Lambda Calculus and the Decision Problem
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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. -
NP-Completeness (Chapter 8)
CSE 421" Algorithms NP-Completeness (Chapter 8) 1 What can we feasibly compute? Focus so far has been to give good algorithms for specific problems (and general techniques that help do this). Now shifting focus to problems where we think this is impossible. Sadly, there are many… 2 History 3 A Brief History of Ideas From Classical Greece, if not earlier, "logical thought" held to be a somewhat mystical ability Mid 1800's: Boolean Algebra and foundations of mathematical logic created possible "mechanical" underpinnings 1900: David Hilbert's famous speech outlines program: mechanize all of mathematics? http://mathworld.wolfram.com/HilbertsProblems.html 1930's: Gödel, Church, Turing, et al. prove it's impossible 4 More History 1930/40's What is (is not) computable 1960/70's What is (is not) feasibly computable Goal – a (largely) technology-independent theory of time required by algorithms Key modeling assumptions/approximations Asymptotic (Big-O), worst case is revealing Polynomial, exponential time – qualitatively different 5 Polynomial Time 6 The class P Definition: P = the set of (decision) problems solvable by computers in polynomial time, i.e., T(n) = O(nk) for some fixed k (indp of input). These problems are sometimes called tractable problems. Examples: sorting, shortest path, MST, connectivity, RNA folding & other dyn. prog., flows & matching" – i.e.: most of this qtr (exceptions: Change-Making/Stamps, Knapsack, TSP) 7 Why "Polynomial"? Point is not that n2000 is a nice time bound, or that the differences among n and 2n and n2 are negligible. Rather, simple theoretical tools may not easily capture such differences, whereas exponentials are qualitatively different from polynomials and may be amenable to theoretical analysis. -
CAS LX 522 Syntax I
It is likely… CAS LX 522 IP This satisfies the EPP in Syntax I both clauses. The main DPj I′ clause has Mary in SpecIP. Mary The embedded clause has Vi+I VP is the trace in SpecIP. V AP Week 14b. PRO and control ti This specific instance of A- A IP movement, where we move a likely subject from an embedded DP I′ clause to a higher clause is tj generally called subject raising. I VP to leave Reluctance to leave Reluctance to leave Now, consider: Reluctant has two θ-roles to assign. Mary is reluctant to leave. One to the one feeling the reluctance (Experiencer) One to the proposition about which the reluctance holds (Proposition) This looks very similar to Mary is likely to leave. Can we draw the same kind of tree for it? Leave has one θ-role to assign. To the one doing the leaving (Agent). How many θ-roles does reluctant assign? In Mary is reluctant to leave, what θ-role does Mary get? IP Reluctance to leave Reluctance… DPi I′ Mary Vj+I VP In Mary is reluctant to leave, is V AP Mary is doing the leaving, gets Agent t Mary is reluctant to leave. j t from leave. i A′ Reluctant assigns its θ- Mary is showing the reluctance, gets θ roles within AP as A θ IP Experiencer from reluctant. required, Mary moves reluctant up to SpecIP in the main I′ clause by Spellout. ? And we have a problem: I vP But what gets the θ-role to Mary appears to be getting two θ-roles, from leave, and what v′ in violation of the θ-criterion. -
ÉCLAT MATHEMATICS JOURNAL Lady Shri Ram College for Women
ECLAT´ MATHEMATICS JOURNAL Lady Shri Ram College For Women Volume X 2018-2019 HYPATIA This journal is dedicated to Hypatia who was a Greek mathematician, astronomer, philosopher and the last great thinker of ancient Alexandria. The Alexandrian scholar authored numerous mathematical treatises, and has been credited with writing commentaries on Diophantuss Arithmetica, on Apolloniuss Conics and on Ptolemys astronomical work. Hypatia is an inspiration, not only as the first famous female mathematician but most importantly as a symbol of learning and science. PREFACE Derived from the French language, Eclat´ means brilliance. Since it’s inception, this journal has aimed at providing a platform for undergraduate students to showcase their understand- ing of diverse mathematical concepts that interest them. As the journal completes a decade this year, it continues to be instrumental in providing an opportunity for students to make valuable contributions to academic inquiry. The work contained in this journal is not original but contains the review research of students. Each of the nine papers included in this year’s edition have been carefully written and compiled to stimulate the thought process of its readers. The four categories discussed in the journal - History of Mathematics, Rigour in Mathematics, Interdisciplinary Aspects of Mathematics and Extension of Course Content - give a comprehensive idea about the evolution of the subject over the years. We would like to express our sincere thanks to the Faculty Advisors of the department, who have guided us at every step leading to the publication of this volume, and to all the authors who have contributed their articles for this volume. -
