David C. Makinson Completeness Theorems, Representation Theorems: What’S the Difference?
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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. -
5 Propositional Logic: Consistency and Completeness
5 Propositional Logic: Consistency and completeness Reading: Metalogic Part II, 24, 15, 28-31 Contents 5.1 Soundness . 61 5.2 Consistency . 62 5.3 Completeness . 63 5.3.1 An Axiomatization of Propositional Logic . 63 5.3.2 Kalmar's Proof: Informal Exposition . 66 5.3.3 Kalmar's Proof . 68 5.4 Homework Exercises . 70 5.4.1 Questions . 70 5.4.2 Answers . 70 5.1 Soundness In this section, we establish the soundness of the system, i.e., Theorem 3 (Soundness). Every theorem is a tautology, i.e., If ` A then j= A. Proof The proof is by induction the length of the proof of A. For the Basis step, we show that each of the axioms is a tautology. For the induction step, we show that if A and A ⊃ B are tautologies, then B is a tautology. Case 1 (PS1): AB B ⊃ A (A ⊃ (B ⊃ A)) TT TT TF TT FT FT FF TT Case 2 (PS2) 62 5 Propositional Logic: Consistency and completeness XYZ ABC B ⊃ C A ⊃ (B ⊃ C) A ⊃ B A ⊃ C Y ⊃ ZX ⊃ (Y ⊃ Z) TTT TTTTTT TTF FFTFFT TFT TTFTTT TFF TTFFTT FTT TTTTTT FTF FTTTTT FFT TTTTTT FFF TTTTTT Case 3 (PS3) AB » B » A » B ⊃∼ A A ⊃ B (» B ⊃∼ A) ⊃ (A ⊃ B) TT FFTTT TF TFFFT FT FTTTT FF TTTTT Case 4 (MP). If A is a tautology, i.e., true for every assignment of truth values to the atomic letters, and if A ⊃ B is a tautology, then there is no assignment which makes A T and B F. -
The Representation Theorem of Persistence Revisited and Generalized
Journal of Applied and Computational Topology https://doi.org/10.1007/s41468-018-0015-3 The representation theorem of persistence revisited and generalized René Corbet1 · Michael Kerber1 Received: 9 November 2017 / Accepted: 15 June 2018 © The Author(s) 2018 Abstract The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of R-modules to R-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimen- sional persistence as a special case. Keywords Computational topology · Persistence modules · Persistent homology · Representation theory Mathematics Subject Classifcation 06F25 · 16D90 · 16W50 · 55U99 · 68W30 1 Introduction Persistent homology, introduced by Edelsbrunner et al. (2002), is a multi-scale extension of classical homology theory. The idea is to track how homological fea- tures appear and disappear in a shape when the scale parameter is increasing. This data can be summarized by a barcode where each bar corresponds to a homology class that appears in the process and represents the range of scales where the class is present. The usefulness of this paradigm in the context of real-world data sets has led to the term topological data analysis; see the surveys (Carlsson 2009; Ghrist 2008; Edelsbrunner and Morozov 2012; Vejdemo-Johansson 2014; Kerber 2016) and textbooks (Edelsbrunner and Harer 2010; Oudot 2015) for various use cases. * René Corbet [email protected] Michael Kerber [email protected] 1 Institute of Geometry, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria Vol.:(0123456789)1 3 R. -
Completeness of the Propositions-As-Types Interpretation of Intuitionistic Logic Into Illative Combinatory Logic
University of Wollongong Research Online Faculty of Engineering and Information Faculty of Engineering and Information Sciences - Papers: Part A Sciences 1-1-1998 Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic Wil Dekkers Catholic University, Netherlands Martin Bunder University of Wollongong, [email protected] Henk Barendregt Catholic University, Netherlands Follow this and additional works at: https://ro.uow.edu.au/eispapers Part of the Engineering Commons, and the Science and Technology Studies Commons Recommended Citation Dekkers, Wil; Bunder, Martin; and Barendregt, Henk, "Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic" (1998). Faculty of Engineering and Information Sciences - Papers: Part A. 1883. https://ro.uow.edu.au/eispapers/1883 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic Abstract Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic prepositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. -
Representations of Direct Products of Finite Groups
Pacific Journal of Mathematics REPRESENTATIONS OF DIRECT PRODUCTS OF FINITE GROUPS BURTON I. FEIN Vol. 20, No. 1 September 1967 PACIFIC JOURNAL OF MATHEMATICS Vol. 20, No. 1, 1967 REPRESENTATIONS OF DIRECT PRODUCTS OF FINITE GROUPS BURTON FEIN Let G be a finite group and K an arbitrary field. We denote by K(G) the group algebra of G over K. Let G be the direct product of finite groups Gι and G2, G — G1 x G2, and let Mi be an irreducible iΓ(G;)-module, i— 1,2. In this paper we study the structure of Mί9 M2, the outer tensor product of Mi and M2. While Mi, M2 is not necessarily an irreducible K(G)~ module, we prove below that it is completely reducible and give criteria for it to be irreducible. These results are applied to the question of whether the tensor product of division algebras of a type arising from group representation theory is a division algebra. We call a division algebra D over K Z'-derivable if D ~ Hom^s) (Mf M) for some finite group G and irreducible ϋΓ(G)- module M. If B(K) is the Brauer group of K, the set BQ(K) of classes of central simple Z-algebras having division algebra components which are iΓ-derivable forms a subgroup of B(K). We show also that B0(K) has infinite index in B(K) if K is an algebraic number field which is not an abelian extension of the rationale. All iΓ(G)-modules considered are assumed to be unitary finite dimensional left if((τ)-modules. -
An Introduction to First-Order Logic
Outline An Introduction to First-Order Logic K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Completeness, Compactness and Inexpressibility Subramani First-Order Logic Outline Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Outline Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Soundness and Completeness Theorem Soundness: If ∆ ⊢ φ, then ∆ |= φ. Theorem Completeness (Godel’s¨ traditional form): If ∆ |= φ, then ∆ ⊢ φ. Theorem Completeness (Godel’s¨ altenate form): If ∆ is consistent, then it has a model. Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Soundness and Completeness (contd.) Theorem The traditional completeness theorem follows from the alternate form of the completeness theorem. Proof. Assume that ∆ |= φ. It follows that any model M that satisfies all the expressions in ∆, also satisfies φ and hence falsifies ¬φ. -
