DOCSLIB.ORG

Enumeration Order Equivalency

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Enumeration Order Equivalency Enumeration Order Equivalency Ali Akbar Safilian (AmirKabir University of Technology, Tehran, Iran [email protected] ) Farzad Didehvar (AmirKabir University of Technology, Tehran, Iran [email protected] ) Let ̻ ɛ ͈ be an infinite recursively enumerable set. There are some total computable functions ͜: ͈ Ŵ ̻ connected with a Turing machine such that ̻ Ɣ ʜ͜ʚ1ʛ, ͜ʚ2ʛ,… ʝ. Definition 2.1 (Listing) A listing of an infinite r.e. set ̻ ɛ ͈ is a bijective computable function ͚: ͈ Ŵ ̻. As mentioned above we can connect every Turing machine with a listing uniquely. Namely, each listing shows the enumeration order of the elements enumerated by the related Turing machine. In the following, we define a reduction on listings and sets based on enumeration orders. Definition 2.2 (Enumeration Order Reducibility on Listings and sets) 1. For listings ͚, ͛: ͈ Ŵ ͈ we say ͚ is “Enumeration order reducible” to ͛ and write ͚ ƙ * ͛, if and only if, Ǣ͝ Ɨ ͞, (͚ʚ͝ʛ Ƙ ͚ʚ͞ʛ ƴ ͛ʚ͝ʛ Ƙ ͛ʚ͞ʛʛ. 2. For r.e. sets ̻, ̼ ɛ ͈, we say ̻ is “Enumeration order reducible” to ̼ and write ̻ ƙ * ̼, if and only if, for any listing ͛ of ̼ there exist some computable function ̀ (from the listings of ̼ to the listings of ̻) such that ̀ ʚ͛ʛ ƙ * ͛. 3. For two listings ͚, ͛ (r.e. sets ̻, ̼), we say ͚ is “Enumeration order equivalent” to ͛ and write ͚ ȸ * ͛ (̻ ȸ * ̼) ,if and only if, both ͚ ƙ * ͛ and ͛ ƙ * ͚ (̻ ƙ * ̼ and ̼ ƙ * ̻) Let ̻ be an r.e. subset of ͈. ʞ̻ʟ * Ɣ ʜ̼ ɛ ͈| ̻ ȸ * ̼ʝ denotes the Enumeration order equivalence class of ̻. Also we call the equivalence classes of enumeration order equivalent sets “Enumeration order degree” and write ͙͛͘ʚ̻ʛ for an r.e. set ̻. The relation ƙ * is preorder, because it has transitive and reflexive properties but it is not a partial order because ̻ ƙ * ̼ and ̼ ƙ * ̻ does not necessarily imply ̻ Ɣ ̼. Lemma 2.3 Any decidable set is enumeration order reducible to every r.e. set. Proof: It is a clear fact that there exists a listing ͚ for any decidable set ̻ such that enumerates the elements of ̻ in the ascendant usual order. Since such listing is enumeration order reducible to every listing, all decidable sets are enumeration order reducible to every r.e. set. □ Depending on Definition 2.2, it is clear that if ̻ Ɨ * ̼ then ̻ is ̼ Ǝ recursive , so the following proposition is valid. Proposition 2.4 If ̻ Ɨ * ̼, then ̻ Ɨ ̼. □ From the above proposition we can conclude the following