Computable Diagonalizations and Turing's Cardinality Paradox
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Epistemological Consequences of the Incompleteness Theorems
Epistemological Consequences of the Incompleteness Theorems Giuseppe Raguní UCAM - Universidad Católica de Murcia, Avenida Jerónimos 135, Guadalupe 30107, Murcia, Spain - [email protected] After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main episte- mological consequences of the first incompleteness Theorem for the two funda- mental arithmetical theories are shown: the non-mechanizability for the truths of the first-order arithmetic and the peculiarities for the model of the second- order arithmetic. Finally, the common epistemological interpretation of the second incompleteness Theorem is corrected, proposing the new Metatheorem of undemonstrability of internal consistency. KEYWORDS: semantics, languages, epistemology, paradoxes, arithmetic, in- completeness, first-order, second-order, consistency. 1 Semantics in the Languages Consider an arbitrary language that, as normally, makes use of a countable1 number of characters. Combining these characters in certain ways, are formed some fundamental strings that we call terms of the language: those collected in a dictionary. When the terms are semantically interpreted, i. e. a certain meaning is assigned to them, we have their distinction in adjectives, nouns, verbs, etc. Then, a proper grammar establishes the rules arXiv:1602.03390v1 [math.GM] 13 Jan 2016 of formation of sentences. While the terms are finite, the combinations of grammatically allowed terms form an infinite-countable amount of possible sentences. In a non-trivial language, the meaning associated to each term, and thus to each ex- pression that contains it, is not always unique. The same sentence can enunciate different things, so representing different propositions. -
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”. -
Algorithms, Turing Machines and Algorithmic Undecidability
U.U.D.M. Project Report 2021:7 Algorithms, Turing machines and algorithmic undecidability Agnes Davidsdottir Examensarbete i matematik, 15 hp Handledare: Vera Koponen Examinator: Martin Herschend April 2021 Department of Mathematics Uppsala University Contents 1 Introduction 1 1.1 Algorithms . .1 1.2 Formalisation of the concept of algorithms . .1 2 Turing machines 3 2.1 Coding of machines . .4 2.2 Unbounded and bounded machines . .6 2.3 Binary sequences representing real numbers . .6 2.4 Examples of Turing machines . .7 3 Undecidability 9 i 1 Introduction This paper is about Alan Turing's paper On Computable Numbers, with an Application to the Entscheidungsproblem, which was published in 1936. In his paper, he introduced what later has been called Turing machines as well as a few examples of undecidable problems. A few of these will be brought up here along with Turing's arguments in the proofs but using a more modern terminology. To begin with, there will be some background on the history of why this breakthrough happened at that given time. 1.1 Algorithms The concept of an algorithm has always existed within the world of mathematics. It refers to a process meant to solve a problem in a certain number of steps. It is often repetitive, with only a few rules to follow. In more recent years, the term also has been used to refer to the rules a computer follows to operate in a certain way. Thereby, an algorithm can be used in a plethora of circumstances. The word might describe anything from the process of solving a Rubik's cube to how search engines like Google work [4]. -
The Strength of Mac Lane Set Theory
The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret -
Type Theory and Applications
Type Theory and Applications Harley Eades [email protected] 1 Introduction There are two major problems growing in two areas. The first is in Computer Science, in particular software engineering. Software is becoming more and more complex, and hence more susceptible to software defects. Software bugs have two critical repercussions: they cost companies lots of money and time to fix, and they have the potential to cause harm. The National Institute of Standards and Technology estimated that software errors cost the United State's economy approximately sixty billion dollars annually, while the Federal Bureau of Investigations estimated in a 2005 report that software bugs cost U.S. companies approximately sixty-seven billion a year [90, 108]. Software bugs have the potential to cause harm. In 2010 there were a approximately a hundred reports made to the National Highway Traffic Safety Administration of potential problems with the braking system of the 2010 Toyota Prius [17]. The problem was that the anti-lock braking system would experience a \short delay" when the brakes where pressed by the driver of the vehicle [106]. This actually caused some crashes. Toyota found that this short delay was the result of a software bug, and was able to repair the the vehicles using a software update [91]. Another incident where substantial harm was caused was in 2002 where two planes collided over Uberlingen¨ in Germany. A cargo plane operated by DHL collided with a passenger flight holding fifty-one passengers. Air-traffic control did not notice the intersecting traffic until less than a minute before the collision occurred. -