Completeness and Compactness of First-Order Tableaux
CS 486: Applied Logic Lecture 18, March 27, 2003 18 Completeness and Compactness of First-Order Tableaux 18.1 Completeness Proving the completeness of a first-order calculus gives us G¨odel’sfamous completeness result. G¨odelproved it for a slightly different proof calculus, and the proof that we will show here goes back to Beth and Hintikka. Let us briefly resume the propositional case. The key to the completeness proof was the use of Hintikka’s lemma, which states that every downward saturated set, finite or not, is satisfiable. We then showed that every open and complete path is in fact a Hintikka sequence. Putting these two things together we reasoned that the root of an open and complete tableau must be satisfiable. Thus a complete tableau for a valid formula cannot be open which means that every tableau for a valid formula will eventually close. We will prove the first order case along these lines, but have to keep in mind that several things have changed. ² The definition of a valuation now includes quantifiers. ² The definition of Hintikka sets must take γ and ± formulas into account. ² The notion of a complete tableau needs to be adjusted, because there is now the possibility of non-terminating proof attempts. Fortunately, we can easily make the necessary adjustments and then proceed as before. First, let us define first-order Hintikka sets. A Hintikka Set for a universe U is a set S of U-formulas such that for all closed U-formulas A, ®, ¯, γ, and ± the following conditions hold. 2 ¯ 62 H0 : A atomic and A S 7! A S 2 2 2 H1 : ® S 7! ®1 S ^ ®2 S 2 2 2 H2 : ¯ S 7! ¯1 S _ ¯2 S 2 2 2 H3 : γ S 7! 8k U. -
Representation Theory of Finite Groups
Representation Theory of Finite Groups Benjamin Steinberg School of Mathematics and Statistics Carleton University [email protected] December 15, 2009 Preface This book arose out of course notes for a fourth year undergraduate/first year graduate course that I taught at Carleton University. The goal was to present group representation theory at a level that is accessible to students who have not yet studied module theory and who are unfamiliar with tensor products. For this reason, the Wedderburn theory of semisimple algebras is completely avoided. Instead, I have opted for a Fourier analysis approach. This sort of approach is normally taken in books with a more analytic flavor; such books, however, invariably contain material on representations of com- pact groups, something that I would also consider beyond the scope of an undergraduate text. So here I have done my best to blend the analytic and the algebraic viewpoints in order to keep things accessible. For example, Frobenius reciprocity is treated from a character point of view to evade use of the tensor product. The only background required for this book is a basic knowledge of linear algebra and group theory, as well as familiarity with the definition of a ring. The proof of Burnside's theorem makes use of a small amount of Galois theory (up to the fundamental theorem) and so should be skipped if used in a course for which Galois theory is not a prerequisite. Many things are proved in more detail than one would normally expect in a textbook; this was done to make things easier on undergraduates trying to learn what is usually considered graduate level material. -
Provability, Soundness and Completeness Deductive Rules of Inference Provide a Mechanism for Deriving True Conclusions from True Premises
CSC 244/444 Lecture Notes Sept. 16, 2021 Provability, Soundness and Completeness Deductive rules of inference provide a mechanism for deriving true conclusions from true premises Rules of inference So far we have treated formulas as \given", and have shown how they can be related to a domain of discourse, and how the truth of a set of premises can guarantee (entail) the truth of a conclusion. However, our goal in logic and particularly in AI is to derive new conclusions from given facts. For this we need rules of inference (and later, strategies for applying such rules so as to derive a desired conclusion, if possible). In general, a \forward" inference rule consists of one or more premises and a con- clusion. Both the premises and the conclusion are generally schemas, i.e., they involve metavariables for formulas or terms that can be particularized in many ways (just as we saw in the case of valid formula schemas). We often put a horizontal line under the premises, and write the conclusion underneath the line. For instance, here is the rule of Modus Ponens φ, φ ) MP : This says that given a premise formula φ, and another formula of form φ ) , we may derive the conclusion . It is intuitively clear that this rule leads from true premises to a true conclusion { but this is an intuition we need to verify by proving the rule sound, as illustrated below. An example of using the rule is this: from Dog(Snoopy), Dog(Snoopy) ) Has-tail(Snoopy), we can conclude Has-tail(Snoopy). -
The Algebraic Theory of Semigroups the Algebraic Theory of Semigroups
http://dx.doi.org/10.1090/surv/007.1 The Algebraic Theory of Semigroups The Algebraic Theory of Semigroups A. H. Clifford G. B. Preston American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 20-XX. International Standard Serial Number 0076-5376 International Standard Book Number 0-8218-0271-2 Library of Congress Catalog Card Number: 61-15686 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1961 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second Edition, 1964 Reprinted with corrections, 1977. The American Mathematical Society retains all rights except those granted to the United States Government. -
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. -
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.