propositions: Proposition 2.5 The enumeration order degree of decidable sets is the least one. The enumeration order degree of decidable sets is denoted by ʞǦʟ * . Also, the enumeration order degree ʞͅʟ0 is the maximal one. □ Proposition 2.7 There are infinite chains on this relation. □ The upcoming theorem is the one of the main theorems in this paper which influences the study of the related equivalency. But before of that we need to depict the following lemma. Lemma 2.8 Consider two listings ͚, ͛ of two r.e. sets ̻, ̼ respectively such that ͚ ƙ * ͛. Assume that we show the sets ̻, ̼ with two sequences of natural numbers ʜ͕$ʝ$Ǩ and ʜ͖$ʝ$Ǩ respectively such that for all ͝ Ǩ ͈: ͕$ Ɨ ͕$ͮͥ and ͖$ Ɨ ͖$ͮͥ, then: ͯͥ ͯͥ 1) ͚ ʚ͕ͥʛ ƙ ͛ ʚ͖ͥʛ and ͯͥ ͯͥ ͯͥ ͯͥ 2) For all ͝ Ƙ 1, if for all ͞ Ɨ ͝: ͚ Ƴ͕%Ʒ Ɣ ͛ ʚ͖%ʛ then ͚ ʚ͕$ʛ ƙ ͛ ʚ͖$ʛ. Proof: ͯͥ ͯͥ First we want to prove that ͚ ʚ͕ͥʛ ƙ ͛ ʚ͖ͥʛ. For the sake of a contradiction, ͯͥ ͯͥ ͯͥ ͯͥ assume that ͚ ʚ͕ͥʛ Ƙ ͛ ʚ͖ͥʛ. Since ͚Ƴ͛ ʚ͖ͥʛƷ Ƙ ͚ʚ͚ ʚ͕ͥʛʛ and ͚ ƙ * ͛, ͯͥ ͯͥ ͛Ƴ͛ ʚ͖ͥʛƷ Ƙ ͛ʚ͚ ʚ͕ͥʛʛ. But this is a contradiction, because ͖ͥ is the least element of ̼. ͯͥ ͯͥ Consider a number ͝ Ƙ 1. Assume that for all ͞ Ɨ ͝: ͚ Ƴ͕%Ʒ Ɣ ͛ ʚ͖%ʛ. We want to ͯͥ ͯͥ ͯͥ prove ͚ ʚ͕$ʛ ƙ ͛ ʚ͖$ʛ. For the sake of a contradiction, assume that ͚ ʚ͕$ʛ Ƙ ͯͥ ͯͥ ͯͥ ͛ ʚ͖$ʛ. It is clear that ͚Ƴ͛ ʚ͖$ʛƷ ǩ ʜ͕ͥ, ͕ͦ, … , ͕$ͯͥʝ and g Ƴ͚ ʚ͕$ʛƷ ǩ ͯͥ ͯͥ ͯͥ ʜ͖ͥ, ͖ͦ, … , ͖$ͯͥʝ. Therefore, ͚Ƴ͛ ʚ͖$ʛƷ Ƙ ͚ʚ͚ ʚ͕$ʛ. Since ͚ ƙ * ͛, ͛Ƴ͛ ʚ͖$ʛƷ Ƙ ͯͥ ͛ʚ͚ ʚ͕$ʛʛ. This is a contradiction. □ Theorem 2.9 Consider two enumeration order equivalent sets ̻ and ̼. There exist two listings ͚, ͛ of ̻, ̼ respectively such that ͚ ȸ * ͛ Proof: Consider a listing ͚ of ̻. According to Definition 2.2, we can deduce that there are two sequences ʜ͚$ʝ$Ǩ and ʜ͛$ʝ$Ǩ of listings of ̻, ̼ respectively such that ͚ͥ Ɣ ͚ and … ƙ * ͛ͦ ƙ * ͚ͦ ƙ * ͛ͥ ƙ * ͚ͥ. Therefore, the following statement is valid: … ƙ * ͚ͧ ƙ * ͚ͦ ƙ * ͚ͥ [1] According to Definition 2.2 (Definition of Enumeration Order Reducibility on r.e. sets) the chain [1] is a computable chain. Based on Lemma 2.8 and this fact that [1] is an infinite chain, we can deduce that ͯͥ ͯͥ there exists ͢ Ǩ ͈ such that for all ͝ ƚ ͢: ͚$ ʚ͕ͥʛ Ɣ ͚$ͮͥ ʚ͕ͥʛ. ͯͥ Assume that there exist ͢, ͟ Ǩ ͈ such that for all ͞ ƙ ͟ and ͝ ƚ ͢: ͚$ Ƴ͕%Ʒ Ɣ ͯͥ ͚$ͮͥ ʚ͕%ʛ. Now consider the chain … ƙ * ͚)ͮͦ ƙ * ͚)ͮͥ ƙ * ͚) [2]. Once again based on Lemma 2.8 and this fact that [2] is an infinite chain, we can gather that there ͯͥ ͯͥ exists ͡ Ǩ ͈ such that for all ͝ ƚ ͡ ƍ ͢: ͚$ ʚ͕&ͮͥʛ Ɣ ͚$ͮͥ ʚ͕&ͮͥʛ. [*] Based on [*], there are ͝, ͞ Ǩ ͈ such that ͝ Ɨ ͞ and ͚$ Ɣ ͚% in the chain [1]. As mentioned above this chain is computable, so ͝ and ͞ can be obtained computably. Since enumeration order reducibility is transitive, ͚$ ƙ * ͛$ ƙ * ͚$, ͚$ ȸ * ͛$. □ Theorem 2.10 For r.e. sets ̻ and ̼ the following statements are equivalent: 1. ̻ ȸ * ̼, 2. For every listing ͚ of ̻ there is a listing ͛ of ̼ such that ͚ ȸ * ͛. Proof: 1⤇⤇⤇2: Let ̻ ȸ * ̼. According to Theorem 2.9, there exist listings ͜ĺ of ̻ and ͛ĺ of ̼ such that ͜ĺ ȸ * ͛ĺ. Assume that ͜ is a listing of ̻. We want to define a listing ͛ of ̼ ɑ ɑͯͥ such that ͜ ȸ * ͛. Consider ͛ Ɣ ͛ ͣ͜ ͣ͜. It is evident that ͛ is a listing of ̼. We claim that ͜~͛. Assume that for , ͞ Ǩ ͈ , ͜ʚ͝ʛ Ɨ ͜ʚ͞ʛ and ͜ʚ͝ʛ Ɣ ͕ͥ Ǩ ̻ , ͜ʚ͞ʛ Ɣ ɑ ͕ͦ Ǩ ̻. Since listings are surjective functions, there exist ͡, ͟ Ǩ ͈ such that ͜ ʚ͟ʛ Ɣ ɑ ͕ͥ and ͜ ʚ͡ʛ Ɣ ͕ͦ. ɑ ɑ ɑ ɑ ɑ ɑͯͥ ɑ ɑͯͥ ͜ʚ͝ʛ Ɨ ͜ʚ͞ʛ ʂ ͜ ʚ͟ʛ Ɨ ͜ ʚ͡ʛ ʂ ͛ ʚ͟ʛ Ɨ ͛ ʚ͡ʛ ʂ ͛ ͣ͜ ʚ͕ͥʛ Ɨ ͛ ͣ͜ ʚ͕ͦʛ ʂ ͛ɑͣ͜ɑͯͥ͜ʚ͝ʛ Ɨ ͛ɑͣ͜ɑͯͥ͜ʚ͞ʛ ʂ ͛ʚ͝ʛ Ɨ ͛ʚ͞ʛ Therefore,,͜ ȸ * ͛. 2⤇⤇⤇1: it is evident. □ We illustrate the above concepts by the following example. Example 2.11 1) Two sets ̻ Ɣ ʜ2͝: ͝ Ǩ ͈ ʝ and ̼ Ɣ ͈ Ǝ ʜ1ʝ are enumeration order equivalent. Proof: Consider two listings ͜ of ̻ and ͛ of ̼ such that for all ͝ Ǩ ͈ , ͜ʚ͝ʛ Ɣ 2͝ and ͛ʚ͝ʛ Ɣ ͝ ƍ 1. It is clear that for all ͝, ͞ Ǩ ͈, ͜ʚ͝ʛ Ɨ ͜ʚ͞ʛ ʃ ͛ʚ͝ʛ Ɨ ͛ʚ͞ʛ □ 2) Two sets ͈ and ͅ are not enumeration order equivalent. Proof: For the sake of a contradiction, assume that these two sets are enumeration order equivalent. The identity function ͘͝: ͈ Ŵ ͈ is a listing of ͈. There exists a listing ͛ of ͅ such that ͜ ȸ * ͛. Therefore, for all ͝,͞Ǩ͈, ͝Ɨ͞ʃ͛ʚ͝ʛ Ɨ ͛ʚ͞ʛ. But this cannot be valid, because ͟ is not a decidable set. This causes a contradiction.□ In the continuation of this section, we want to explore some relationships between the enumeration order equivalency on sets and both one-one reducibility & Turing- reducibility. Lemma 2.12 If two sets belong to same one-one reducibility equivalence class, then they do not belong necessarily to same degree of enumeration order. Proof: Consider an r.e. non-decidable set ̻. There are two cases. Case 1: 1 Ǩ ̻, Case 2: 1 ǩ ̻. For simplicity, we assume case 2. Two sets ̻ and ̼ Ɣ ̻ ȃ ʜ1ʝ belong to same one- one reducibility equivalence class. Let ̻ Ɣ ʜ͕ͥ, ͕ͦ, … , ͕), … ʝ in which ͕ͥ Ɨ ͕ͦ Ɨ ˆ Ɨ ͕) Ɨ ˆ and ͚: ͈ Ŵ ̼ is a listing of ̼ in which ͚ʚ1ʛ Ɣ 1. For the sake of a contradiction, assume that two sets ̻ and ̼ are enumeration order equivalent. Therefore, there exists a listing ͛ of ̻ such that ͚ ȸ * ͛.
