of Computation Prof. Michael Mascagni Florida State University Department of Computer Science 1 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) Enumeration Theorem Definition. We write Wn = fx 2 N j Φ(x; n) #g: Then we have Theorem 4.6. A set B is r.e. if and only if there is an n for which B = Wn. Proof. This is simply by the definition ofΦ( x; n). 2 Note that W0; W1; W2;::: is an enumeration of all r.e. sets. 2 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) The Set K Let K = fn 2 N j n 2 Wng: Now n 2 K , Φ(n; n) #, HALT(n; n) This, K is the set of all numbers n such that program number n eventually halts on input n. 3 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) K Is r.e. but Not Recursive Theorem 4.7. K is r.e. but not recursive. Proof. By the universality theorem, Φ(n; n) is partially computable, hence K is r.e. If K¯ were also r.e., then by the enumeration theorem, K¯ = Wi for some i. We then arrive at i 2 K , i 2 Wi , i 2 K¯ which is a contradiction. -
Logique Combinatoire Et Physique Quantique
Eigenlogic Zeno Toffano [email protected] CentraleSupélec, Gif-sur-Yvette, France Laboratoire des Signaux et Systèmes, UMR8506-CNRS Université Paris-Saclay, France Zeno Toffano : “Eigenlogic” 1 UNILOG 2018 (6th World Congress and School on Universal Logic), Workshop Logic & Physics, Vichy (F), June 22, 2018 statement facts in quantum physics Quantum Physics started in 1900 with the quantification of radiation (Planck’s constant ℎ = 6.63 10−34 J.s) It is the most successful theory in physics and explains: nuclear energy, semiconductors, magnetism, lasers, quantum gravity… The theory has an increasing influence outside physics: computer science, AI, communications, biology, cognition… Nowadays there is a great quantum revival (second quantum revolution) which addresses principally: quantum entanglement (Bell inequalites, teleportation…), quantum superposition (Schrödinger cat), decoherence non-locality, non- separability, non-contextuality, non-classicality, non-… A more basic aspect is quantification: a measurement can only give one of the (real) eigenvalues of an observable. The associated eigenvector is the resulting quantum state. Measurements on non-eigenstates are indeterminate, the outcome probability is given by the Born rule. The eigenvalues are the spectrum (e.g. energies for the Hamiltonian). In physics it is natural to “work” in different representation eignespaces (of the considered observable) e.g: semiconductors “make sense” in the reciprocal lattice (momentum space 푝 , spatial Fourier transform of coordinate -
The Interplay Between Computability and Incomputability Draft 619.Tex
The Interplay Between Computability and Incomputability Draft 619.tex Robert I. Soare∗ January 7, 2008 Contents 1 Calculus, Continuity, and Computability 3 1.1 When to Introduce Relative Computability? . 4 1.2 Between Computability and Relative Computability? . 5 1.3 The Development of Relative Computability . 5 1.4 Turing Introduces Relative Computability . 6 1.5 Post Develops Relative Computability . 6 1.6 Relative Computability in Real World Computing . 6 2 Origins of Computability and Incomputability 6 2.1 G¨odel’s Incompleteness Theorem . 8 2.2 Incomputability and Undecidability . 9 2.3 Alonzo Church . 9 2.4 Herbrand-G¨odel Recursive Functions . 10 2.5 Stalemate at Princeton Over Church’s Thesis . 11 2.6 G¨odel’s Thoughts on Church’s Thesis . 11 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18–23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene,klee Alan Turing, Kurt G¨odel, Emil Post, computability, incomputabil- ity, undecidability, Church-Turing Thesis (CTT), Post-Church Second Thesis on relative computability, computable approximations, Limit Lemma, effectively continuous func- tions, computability in analysis, strong reducibilities. 1 3 Turing Breaks the Stalemate 12 3.1 Turing’s Machines and Turing’s Thesis . 12 3.2 G¨odel’s Opinion of Turing’s Work . 13 3.3 Kleene Said About Turing . 14 3.4 Church Said About Turing . 15 3.5 Naming the Church-Turing Thesis . 15 4 Turing Defines Relative Computability 17 4.1 Turing’s Oracle Machines . -
