On Consistency and Existence in Mathematics
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Computational Complexity Theory Introduction to Computational
31/03/2012 Computational complexity theory Introduction to computational complexity theory Complexity (computability) theory theory deals with two aspects: Algorithm’s complexity. Problem’s complexity. References S. Cook, « The complexity of Theorem Proving Procedures », 1971. Garey and Johnson, « Computers and Intractability, A guide to the theory of NP-completeness », 1979. J. Carlier et Ph. Chrétienne « Problèmes d’ordonnancements : algorithmes et complexité », 1988. 1 31/03/2012 Basic Notions • Some problem is a “question” characterized by parameters and needs an answer. – Parameters description; – Properties that a solutions must satisfy; – An instance is obtained when the parameters are fixed to some values. • An algorithm: a set of instructions describing how some task can be achieved or a problem can be solved. • A program : the computational implementation of an algorithm. Algorithm’s complexity (I) • There may exists several algorithms for the same problem • Raised questions: – Which one to choose ? – How they are compared ? – How measuring the efficiency ? – What are the most appropriate measures, running time, memory space ? 2 31/03/2012 Algorithm’s complexity (II) • Running time depends on: – The data of the problem, – Quality of program..., – Computer type, – Algorithm’s efficiency, – etc. • Proceed by analyzing the algorithm: – Search for some n characterizing the data. – Compute the running time in terms of n. – Evaluating the number of elementary operations, (elementary operation = simple instruction of a programming language). Algorithm’s evaluation (I) • Any algorithm is composed of two main stages: initialization and computing one • The complexity parameter is the size data n (binary coding). Definition: Let be n>0 andT(n) the running time of an algorithm expressed in terms of the size data n, T(n) is of O(f(n)) iff n0 and some constant c such that: n n0, we have T(n) c f(n). -
Submitted Thesis
Saarland University Faculty of Mathematics and Computer Science Bachelor’s Thesis Synthetic One-One, Many-One, and Truth-Table Reducibility in Coq Author Advisor Felix Jahn Yannick Forster Reviewers Prof. Dr. Gert Smolka Prof. Dr. Markus Bläser Submitted: September 15th, 2020 ii Eidesstattliche Erklärung Ich erkläre hiermit an Eides statt, dass ich die vorliegende Arbeit selbstständig ver- fasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverständniserklärung Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit veröffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrücken, September 15th, 2020 Abstract Reducibility is an essential concept for undecidability proofs in computability the- ory. The idea behind reductions was conceived by Turing, who introduced the later so-called Turing reduction based on oracle machines. In 1944, Post furthermore in- troduced with one-one, many-one, and truth-table reductions in comparison to Tur- ing reductions more specific reducibility notions. Post then also started to analyze the structure of the different reducibility notions and their computability degrees. Most undecidable problems were reducible from the halting problem, since this was exactly the method to show them undecidable. However, Post was able to con- struct also semidecidable but undecidable sets that do not one-one, many-one, or truth-table reduce from the halting problem. -
MOTIVATING HILBERT SYSTEMS Hilbert Systems Are Formal Systems
MOTIVATING HILBERT SYSTEMS Hilbert systems are formal systems that encode the notion of a mathematical proof, which are used in the branch of mathematics known as proof theory. There are other formal systems that proof theorists use, with their own advantages and disadvantages. The advantage of Hilbert systems is that they are simpler than the alternatives in terms of the number of primitive notions they involve. Formal systems for proof theory work by deriving theorems from axioms using rules of inference. The distinctive characteristic of Hilbert systems is that they have very few primitive rules of inference|in fact a Hilbert system with just one primitive rule of inference, modus ponens, suffices to formalize proofs using first- order logic, and first-order logic is sufficient for all mathematical reasoning. Modus ponens is the rule of inference that says that if we have theorems of the forms p ! q and p, where p and q are formulas, then χ is also a theorem. This makes sense given the interpretation of p ! q as meaning \if p is true, then so is q". The simplicity of inference in Hilbert systems is compensated for by a somewhat more complicated set of axioms. For minimal propositional logic, the two axiom schemes below suffice: (1) For every pair of formulas p and q, the formula p ! (q ! p) is an axiom. (2) For every triple of formulas p, q and r, the formula (p ! (q ! r)) ! ((p ! q) ! (p ! r)) is an axiom. One thing that had always bothered me when reading about Hilbert systems was that I couldn't see how people could come up with these axioms other than by a stroke of luck or genius. -