Load more

Recommended publications
  • “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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Relationships Between Computability-Theoretic Properties of Problems
    RELATIONSHIPS BETWEEN COMPUTABILITY-THEORETIC PROPERTIES OF PROBLEMS ROD DOWNEY, NOAM GREENBERG, MATTHEW HARRISON-TRAINOR, LUDOVIC PATEY, AND DAN TURETSKY Abstract. A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solu- tion relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set A and a computable instance of a problem P, to find a solution relative to which A is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent 0 to admit avoidance of 1 cone, of ! cones, of 1 hyperimmunity or of 1 non-Σ1 definition. We also prove that the hierarchies of preservation of hyperimmunity 0 and non-Σ1 definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting. 1. Introduction In this article, we classify computability-theoretic preservation properties studied in reverse mathematics, namely cone avoidance, preservation of hyperimmunities, 0 preservation of non-Σ1 definitions, among others. Many of these preservation prop- erties have already been separated using natural problems in reverse mathematics { that is, there is a natural problem which is known to admit preservation of one property but not preservation of the other.
    [Show full text]
  • 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 .
    [Show full text]
  • Computable Functions A
    http://dx.doi.org/10.1090/stml/019 STUDENT MATHEMATICAL LIBRARY Volume 19 Computable Functions A. Shen N. K. Vereshchagin Translated by V. N. Dubrovskii #AMS AMERICAN MATHEMATICAL SOCIETY Editorial Board Robert Devaney Carl Pomerance Daniel L. Goroff Hung-Hsi Wu David Bressoud, Chair H. K. BepemarHH, A. Illem* BMHHCJIHMME ^YHKUMM MIIHMO, 1999 Translated from the Russian by V. N. Dubrovskii 2000 Mathematics Subject Classification. Primary 03-01, 03Dxx. Library of Congress Cataloging-in-Publication Data Vereshchagin, Nikolai Konstantinovich, 1958- Computable functions / A. Shen, N.K. Vereshchagin ; translated by V.N. Dubrovskii. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 19) Authors' names on t.p. of translation reversed from original. Includes bibliographical references and index. ISBN 0-8218-2732-4 (alk. paper) 1. Computable functions. I. Shen, A. (Alexander), 1958- II. Title. III. Se• ries. QA9.59 .V47 2003 511.3—dc21 2002038567 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294, USA. Requests can also be made by e-mail to reprint-permissionQams.org. © 2003 by the American Mathematical Society.
    [Show full text]
  • 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 .
    [Show full text]
  • 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 .
    [Show full text]
  • 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".
    [Show full text]
  • Co~Nparison of Polynomial- Time Reducibilities Richard Ladner University of Washington Seattle, Wash. Nanay Lynch University Of
    Co~nparison of Polynomial- Time Reducibilities interesting relationships. Richard Ladner University of Washington I. Introduction Seattle, Wash. Computation-resource-bounded reducibil- Nanay Lynch University of Southern California ities play a role in the theory of computational Los Angeles, Cal. complexity which is analogous to, and perhaps AlaD Selman Florida State University as important as, the various kinds of effective Tallahas see, Fla. reducibilities used in recursive function theory. Abstract Just as the effective reducibilities are used to Comparison of the polynomial-time-bounded classify problems according to their degrees reducibilities introduced by Cook [1] and of unsolvability, [9] space- and time- bou6ded t<arp [4] leads naturally to the definition of reducibilities may be used to classify problems several intermediate truth-table reducibilities. according to their complexity level. We give definitions and comparisons for these Tne most fruitful resource-bounded reducibilities ; we note, in particular, that reducibilities thus far have been the polynomial- all reducibilities of this type which do not time-bounded reducibilities of Cook [I] and have obvious implication relationships are I<arp [4], corresponding respectively to in fact distinct in a strong sense. Proofs Turing and many-one reducibilities in recursive are by simultaneous diagonalization and function theory. Other resource-bounded encoding constructions. reducibilities have been defined and used as well Work of Meyer and Stockmeyer [7] and [3] [5] [6] [8]; they differ from Cook's or Karp's Gill [Z] then leads u.,~ to define nondeterministic only in the bound on time or space allowed versions of all of our reducibilities. Although for the reduction, and thus they also correspond many of the definitions degenerate, comparison to Turing or many-one reducibility.
    [Show full text]
  • 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.
    [Show full text]