Introduction to the Theory of Computation Computability, Complexity, and the Lambda Calculus Some Notes for CIS262
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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. -
Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
Slides 6, HT 2019 Space Complexity
Computational Complexity; slides 6, HT 2019 Space complexity Prof. Paul W. Goldberg (Dept. of Computer Science, University of Oxford) HT 2019 Paul Goldberg Space complexity 1 / 51 Road map I mentioned classes like LOGSPACE (usually calledL), SPACE(f (n)) etc. How do they relate to each other, and time complexity classes? Next: Various inclusions can be proved, some more easy than others; let's begin with \low-hanging fruit"... e.g., I have noted: TIME(f (n)) is a subset of SPACE(f (n)) (easy!) We will see e.g.L is a proper subset of PSPACE, although it's unknown how they relate to various intermediate classes, e.g.P, NP Various interesting problems are complete for PSPACE, EXPTIME, and some of the others. Paul Goldberg Space complexity 2 / 51 Convention: In this section we will be using Turing machines with a designated read only input tape. So, \logarithmic space" becomes meaningful. Space Complexity So far, we have measured the complexity of problems in terms of the time required to solve them. Alternatively, we can measure the space/memory required to compute a solution. Important difference: space can be re-used Paul Goldberg Space complexity 3 / 51 Space Complexity So far, we have measured the complexity of problems in terms of the time required to solve them. Alternatively, we can measure the space/memory required to compute a solution. Important difference: space can be re-used Convention: In this section we will be using Turing machines with a designated read only input tape. So, \logarithmic space" becomes meaningful. Paul Goldberg Space complexity 3 / 51 Definition. -
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. -
Week 1: an Overview of Circuit Complexity 1 Welcome 2
Topics in Circuit Complexity (CS354, Fall’11) Week 1: An Overview of Circuit Complexity Lecture Notes for 9/27 and 9/29 Ryan Williams 1 Welcome The area of circuit complexity has a long history, starting in the 1940’s. It is full of open problems and frontiers that seem insurmountable, yet the literature on circuit complexity is fairly large. There is much that we do know, although it is scattered across several textbooks and academic papers. I think now is a good time to look again at circuit complexity with fresh eyes, and try to see what can be done. 2 Preliminaries An n-bit Boolean function has domain f0; 1gn and co-domain f0; 1g. At a high level, the basic question asked in circuit complexity is: given a collection of “simple functions” and a target Boolean function f, how efficiently can f be computed (on all inputs) using the simple functions? Of course, efficiency can be measured in many ways. The most natural measure is that of the “size” of computation: how many copies of these simple functions are necessary to compute f? Let B be a set of Boolean functions, which we call a basis set. The fan-in of a function g 2 B is the number of inputs that g takes. (Typical choices are fan-in 2, or unbounded fan-in, meaning that g can take any number of inputs.) We define a circuit C with n inputs and size s over a basis B, as follows. C consists of a directed acyclic graph (DAG) of s + n + 2 nodes, with n sources and one sink (the sth node in some fixed topological order on the nodes). -
Artificial Consciousness and the Consciousness-Attention Dissociation
Consciousness and Cognition 45 (2016) 210–225 Contents lists available at ScienceDirect Consciousness and Cognition journal homepage: www.elsevier.com/locate/concog Review article Artificial consciousness and the consciousness-attention dissociation ⇑ Harry Haroutioun Haladjian a, , Carlos Montemayor b a Laboratoire Psychologie de la Perception, CNRS (UMR 8242), Université Paris Descartes, Centre Biomédical des Saints-Pères, 45 rue des Saints-Pères, 75006 Paris, France b San Francisco State University, Philosophy Department, 1600 Holloway Avenue, San Francisco, CA 94132 USA article info abstract Article history: Artificial Intelligence is at a turning point, with a substantial increase in projects aiming to Received 6 July 2016 implement sophisticated forms of human intelligence in machines. This research attempts Accepted 12 August 2016 to model specific