The Power of Principles: Physics Revealed
The Passions of Logic: Appreciating Analytic Philosophy A CONVERSATION WITH Scott Soames This eBook is based on a conversation between Scott Soames of University of Southern California (USC) and Howard Burton that took place on September 18, 2014. Chapters 4a, 5a, and 7a are not included in the video version. Edited by Howard Burton Open Agenda Publishing © 2015. All rights reserved. Table of Contents Biography 4 Introductory Essay The Utility of Philosophy 6 The Conversation 1. Analytic Sociology 9 2. Mathematical Underpinnings 15 3. What is Logic? 20 4. Creating Modernity 23 4a. Understanding Language 27 5. Stumbling Blocks 30 5a. Re-examining Information 33 6. Legal Applications 38 7. Changing the Culture 44 7a. Gödelian Challenges 47 Questions for Discussion 52 Topics for Further Investigation 55 Ideas Roadshow • Scott Soames • The Passions of Logic Biography Scott Soames is Distinguished Professor of Philosophy and Director of the School of Philosophy at the University of Southern California (USC). Following his BA from Stanford University (1968) and Ph.D. from M.I.T. (1976), Scott held professorships at Yale (1976-1980) and Princeton (1980-204), before moving to USC in 2004. Scott’s numerous awards and fellowships include USC’s Albert S. Raubenheimer Award, a John Simon Guggenheim Memorial Foun- dation Fellowship, Princeton University’s Class of 1936 Bicentennial Preceptorship and a National Endowment for the Humanities Research Fellowship. His visiting positions include University of Washington, City University of New York and the Catholic Pontifical University of Peru. He was elected to the American Academy of Arts and Sciences in 2010. In addition to a wide array of peer-reviewed articles, Scott has authored or co-authored numerous books, including Rethinking Language, Mind and Meaning (Carl G. -
Computability (Section 12.3) Computability
Computability (Section 12.3) Computability • Some problems cannot be solved by any machine/algorithm. To prove such statements we need to effectively describe all possible algorithms. • Example (Turing Machines): Associate a Turing machine with each n 2 N as follows: n $ b(n) (the binary representation of n) $ a(b(n)) (b(n) split into 7-bit ASCII blocks, w/ leading 0's) $ if a(b(n)) is the syntax of a TM then a(b(n)) else (0; a; a; S,halt) fi • So we can effectively describe all possible Turing machines: T0; T1; T2;::: Continued • Of course, we could use the same technique to list all possible instances of any computational model. For example, we can effectively list all possible Simple programs and we can effectively list all possible partial recursive functions. • If we want to use the Church-Turing thesis, then we can effectively list all possible solutions (e.g., Turing machines) to every intuitively computable problem. Decidable. • Is an arbitrary first-order wff valid? Undecidable and partially decidable. • Does a DFA accept infinitely many strings? Decidable • Does a PDA accept a string s? Decidable Decision Problems • A decision problem is a problem that can be phrased as a yes/no question. Such a problem is decidable if an algorithm exists to answer yes or no to each instance of the problem. Otherwise it is undecidable. A decision problem is partially decidable if an algorithm exists to halt with the answer yes to yes-instances of the problem, but may run forever if the answer is no. -
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. -
Syntax and Semantics of Propositional Logic
INTRODUCTIONTOLOGIC Lecture 2 Syntax and Semantics of Propositional Logic. Dr. James Studd Logic is the beginning of wisdom. Thomas Aquinas Outline 1 Syntax vs Semantics. 2 Syntax of L1. 3 Semantics of L1. 4 Truth-table methods. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. -
Homework 4 Solutions Uploaded 4:00Pm on Dec 6, 2017 Due: Monday Dec 4, 2017
CS3510 Design & Analysis of Algorithms Section A Homework 4 Solutions Uploaded 4:00pm on Dec 6, 2017 Due: Monday Dec 4, 2017 This homework has a total of 3 problems on 4 pages. Solutions should be submitted to GradeScope before 3:00pm on Monday Dec 4. The problem set is marked out of 20, you can earn up to 21 = 1 + 8 + 7 + 5 points. If you choose not to submit a typed write-up, please write neat and legibly. Collaboration is allowed/encouraged on problems, however each student must independently com- plete their own write-up, and list all collaborators. No credit will be given to solutions obtained verbatim from the Internet or other sources. Modifications since version 0 1. 1c: (changed in version 2) rewored to showing NP-hardness. 2. 2b: (changed in version 1) added a hint. 3. 3b: (changed in version 1) added comment about the two loops' u variables being different due to scoping. 0. [1 point, only if all parts are completed] (a) Submit your homework to Gradescope. (b) Student id is the same as on T-square: it's the one with alphabet + digits, NOT the 9 digit number from student card. (c) Pages for each question are separated correctly. (d) Words on the scan are clearly readable. 1. (8 points) NP-Completeness Recall that the SAT problem, or the Boolean Satisfiability problem, is defined as follows: • Input: A CNF formula F having m clauses in n variables x1; x2; : : : ; xn. There is no restriction on the number of variables in each clause. -