On the Progression of Finding the Smallest Known Universal Turing Machine
On the progression of finding the smallest known universal Turing machine Fakult¨atf¨urElektrotechnik und Informatik der Gottfried Wilhelm Leibniz Universit¨atHannover Institut f¨urTheoretische Informatik Abschlussarbeit zur Erlangung des akademischen Grades Bachelor of Science vorgelegt von Laura Strieker 10004978 im Februar 2020 Erstprufer:¨ Prof. Dr. H. Vollmer Zweitprufer:¨ Dr. A. Meier Betreuer: Prof. Dr. H. Vollmer Erkl¨arungder Selbstst¨andigkeit Hiermit versichere ich, die vorliegende Bachelorarbeit selbstst¨andigverfasst und keine an- deren als die angegebenen Quellen und Hilfsmittel verwendet zu haben. Die Arbeit hat in gleicher oder ¨ahnlicher Form noch keinem anderen Pr¨ufungsamt vorgelegen. Hannover, den 21. Februar 2020 Vorname Nachname Contents 1 Introduction1 2 Fundamentals3 2.1 Universal Turing machines . .4 2.2 Notation . .4 2.2.1 Symbols, States and Transitions . .5 2.2.2 Transition tables . .5 3 The first universal Turing machine7 3.1 The first definition of Turing machines . .7 3.2 Standard Description . .7 3.3 Turing’s UTM . .8 3.3.1 Rough outline . .8 3.3.2 Detailed description . .9 4 The idea of small universal Turing machines 11 4.1 A universal Turing machine with 2 states . 11 4.2 Impossibility of a one state universal Turing machine . 13 4.3 A universal Turing machine with two tape symbols . 13 5 A competition arises 15 5.1 Watanabe’s contribution . 15 5.1.1 The (5,8)-UTM . 15 5.1.2 The (5,6)-UTM . 17 5.2 Minsky’s contribution . 17 5.2.1 Tag systems . 17 5.2.2 Universality through tag systems . 18 5.2.3 The (6,6)-UTM . -
METALOGIC METALOGIC an Introduction to the Metatheory of Standard First Order Logic
METALOGIC METALOGIC An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter Senior Lecturer in the Department of Logic and Metaphysics University of St Andrews PALGRA VE MACMILLAN © Geoffrey Hunter 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published 1971 by MACMILLAN AND CO LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 11589 9 (hard cover) 333 11590 2 (paper cover) ISBN 978-0-333-11590-9 ISBN 978-1-349-15428-9 (eBook) DOI 10.1007/978-1-349-15428-9 The Papermac edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent, in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. To my mother and to the memory of my father, Joseph Walter Hunter Contents Preface xi Part One: Introduction: General Notions 1 Formal languages 4 2 Interpretations of formal languages. Model theory 6 3 Deductive apparatuses. Formal systems. Proof theory 7 4 'Syntactic', 'Semantic' 9 5 Metatheory. The metatheory of logic 10 6 Using and mentioning. Object language and metalang- uage. Proofs in a formal system and proofs about a formal system. Theorem and metatheorem 10 7 The notion of effective method in logic and mathematics 13 8 Decidable sets 16 9 1-1 correspondence. -
Estimating Algorithmic Information Using Quantum Computing for Genomics Applications
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 February 2021 doi:10.20944/preprints202102.0135.v1 Article Estimating Algorithmic Information using Quantum Computing for Genomics Applications Aritra Sarkar1* 0000-0002-3026-6892, Zaid Al-Ars1 0000-0001-7670-8572 and Koen Bertels2 0000-0001-9310-4885 1 Department of Quantum & Computer Engineering, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands 2 Department of Informatics Engineering, Faculty of Engineering, University of Porto, Porto, Portugal * Correspondence: [email protected] Abstract: Inferring algorithmic structure in data is essential for discovering causal generative models. In this research, we present a quantum computing framework using the circuit model, for estimating algorithmic information metrics. The canonical computation model of the Turing machine is restricted in time and space resources, to make the target metrics computable under realistic assumptions. The universal prior distribution for the automata is obtained as a quantum superposition, which is further conditioned to estimate the metrics. Specific cases are explored where the quantum implementation offers polynomial advantage, in contrast to an indispensable exhaustive enumeration in the corresponding classical case. The unstructured output data and the computational irreducibility of Turing machines make this algorithm impossible to approximate using heuristics. Thus, exploring the space of program-output relations is one of the most promising problems for demonstrating quantum supremacy using Grover search that cannot be dequantized. Experimental use cases for quantum acceleration are developed for self-replicating programs and algorithmic complexity of short strings. With quantum computing hardware rapidly attaining technological maturity, we discuss how this framework will have significant advantage for various genomics applications in meta-biology, phylogenetic tree analysis, protein-protein interaction mapping and synthetic biology. -