Module 34: Reductions and NP-Complete Proofs
Module 34: Reductions and NP-Complete Proofs This module 34 focuses on reductions and proof of NP-Complete problems. The module illustrates the ways of reducing one problem to another. Then, the module illustrates the outlines of proof of NP- Complete problems with some examples. The objectives of this module are To understand the concept of Reductions among problems To Understand proof of NP-Complete problems To overview proof of some Important NP-Complete problems. Computational Complexity There are two types of complexity theories. One is related to algorithms, known as algorithmic complexity theory and another related to complexity of problems called computational complexity theory [2,3]. Algorithmic complexity theory [3,4] aims to analyze the algorithms in terms of the size of the problem. In modules 3 and 4, we had discussed about these methods. The size is the length of the input. It can be recollected from module 3 that the size of a number n is defined to be the number of binary bits needed to write n. For example, Example: b(5) = b(1012) = 3. In other words, the complexity of the algorithm is stated in terms of the size of the algorithm. Asymptotic Analysis [1,2] refers to the study of an algorithm as the input size reaches a limit and analyzing the behaviour of the algorithms. The asymptotic analysis is also science of approximation where the behaviour of the algorithm is expressed in terms of notations such as big-oh, Big-Omega and Big- Theta. Computational complexity theory is different. Computational complexity aims to determine complexity of the problems itself. -
David Hilbert's Contributions to Logical Theory
David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques. -
Decidable. (Recall It’S Not “Regular”.)
15-251: Great Theoretical Ideas in Computer Science Fall 2016 Lecture 6 September 15, 2016 Turing & the Uncomputable Comparing the cardinality of sets 퐴 ≤ 퐵 if there is an injection (one-to-one map) from 퐴 to 퐵 퐴 ≥ 퐵 if there is a surjection (onto map) from 퐴 to 퐵 퐴 = 퐵 if there is a bijection from 퐴 to 퐵 퐴 > |퐵| if there is no surjection from 퐵 to 퐴 (or equivalently, there is no injection from 퐴 to 퐵) Countable and uncountable sets countable countably infinite uncountable One slide guide to countability questions You are given a set 퐴 : is it countable or uncountable 퐴 ≤ |ℕ| or 퐴 > |ℕ| 퐴 ≤ |ℕ| : • Show directly surjection from ℕ to 퐴 • Show that 퐴 ≤ |퐵| where 퐵 ∈ {ℤ, ℤ x ℤ, ℚ, Σ∗, ℚ[x], …} 퐴 > |ℕ| : • Show directly using a diagonalization argument • Show that 퐴 ≥ | 0,1 ∞| Proving sets countable using computation For example, f(n) = ‘the nth prime’. You could write a program (Turing machine) to compute f. So this is a well-defined rule. Or: f(n) = the nth rational in our listing of ℚ. (List ℤ2 via the spiral, omit the terms p/0, omit rationals seen before…) You could write a program to compute this f. Poll Let 퐴 be the set of all languages over Σ = 1 ∗ Select the correct ones: - A is finite - A is infinite - A is countable - A is uncountable Another thing to remember from last week Encoding different objects with strings Fix some alphabet Σ . We use the ⋅ notation to denote the encoding of an object as a string in Σ∗ Examples: is the encoding a TM 푀 is the encoding a DFA 퐷 is the encoding of a pair of TMs 푀1, 푀2 is the encoding a pair 푀, 푥, where 푀 is a TM, and 푥 ∈ Σ∗ is an input to 푀 Uncountable to uncomputable The real number 1/7 is “computable”. -
An Anti-Locally-Nameless Approach to Formalizing Quantifiers Olivier Laurent
An Anti-Locally-Nameless Approach to Formalizing Quantifiers Olivier Laurent To cite this version: Olivier Laurent. An Anti-Locally-Nameless Approach to Formalizing Quantifiers. Certified Programs and Proofs, Jan 2021, virtual, Denmark. pp.300-312, 10.1145/3437992.3439926. hal-03096253 HAL Id: hal-03096253 https://hal.archives-ouvertes.fr/hal-03096253 Submitted on 4 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. An Anti-Locally-Nameless Approach to Formalizing Quantifiers Olivier Laurent Univ Lyon, EnsL, UCBL, CNRS, LIP LYON, France [email protected] Abstract all cases have been covered. This works perfectly well for We investigate the possibility of formalizing quantifiers in various propositional systems, but as soon as (first-order) proof theory while avoiding, as far as possible, the use of quantifiers enter the picture, we have to face one ofthe true binding structures, α-equivalence or variable renam- nightmares of formalization: quantifiers are binders... The ings. We propose a solution with two kinds of variables in question of the formalization of binders does not yet have an 1 terms and formulas, as originally done by Gentzen. In this answer which makes perfect consensus . -
Computability on the Integers