forms of intelligence through brute-force search heuristics and also reproduce features of human perception and cognition, including emotions. Such goals have implications for artificial consciousness, with some arguing that it will be achievable Keywords: once we overcome short-term engineering challenges. We believe, however, that phenom- Artificial intelligence enal consciousness cannot be implemented in machines. This becomes clear when consid- Artificial consciousness ering emotions and examining the dissociation between consciousness and attention in Consciousness Visual attention humans. While we may be able to program ethical behavior based on rules and machine Phenomenology learning, we will never be able to reproduce emotions or empathy by programming such Emotions control systems—these will be merely simulations. Arguments in favor of this claim include Empathy considerations about evolution, the neuropsychological aspects of emotions, and the disso- ciation between attention and consciousness found in humans. -
Chapter 24 Conp, Self-Reductions
Chapter 24 coNP, Self-Reductions CS 473: Fundamental Algorithms, Spring 2013 April 24, 2013 24.1 Complementation and Self-Reduction 24.2 Complementation 24.2.1 Recap 24.2.1.1 The class P (A) A language L (equivalently decision problem) is in the class P if there is a polynomial time algorithm A for deciding L; that is given a string x, A correctly decides if x 2 L and running time of A on x is polynomial in jxj, the length of x. 24.2.1.2 The class NP Two equivalent definitions: (A) Language L is in NP if there is a non-deterministic polynomial time algorithm A (Turing Machine) that decides L. (A) For x 2 L, A has some non-deterministic choice of moves that will make A accept x (B) For x 62 L, no choice of moves will make A accept x (B) L has an efficient certifier C(·; ·). (A) C is a polynomial time deterministic algorithm (B) For x 2 L there is a string y (proof) of length polynomial in jxj such that C(x; y) accepts (C) For x 62 L, no string y will make C(x; y) accept 1 24.2.1.3 Complementation Definition 24.2.1. Given a decision problem X, its complement X is the collection of all instances s such that s 62 L(X) Equivalently, in terms of languages: Definition 24.2.2. Given a language L over alphabet Σ, its complement L is the language Σ∗ n L. 24.2.1.4 Examples (A) PRIME = nfn j n is an integer and n is primeg o PRIME = n n is an integer and n is not a prime n o PRIME = COMPOSITE . -
Software for Numerical Computation
Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1977 Software for Numerical Computation John R. Rice Purdue University, [email protected] Report Number: 77-214 Rice, John R., "Software for Numerical Computation" (1977). Department of Computer Science Technical Reports. Paper 154. https://docs.lib.purdue.edu/cstech/154 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. SOFTWARE FOR NUMERICAL COMPUTATION John R. Rice Department of Computer Sciences Purdue University West Lafayette, IN 47907 CSD TR #214 January 1977 SOFTWARE FOR NUMERICAL COMPUTATION John R. Rice Mathematical Sciences Purdue University CSD-TR 214 January 12, 1977 Article to appear in the book: Research Directions in Software Technology. SOFTWARE FOR NUMERICAL COMPUTATION John R. Rice Mathematical Sciences Purdue University INTRODUCTION AND MOTIVATING PROBLEMS. The purpose of this article is to examine the research developments in software for numerical computation. Research and development of numerical methods is not intended to be discussed for two reasons. First, a reasonable survey of the research in numerical methods would require a book. The COSERS report [Rice et al, 1977] on Numerical Computation does such a survey in about 100 printed pages and even so the discussion of many important fields (never mind topics) is limited to a few paragraphs. Second, the present book is focused on software and thus it is natural to attempt to separate software research from numerical computation research. This, of course, is not easy as the two are intimately intertwined. -
NP-Completeness Part I
