CST Part IB [4] Computation Theory
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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. -
The Consistency of Arithmetic
The Consistency of Arithmetic Timothy Y. Chow July 11, 2018 In 2010, Vladimir Voevodsky, a Fields Medalist and a professor at the Insti- tute for Advanced Study, gave a lecture entitled, “What If Current Foundations of Mathematics Are Inconsistent?” Voevodsky invited the audience to consider seriously the possibility that first-order Peano arithmetic (or PA for short) was inconsistent. He briefly discussed two of the standard proofs of the consistency of PA (about which we will say more later), and explained why he did not find either of them convincing. He then said that he was seriously suspicious that an inconsistency in PA might someday be found. About one year later, Voevodsky might have felt vindicated when Edward Nelson, a professor of mathematics at Princeton University, announced that he had a proof not only that PA was inconsistent, but that a small fragment of primitive recursive arithmetic (PRA)—a system that is widely regarded as implementing a very modest and “safe” subset of mathematical reasoning—was inconsistent [11]. However, a fatal error in his proof was soon detected by Daniel Tausk and (independently) Terence Tao. Nelson withdrew his claim, remarking that the consistency of PA remained “an open problem.” For mathematicians without much training in formal logic, these claims by Voevodsky and Nelson may seem bewildering. While the consistency of some axioms of infinite set theory might be debatable, is the consistency of PA really “an open problem,” as Nelson claimed? Are the existing proofs of the con- sistency of PA suspect, as Voevodsky claimed? If so, does this mean that we cannot be sure that even basic mathematical reasoning is consistent? This article is an expanded version of an answer that I posted on the Math- Overflow website in response to the question, “Is PA consistent? do we know arXiv:1807.05641v1 [math.LO] 16 Jul 2018 it?” Since the question of the consistency of PA seems to come up repeat- edly, and continues to generate confusion, a more extended discussion seems worthwhile. -
The Cyber Entscheidungsproblem: Or Why Cyber Can’T Be Secured and What Military Forces Should Do About It
THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier JCSP 41 PCEMI 41 Exercise Solo Flight Exercice Solo Flight Disclaimer Avertissement Opinions expressed remain those of the author and Les opinons exprimées n’engagent que leurs auteurs do not represent Department of National Defence or et ne reflètent aucunement des politiques du Canadian Forces policy. This paper may not be used Ministère de la Défense nationale ou des Forces without written permission. canadiennes. Ce papier ne peut être reproduit sans autorisation écrite. © Her Majesty the Queen in Right of Canada, as © Sa Majesté la Reine du Chef du Canada, représentée par represented by the Minister of National Defence, 2015. le ministre de la Défense nationale, 2015. CANADIAN FORCES COLLEGE – COLLÈGE DES FORCES CANADIENNES JCSP 41 – PCEMI 41 2014 – 2015 EXERCISE SOLO FLIGHT – EXERCICE SOLO FLIGHT THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier “This paper was written by a student “La présente étude a été rédigée par un attending the Canadian Forces College stagiaire du Collège des Forces in fulfilment of one of the requirements canadiennes pour satisfaire à l'une des of the Course of Studies. The paper is a exigences du cours. L'étude est un scholastic document, and thus contains document qui se rapporte au cours et facts and opinions, which the author contient donc des faits et des opinions alone considered appropriate and que seul l'auteur considère appropriés et correct for the subject. -
A Probabilistic Anytime Algorithm for the Halting Problem
Computability 7 (2018) 259–271 259 DOI 10.3233/COM-170073 IOS Press A probabilistic anytime algorithm for the halting problem Cristian S. Calude Department of Computer Science, University of Auckland, Auckland, New Zealand [email protected] www.cs.auckland.ac.nz/~cristian Monica Dumitrescu Faculty of Mathematics and Computer Science, Bucharest University, Romania [email protected] http://goo.gl/txsqpU The first author dedicates his contribution to this paper to the memory of his collaborator and friend S. Barry Cooper (1943–2015). Abstract. TheHaltingProblem,the most (in)famous undecidableproblem,has important applications in theoretical andapplied computer scienceand beyond, hencethe interest in its approximate solutions. Experimental results reportedonvarious models of computationsuggest that haltingprogramsare not uniformly distributed– running times play an important role.A