Stephen Cole Kleene
STEPHEN COLE KLEENE American mathematician and logician Stephen Cole Kleene (January 5, 1909 – January 25, 1994) helped lay the foundations for modern computer science. In the 1930s, he and Alonzo Church, developed the lambda calculus, considered to be the mathematical basis for programming language. It was an effort to express all computable functions in a language with a simple syntax and few grammatical restrictions. Kleene was born in Hartford, Connecticut, but his real home was at his paternal grandfather’s farm in Maine. After graduating summa cum laude from Amherst (1930), Kleene went to Princeton University where he received his PhD in 1934. His thesis, A Theory of Positive Integers in Formal Logic, was supervised by Church. Kleene joined the faculty of the University of Wisconsin in 1935. During the next several years he spent time at the Institute for Advanced Study, taught at Amherst College and served in the U.S. Navy, attaining the rank of Lieutenant Commander. In 1946, he returned to Wisconsin, where he stayed until his retirement in 1979. In 1964 he was named the Cyrus C. MacDuffee professor of mathematics. Kleene was one of the pioneers of 20th century mathematics. His research was devoted to the theory of algorithms and recursive functions (that is, functions defined in a finite sequence of combinatorial steps). Together with Church, Kurt Gödel, Alan Turing, and others, Kleene developed the branch of mathematical logic known as recursion theory, which helped provide the foundations of theoretical computer science. The Kleene Star, Kleene recursion theorem and the Ascending Kleene Chain are named for him. -
Russell's Paradox in Appendix B of the Principles of Mathematics
HISTORY AND PHILOSOPHY OF LOGIC, 22 (2001), 13± 28 Russell’s Paradox in Appendix B of the Principles of Mathematics: Was Frege’s response adequate? Ke v i n C. Kl e m e n t Department of Philosophy, University of Massachusetts, Amherst, MA 01003, USA Received March 2000 In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. 1. Introduction Russell’s letter to Frege from June of 1902 in which he related his discovery of the Russell paradox was a monumental event in the history of logic. However, in the ensuing correspondence between Russell and Frege, the Russell paradox was not the only antinomy discussed. -
Syntax: Recursion, Conjunction, and Constituency
Syntax: Recursion, Conjunction, and Constituency Course Readings Recursion Syntax: Conjunction Recursion, Conjunction, and Constituency Tests Auxiliary Verbs Constituency . Syntax: Course Readings Recursion, Conjunction, and Constituency Course Readings Recursion Conjunction Constituency Tests The following readings have been posted to the Moodle Auxiliary Verbs course site: I Language Files: Chapter 5 (pp. 204-215, 216-220) I Language Instinct: Chapter 4 (pp. 74-99) . Syntax: An Interesting Property of our PS Rules Recursion, Conjunction, and Our Current PS Rules: Constituency S ! f NP , CP g VP NP ! (D) (A*) N (CP) (PP*) Course Readings VP ! V (NP) f (NP) (CP) g (PP*) Recursion PP ! P (NP) Conjunction CP ! C S Constituency Tests Auxiliary Verbs An Interesting Feature of These Rules: As we saw last time, these rules allow sentences to contain other sentences. I A sentence must have a VP in it. I A VP can have a CP in it. I A CP must have an S in it. Syntax: An Interesting Property of our PS Rules Recursion, Conjunction, and Our Current PS Rules: Constituency S ! f NP , CP g VP NP ! (D) (A*) N (CP) (PP*) Course Readings VP ! V (NP) f (NP) (CP) g (PP*) Recursion ! PP P (NP) Conjunction CP ! C S Constituency Tests Auxiliary Verbs An Interesting Feature of These Rules: As we saw last time, these rules allow sentences to contain other sentences. S NP VP N V CP Dave thinks C S that . he. is. cool. Syntax: An Interesting Property of our PS Rules Recursion, Conjunction, and Our Current PS Rules: Constituency S ! f NP , CP g VP NP ! (D) (A*) N (CP) (PP*) Course Readings VP ! V (NP) f (NP) (CP) g (PP*) Recursion PP ! P (NP) Conjunction CP ! CS Constituency Tests Auxiliary Verbs Another Interesting Feature of These Rules: These rules also allow noun phrases to contain other noun phrases.