Introduction to the Theory of Computation Computability, Complexity, and the Lambda Calculus Some Notes for CIS262
Introduction to the Theory of Computation Computability, Complexity, And the Lambda Calculus Some Notes for CIS262 Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] c Jean Gallier Please, do not reproduce without permission of the author April 28, 2020 2 Contents Contents 3 1 RAM Programs, Turing Machines 7 1.1 Partial Functions and RAM Programs . 10 1.2 Definition of a Turing Machine . 15 1.3 Computations of Turing Machines . 17 1.4 Equivalence of RAM programs And Turing Machines . 20 1.5 Listable Languages and Computable Languages . 21 1.6 A Simple Function Not Known to be Computable . 22 1.7 The Primitive Recursive Functions . 25 1.8 Primitive Recursive Predicates . 33 1.9 The Partial Computable Functions . 35 2 Universal RAM Programs and the Halting Problem 41 2.1 Pairing Functions . 41 2.2 Equivalence of Alphabets . 48 2.3 Coding of RAM Programs; The Halting Problem . 50 2.4 Universal RAM Programs . 54 2.5 Indexing of RAM Programs . 59 2.6 Kleene's T -Predicate . 60 2.7 A Non-Computable Function; Busy Beavers . 62 3 Elementary Recursive Function Theory 67 3.1 Acceptable Indexings . 67 3.2 Undecidable Problems . 70 3.3 Reducibility and Rice's Theorem . 73 3.4 Listable (Recursively Enumerable) Sets . 76 3.5 Reducibility and Complete Sets . 82 4 The Lambda-Calculus 87 4.1 Syntax of the Lambda-Calculus . 89 4.2 β-Reduction and β-Conversion; the Church{Rosser Theorem . 94 4.3 Some Useful Combinators . -
Foundations of Mathematics
Foundations of Mathematics Fall 2019 ii Contents 1 Propositional Logic 1 1.1 The basic definitions . .1 1.2 Disjunctive Normal Form Theorem . .3 1.3 Proofs . .4 1.4 The Soundness Theorem . 11 1.5 The Completeness Theorem . 12 1.6 Completeness, Consistency and Independence . 14 2 Predicate Logic 17 2.1 The Language of Predicate Logic . 17 2.2 Models and Interpretations . 19 2.3 The Deductive Calculus . 21 2.4 Soundness Theorem for Predicate Logic . 24 3 Models for Predicate Logic 27 3.1 Models . 27 3.2 The Completeness Theorem for Predicate Logic . 27 3.3 Consequences of the completeness theorem . 31 4 Computability Theory 33 4.1 Introduction and Examples . 33 4.2 Finite State Automata . 34 4.3 Exercises . 37 4.4 Turing Machines . 38 4.5 Recursive Functions . 43 4.6 Exercises . 48 iii iv CONTENTS Chapter 1 Propositional Logic 1.1 The basic definitions Propositional logic concerns relationships between sentences built up from primitive proposition symbols with logical connectives. The symbols of the language of predicate calculus are 1. Logical connectives: ,&, , , : _ ! $ 2. Punctuation symbols: ( , ) 3. Propositional variables: A0, A1, A2,.... A propositional variable is intended to represent a proposition which can either be true or false. Restricted versions, , of the language of propositional logic can be constructed by specifying a subset of the propositionalL variables. In this case, let PVar( ) denote the propositional variables of . L L Definition 1.1.1. The collection of sentences, denoted Sent( ), of a propositional language is defined by recursion. L L 1. The basis of the set of sentences is the set PVar( ) of propositional variables of . -
Axioms" for Computational Complexity and Computation of Finite Functions