reason is thataprogram whicheventually stopsbut does not halt “quickly”, stops ata timewhich is algorithmically compressible. In this paperweworkwith running times to defineaclass of computable probability distributions on theset of haltingprograms in ordertoconstructananytimealgorithmfor theHaltingproblem withaprobabilisticevaluationofthe errorofthe decision. Keywords: HaltingProblem,anytimealgorithm, running timedistribution 1. Introduction The Halting Problem asks to decide, from a description of an arbitrary program and an input, whether the computation of the program on that input will eventually stop or continue forever. In 1936 Alonzo Church, and independently Alan Turing, proved that (in Turing’s formulation) an algorithm to solve the Halting Problem for all possible program-input pairs does not exist; two equivalent models have been used to describe computa- tion by algorithms (an informal notion), the lambda calculus by Church and Turing machines by Turing. The Halting Problem is historically the first proved undecidable problem; it has many applications in mathematics, logic and theoretical as well as applied computer science, mathematics, physics, biology, etc. -
Arxiv:1311.3168V22 [Math.LO] 24 May 2021 the Foundations Of
The Foundations of Mathematics in the Physical Reality Doeko H. Homan May 24, 2021 It is well-known that the concept set can be used as the foundations for mathematics, and the Peano axioms for the set of all natural numbers should be ‘considered as the fountainhead of all mathematical knowledge’ (Halmos [1974] page 47). However, natural numbers should be defined, thus ‘what is a natural number’, not ‘what is the set of all natural numbers’. Set theory should be as intuitive as possible. Thus there is no ‘empty set’, and a single shoe is not a singleton set but an individual, a pair of shoes is a set. In this article we present an axiomatic definition of sets with individuals. Natural numbers and ordinals are defined. Limit ordinals are ‘first numbers’, that is a first number of the Peano axioms. For every natural number m we define ‘ωm-numbers with a first number’. Every ordinal is an ordinal with a first ω0-number. Ordinals with a first number satisfy the Peano axioms. First ωω-numbers are defined. 0 is a first ωω-number. Then we prove a first ωω-number notequal 0 belonging to ordinal γ is an impassable barrier for counting down γ to 0 in a finite number of steps. 1 What is a set? At an early age you develop the idea of ‘what is a set’. You belong to a arXiv:1311.3168v22 [math.LO] 24 May 2021 family or a clan, you belong to the inhabitants of a village or a particular region. Experience shows there are objects that constitute a set. -
The Continuum Hypothesis, Part I, Volume 48, Number 6
fea-woodin.qxp 6/6/01 4:39 PM Page 567 The Continuum Hypothesis, Part I W. Hugh Woodin Introduction of these axioms and related issues, see [Kanamori, Arguably the most famous formally unsolvable 1994]. problem of mathematics is Hilbert’s first prob- The independence of a proposition φ from the axioms of set theory is the arithmetic statement: lem: ZFC does not prove φ, and ZFC does not prove φ. ¬ Cantor’s Continuum Hypothesis: Suppose that Of course, if ZFC is inconsistent, then ZFC proves X R is an uncountable set. Then there exists a bi- anything, so independence can be established only ⊆ jection π : X R. by assuming at the very least that ZFC is consis- → tent. Sometimes, as we shall see, even stronger as- This problem belongs to an ever-increasing list sumptions are necessary. of problems known to be unsolvable from the The first result concerning the Continuum (usual) axioms of set theory. Hypothesis, CH, was obtained by Gödel. However, some of these problems have now Theorem (Gödel). Assume ZFC is consistent. Then been solved. But what does this actually mean? so is ZFC + CH. Could the Continuum Hypothesis be similarly solved? These questions are the subject of this ar- The modern era of set theory began with Cohen’s ticle, and the discussion will involve ingredients discovery of the method of forcing and his appli- from many of the current areas of set theoretical cation of this new method to show: investigation. Most notably, both Large Cardinal Theorem (Cohen). Assume ZFC is consistent. Then Axioms and Determinacy Axioms play central roles. -
Self-Referential Basis of Undecidable Dynamics: from the Liar Paradox and the Halting Problem to the Edge of Chaos
Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos Mikhail Prokopenko1, Michael Harre´1, Joseph Lizier1, Fabio Boschetti2, Pavlos Peppas3;4, Stuart Kauffman5 1Centre for Complex Systems, Faculty of Engineering and IT The University of Sydney, NSW 2006, Australia 2CSIRO Oceans and Atmosphere, Floreat, WA 6014, Australia 3Department of Business Administration, University of Patras, Patras 265 00, Greece 4University of Pennsylvania, Philadelphia, PA 19104, USA 5University of Pennsylvania, USA [email protected] Abstract In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical sys- tems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these com- putational frameworks. Some of these interconnections are well-known, while some are clarified in this study as a result of a fine-grained comparison between recursive formal systems, Turing machines, and Cellular Automata (CAs). In particular, we elaborate on the diagonalization argument applied to distributed computation carried out by CAs, illustrating the key elements of Godel’s¨ proof for CAs. The comparative analysis emphasizes three factors which underlie the capacity to generate undecidable dynamics within the examined computational frameworks: (i) the program-data duality; (ii) the potential to access an infinite computational medium; and (iii) the ability to im- plement negation. The considered adaptations of Godel’s¨ proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability. 1 Introduction It is well-known that there are deep connections between dynamical systems, algorithms, and formal systems. -
Halting Problems: a Historical Reading of a Paradigmatic Computer Science Problem
PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol Halting problems: a historical reading of a paradigmatic computer science problem Liesbeth De Mol1 and Edgar G. Daylight2 1 CNRS, UMR 8163 Savoirs, Textes, Langage, Universit´e de Lille, [email protected] 2 Siegen University, [email protected] 1 PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol and E.G. Daylight Introduction (1) \The improbable symbolism of Peano, Russel, and Whitehead, the analysis of proofs by flowcharts spearheaded by Gentzen, the definition of computability by Church and Turing, all inventions motivated by the purest of mathematics, mark the beginning of the computer revolution. Once more, we find a confirmation of the sentence Leonardo jotted despondently on one of those rambling sheets where he confided his innermost thoughts: `Theory is the captain, and application the soldier.' " (Metropolis, Howlett and Rota, 1980) Introduction 2 PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol and E.G. Daylight Introduction (2) Why is this `improbable' symbolism considered relevant in comput- ing? ) Different non-excluding answers... 1. (the socio-historical answers) studying social and institutional developments in computing to understand why logic, or, theory, was/is considered to be the captain (or not) e.g. need for logic framed in CS's struggle for disciplinary identity and independence (cf (Tedre 2015)) 2. (the philosophico-historical answers) studying history of computing on a more technical level to understand why and how logic (or theory) are in- troduced in the computing practices { question: is there something to com- puting which makes logic epistemologically relevant to it? ) significance of combining the different answers (and the respective approaches they result in) ) In this talk: focus on paradigmatic \problem" of (theoretical) computer science { the halting problem Introduction 3 PROGRAMme Autumn workshop 2018, Bertinoro L. -
Lambda Calculus and the Decision Problem
Lambda Calculus and the Decision Problem William Gunther November 8, 2010 Abstract In 1928, Alonzo Church formalized the λ-calculus, which was an attempt at providing a foundation for mathematics. At the heart of this system was the idea of function abstraction and application. In this talk, I will outline the early history and motivation for λ-calculus, especially in recursion theory. I will discuss the basic syntax and the primary rule: β-reduction. Then I will discuss how one would represent certain structures (numbers, pairs, boolean values) in this system. This will lead into some theorems about fixed points which will allow us have some fun and (if there is time) to supply a negative answer to Hilbert's Entscheidungsproblem: is there an effective way to give a yes or no answer to any mathematical statement. 1 History In 1928 Alonzo Church began work on his system. There were two goals of this system: to investigate the notion of 'function' and to create a powerful logical system sufficient for the foundation of mathematics. His main logical system was later (1935) found to be inconsistent by his students Stephen Kleene and J.B.Rosser, but the 'pure' system was proven consistent in 1936 with the Church-Rosser Theorem (which more details about will be discussed later). An application of λ-calculus is in the area of computability theory. There are three main notions of com- putability: Hebrand-G¨odel-Kleenerecursive functions, Turing Machines, and λ-definable functions. Church's Thesis (1935) states that λ-definable functions are exactly the functions that can be “effectively computed," in an intuitive way. -