Logical Foundations of Physics. Part I. Zeno's Dichotomy Paradox
LOGICAL FOUNDATIONS OF PHYSICS. PART I. ZENO'S DICHOTOMY PARADOX, SUPERTASKS AND THE FAILURE OF CLASSICAL INFINITARY REASONING RADHAKRISHNAN SRINIVASAN∗ C-113, Symphony CHS Ltd. Jain Mandir Marg, Nahar's Amrit Shakti Township Chandivali, Mumbai 400072, India rk [email protected] March 5, 2019 Abstract Several variants of Zeno's dichotomy paradox are considered, with the objective of exploring the logical foundations of physics. It is shown that Zeno's dichotomy paradox leads to contradictions at the metamathemat- ical (as opposed to formal) level in the basic classical infinitary reasoning that is routinely used in theoretical physics. Both Newtonian mechanics and special relativity theory suffer from these metamathematical incon- sistencies, which occur essentially because the classical refutation of the dichotomy paradox requires supertasks to be completed. In previous pa- pers, Non-Aristotelian Finitary Logic (NAFL) was proposed as a logical foundation for some of the basic principles of quantum mechanics, such as, quantum superposition and entanglement. We outline how the fini- tistic and paraconsistent reasoning used in NAFL helps in resolving the metamathematical inconsistencies that arise from the dichotomy paradox. 1 Zeno's dichotomy paradox Probably the first serious challenge to the use of classical infinitary reasoning in physics came from Zeno's paradoxes of motion, in particular, the dichotomy paradox, originally proposed by Zeno in the fifth century BC [1, 2, 3]. Zeno's dichotomy paradox has remained completely unresolved for over 2400 years till date, despite claims to the contrary. And contrary to conventional wisdom, ∗Permanent address (for correspondence): 1102 Tejas Heights, Plot 245, Mumbai Tamil Sangam Marg, Sion East, Mumbai 400022, India. -
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. -
Axiomatic Set Theory As a Basis for the Construction of Mathematics"
"Axiomatic set theory as a basis for the construction of mathematics"# A diosertatien submitted in support of candidature for the Master of Science degree of the University of London (Faculty of Science) by BRIAN ROTMAN March, 1962% ir • ProQuest Number: 10097259 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10097259 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT It is widely known that one of the major tasks of •Foundations* is to construct a formal system which can he said to contain the whole of mathematics# For various reasons axiomatic set theory is a very suitable • choice for such a system and it is one which has proved acceptable to both logicians and mathematicians# The particular demands of mathematicians and logicians, however, are not the same# As a result there exist at the moment two different formulations of set theory idiich can be rou^ily said to cater for logicians and mathematicians respectively*# It is these systems which are the subject of this dissertation# The -
Samplestanford University, Stanford, CA 94305 Turing Machines Copyright C 2021 by the Author
pdf version of the entry Turing Machines https://plato.stanford.edu/archives/fall2021/entries/turing-machine/ from the Fall 2021 Edition of the Stanford Encyclopedia of Philosophy Edward N. Zalta Uri Nodelman Principal Editor Senior Editor Colin Allen Hannah Kim Paul Oppenheimer Associate Editor Associate Editor Associate Editor Faculty Sponsors: R. Lanier Anderson & Thomas Icard Editorial Board: https://plato.stanford.edu/board.html Library of Congress ISSN: 1095-5054 Notice: This PDF version was distributed by request to mem- bers of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the SEP Society and obtain authorized PDF versions of SEP entries, please visit https://leibniz.stanford.edu/friends/ . Stanford Encyclopedia of Philosophy Copyright c 2021 by the publisher The Metaphysics Research Lab Department of Philosophy SAMPLEStanford University, Stanford, CA 94305 Turing Machines Copyright c 2021 by the author Liesbeth De Mol All rights reserved. Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/ Turing Machines First published Mon Sep 24, 2018 Turing machines, first described by Alan Turing in Turing 1936–7, are simple abstract computational devices intended to help investigate the extent and limitations of what can be computed. Turing’s ‘automatic machines’, as he termed them in 1936, were specifically devised for the computing of real numbers. They were first named ‘Turing machines’ by Alonzo Church in a review of Turing’s paper (Church 1937). Today, they are considered to be one of the foundational models of computability and (theoretical) computer science.[1] 1.