NP-Completeness Part I Outline for Today ● Recap from Last Time ● Welcome back from break! Let's make sure we're all on the same page here. ● Polynomial-Time Reducibility ● Connecting problems together. ● NP-Completeness ● What are the hardest problems in NP? ● The Cook-Levin Theorem ● A concrete NP-complete problem. Recap from Last Time The Limits of Computability EQTM EQTM co-RE R RE LD LD ADD HALT ATM HALT ATM 0*1* The Limits of Efficient Computation P NP R P and NP Refresher ● The class P consists of all problems solvable in deterministic polynomial time. ● The class NP consists of all problems solvable in nondeterministic polynomial time. ● Equivalently, NP consists of all problems for which there is a deterministic, polynomial-time verifier for the problem. Reducibility Maximum Matching ● Given an undirected graph G, a matching in G is a set of edges such that no two edges share an endpoint. ● A maximum matching is a matching with the largest number of edges. AA maximummaximum matching.matching. Maximum Matching ● Jack Edmonds' paper “Paths, Trees, and Flowers” gives a polynomial-time algorithm for finding maximum matchings. ● (This is the same Edmonds as in “Cobham- Edmonds Thesis.) ● Using this fact, what other problems can we solve? Domino Tiling Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling The Setup ● To determine whether you can place at least k dominoes on a crossword grid, do the following: ● Convert the grid into a graph: each empty cell is a node, and any two adjacent empty cells have an edge between them. -
Computability and Complexity
Computability and Complexity Lecture Notes Herbert Jaeger, Jacobs University Bremen Version history Jan 30, 2018: created as copy of CC lecture notes of Spring 2017 Feb 16, 2018: cleaned up font conversion hickups up to Section 5 Feb 23, 2018: cleaned up font conversion hickups up to Section 6 Mar 15, 2018: cleaned up font conversion hickups up to Section 7 Apr 5, 2018: cleaned up font conversion hickups up to the end of these lecture notes 1 1 Introduction 1.1 Motivation This lecture will introduce you to the theory of computation and the theory of computational complexity. The theory of computation offers full answers to the questions, • what problems can in principle be solved by computer programs? • what functions can in principle be computed by computer programs? • what formal languages can in principle be decided by computer programs? Full answers to these questions have been found in the last 70 years or so, and we will learn about them. (And it turns out that these questions are all the same question). The theory of computation is well-established, transparent, and basically simple (you might disagree at first). The theory of complexity offers many insights to questions like • for a given problem / function / language that has to be solved / computed / decided by a computer program, how long does the fastest program actually run? • how much memory space has to be used at least? • can you speed up computations by using different computer architectures or different programming approaches? The theory of complexity is historically younger than the theory of computation – the first surge of results came in the 60ties of last century. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
Quantum Hypercomputation—Hype Or Computation?
Quantum Hypercomputation—Hype or Computation? Amit Hagar,∗ Alex Korolev† February 19, 2007 Abstract A recent attempt to compute a (recursion–theoretic) non–computable func- tion using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from. ∗HPS Department, Indiana University, Bloomington, IN, 47405. [email protected] †Philosophy Department, University of BC, Vancouver, BC. [email protected] 1 1 Introduction Combining physics, mathematics and computer science, quantum computing has developed in the past two decades from a visionary idea (Feynman 1982) to one of the most exciting areas of quantum mechanics (Nielsen and Chuang 2000). The recent excitement in this lively and fashionable domain of research was triggered by Peter Shor (1994) who showed how a quantum computer could exponentially speed–up classical computation and factor numbers much more rapidly (at least in terms of the number of computational steps involved) than any known classical algorithm. Shor’s algorithm was soon followed by several other algorithms that aimed to solve combinatorial and algebraic problems, and in the last few years the- oretical study of quantum systems serving as computational devices has achieved tremendous progress. According to one authority in the field (Aharonov 1998, Abstract), we now have strong theoretical evidence that quantum computers, if built, might be used as powerful computational tool, capable of per- forming tasks which seem intractable for classical computers.