Undecidable Problems: a Sampler (.Pdf)
UNDECIDABLE PROBLEMS: A SAMPLER BJORN POONEN Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathemat- ics. 1. Introduction The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. The criterion for selection of a problem in this survey is simply that the author finds it entertaining! We do not pretend that our list of undecidable problems is complete in any sense. And some of the problems we consider turn out to be decidable or to have unknown decidability status. For another survey of undecidable problems, see [Dav77]. 2. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number @0 strictly between @0 and 2 , is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [G¨od40,Coh63, Coh64].) The first examples of statements independent of a \natural" axiom system were constructed by K. G¨odel[G¨od31]. 2. Decision problem: A family of problems with YES/NO answers is called unde- cidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert's tenth problem, to decide whether a mul- tivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 2.1. -
Lambda Calculus and Computation 6.037 – Structure and Interpretation of Computer Programs
Lambda Calculus and Computation 6.037 { Structure and Interpretation of Computer Programs Benjamin Barenblat [email protected] Massachusetts Institute of Technology With material from Mike Phillips, Nelson Elhage, and Chelsea Voss January 30, 2019 Benjamin Barenblat 6.037 Lambda Calculus and Computation : Build a calculating machine that gives a yes/no answer to all mathematical questions. Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Theorem (Church, Turing, 1936): These models of computation can't solve every problem. Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928) Benjamin Barenblat 6.037 Lambda Calculus and Computation Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Theorem (Church, Turing, 1936): These models of computation can't solve every problem. Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928): Build a calculating machine that gives a yes/no answer to all mathematical questions. Benjamin Barenblat 6.037 Lambda Calculus and Computation Theorem (Church, Turing, 1936): These models of computation can't solve every problem. Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928): Build a calculating machine that gives a yes/no answer to all mathematical questions. Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Benjamin Barenblat 6.037 Lambda Calculus and Computation Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928): Build a calculating machine that gives a yes/no answer to all mathematical questions. Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Theorem (Church, Turing, 1936): These models of computation can't solve every problem. -
THE PEANO AXIOMS 1. Introduction We Begin Our Exploration
CHAPTER 1: THE PEANO AXIOMS MATH 378, CSUSM. SPRING 2015. 1. Introduction We begin our exploration of number systems with the most basic number system: the natural numbers N. Informally, natural numbers are just the or- dinary whole numbers 0; 1; 2;::: starting with 0 and continuing indefinitely.1 For a formal description, see the axiom system presented in the next section. Throughout your life you have acquired a substantial amount of knowl- edge about these numbers, but do you know the reasons behind your knowl- edge? Why is addition commutative? Why is multiplication associative? Why does the distributive law hold? Why is it that when you count a finite set you get the same answer regardless of the order in which you count the elements? In this and following chapters we will systematically prove basic facts about the natural numbers in an effort to answer these kinds of ques- tions. Sometimes we will encounter more than one answer, each yielding its own insights. You might see an informal explanation and then a for- mal explanation, or perhaps you will see more than one formal explanation. For instance, there will be a proof for the commutative law of addition in Chapter 1 using induction, and then a more insightful proof in Chapter 2 involving the counting of finite sets. We will use the axiomatic method where we start with a few axioms and build up the theory of the number systems by proving that each new result follows from earlier results. In the first few chapters of these notes there will be a strong temptation to use unproved facts about arithmetic and numbers that are so familiar to us that they are practically part of our mental DNA.