CORE Metadata, citation and similar papers at core.ac.uk Provided by Purdue E-Pubs Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1970 A Note on "Axioms" for Computational Complexity and Computation of Finite Functions Paul R. Young Report Number: 69-039 Young, Paul R., "A Note on "Axioms" for Computational Complexity and Computation of Finite Functions" (1970). Department of Computer Science Technical Reports. Paper 315. https://docs.lib.purdue.edu/cstech/315 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. A NOTE ON "AXIOMS" FOR COMPUTATIONAL COMPLEXITY AND COMPUTATION OF FINITE FUNCTIONS Paul R . Young April 1970 CSD TR 39 (Revised version) Computer Sciences Department Purdue University Lafayette, Indiana 47907 Short title: ON THE COMPUTATION OF FINITE FUNCTIONS List of Symbols Greek: \ Math: L * z , i, ABSTRACT Recent studies of computational complexity have focused on "axioms" which characterize the "difficul ty of a computation" (Blum , 1967a) or the measure of the "size of a program ," (Blum 1967b and Pager 1969). In this paper we w ish to carefully exam ine the consequences of hypothesizing a relation whi ch connects measures of size and of difficulty of computat ion. The relation is mot ivated by the fact that computations are performed "a few instructions at a time" so that if one has a bound on the difficulty of a comput a t ion, one also has a bound on the "number of executed instruct ions". -
The Entscheidungsproblem and Alan Turing
The Entscheidungsproblem and Alan Turing Author: Laurel Brodkorb Advisor: Dr. Rachel Epstein Georgia College and State University December 18, 2019 1 Abstract Computability Theory is a branch of mathematics that was developed by Alonzo Church, Kurt G¨odel,and Alan Turing during the 1930s. This paper explores their work to formally define what it means for something to be computable. Most importantly, this paper gives an in-depth look at Turing's 1936 paper, \On Computable Numbers, with an Application to the Entscheidungsproblem." It further explores Turing's life and impact. 2 Historic Background Before 1930, not much was defined about computability theory because it was an unexplored field (Soare 4). From the late 1930s to the early 1940s, mathematicians worked to develop formal definitions of computable functions and sets, and applying those definitions to solve logic problems, but these problems were not new. It was not until the 1800s that mathematicians began to set an axiomatic system of math. This created problems like how to define computability. In the 1800s, Cantor defined what we call today \naive set theory" which was inconsistent. During the height of his mathematical period, Hilbert defended Cantor but suggested an axiomatic system be established, and characterized the Entscheidungsproblem as the \fundamental problem of mathemat- ical logic" (Soare 227). The Entscheidungsproblem was proposed by David Hilbert and Wilhelm Ackerman in 1928. The Entscheidungsproblem, or Decision Problem, states that given all the axioms of math, there is an algorithm that can tell if a proposition is provable. During the 1930s, Alonzo Church, Stephen Kleene, Kurt G¨odel,and Alan Turing worked to formalize how we compute anything from real numbers to sets of functions to solve the Entscheidungsproblem. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Interactions between computability theory and set theory Permalink https://escholarship.org/uc/item/5hn678b1 Author Schweber, Noah Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Interactions between computability theory and set theory by Noah Schweber A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Antonio Montalban, Chair Professor Leo Harrington Professor Wesley Holliday Spring 2016 Interactions between computability theory and set theory Copyright 2016 by Noah Schweber i To Emma ii Contents Contents ii 1 Introduction1 1.1 Higher reverse mathematics ........................... 1 1.2 Computability in generic extensions....................... 3 1.3 Further results .................................. 5 2 Higher reverse mathematics, 1/28 2.1 Introduction.................................... 8 2.2 Reverse mathematics beyond type 1....................... 11 2.3 Separating clopen and open determinacy.................... 25 2.4 Conclusion..................................... 38 3 Higher reverse mathematics, 2/2 41 3 3.1 The strength of RCA0 ............................... 41 3.2 Choice principles ................................. 50 4 Computable structures in generic extensions 54 4.1 Introduction...................................