Computability on the integers Laurent Bienvenu ( LIAFA, CNRS & Université de Paris 7 ) EJCIM 2011 Amiens, France 30 mars 2011 1. Basic objects of computability The formalization and study of the notion of computable function is what computability theory is about. As opposed to complexity theory, we do not care about efficiency, just about feasibility. Computable. functions What does it means for a function f : N ! N to be computable? 1. Basic objects of computability 3/79 As opposed to complexity theory, we do not care about efficiency, just about feasibility. Computable. functions What does it means for a function f : N ! N to be computable? The formalization and study of the notion of computable function is what computability theory is about. 1. Basic objects of computability 3/79 Computable. functions What does it means for a function f : N ! N to be computable? The formalization and study of the notion of computable function is what computability theory is about. As opposed to complexity theory, we do not care about efficiency, just about feasibility. 1. Basic objects of computability 3/79 But for us, it is now obvious that computable = realizable by a program / algorithm. Surprisingly, the first acceptable formalization (Turing machines) is still one of the best (if not the best) we know today. The. intuition At the time these questions were first considered (1930’s), computers did not exist (at least in the modern sense). 1. Basic objects of computability 4/79 Surprisingly, the first acceptable formalization (Turing machines) is still one of the best (if not the best) we know today. -
Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg
Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By “elementary” plane geometry I mean the geometry of lines and circles—straight- edge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incom- pleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry. 1. AXIOMATIC DEVELOPMENT 1.0. Viewpoint. Euclid’s Elements was the first axiomatic presentation of mathemat- ics, based on his five postulates plus his “common notions.” It wasn’t until the end of the nineteenth century that rigorous revisions of Euclid’s axiomatics were presented, filling in the many gaps in his definitions and proofs. The revision with the great- est influence was that by David Hilbert starting in 1899, which will be discussed below. Hilbert not only made Euclid’s geometry rigorous, he investigated the min- imal assumptions needed to prove Euclid’s results, he showed the independence of some of his own axioms from the others, he presented unusual models to show certain statements unprovable from others, and in subsequent editions he explored in his ap- pendices many other interesting topics, including his foundation for plane hyperbolic geometry without bringing in real numbers. -
Computability Theory
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2019 Computability Theory This section is partly inspired by the material in \A Course in Mathematical Logic" by Bell and Machover, Chap 6, sections 1-10. Other references: \Introduction to the theory of computation" by Michael Sipser, and \Com- putability, Complexity, and Languages" by M. Davis and E. Weyuker. Our first goal is to give a formal definition for what it means for a function on N to be com- putable by an algorithm. Historically the first convincing such definition was given by Alan Turing in 1936, in his paper which introduced what we now call Turing machines. Slightly before Turing, Alonzo Church gave a definition based on his lambda calculus. About the same time G¨odel,Herbrand, and Kleene developed definitions based on recursion schemes. Fortunately all of these definitions are equivalent, and each of many other definitions pro- posed later are also equivalent to Turing's definition. This has lead to the general belief that these definitions have got it right, and this assertion is roughly what we now call \Church's Thesis". A natural definition of computable function f on N allows for the possibility that f(x) may not be defined for all x 2 N, because algorithms do not always halt. Thus we will use the symbol 1 to mean “undefined". Definition: A partial function is a function n f :(N [ f1g) ! N [ f1g; n ≥ 0 such that f(c1; :::; cn) = 1 if some ci = 1. In the context of computability theory, whenever we refer to a function on N, we mean a partial function in the above sense. -
Foundations of Geometry
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement. -
Mechanism, Mentalism, and Metamathematics Synthese Library
MECHANISM, MENTALISM, AND METAMATHEMATICS SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Florida State University Editors: ROBER T S. COHEN, Boston University DONALD DAVIDSON, University o/Chicago GABRIEL NUCHELMANS, University 0/ Leyden WESLEY C. SALMON, University 0/ Arizona VOLUME 137 JUDSON CHAMBERS WEBB Boston University. Dept. 0/ Philosophy. Boston. Mass .• U.S.A. MECHANISM, MENT ALISM, AND MET AMA THEMA TICS An Essay on Finitism i Springer-Science+Business Media, B.V. Library of Congress Cataloging in Publication Data Webb, Judson Chambers, 1936- CII:J Mechanism, mentalism, and metamathematics. (Synthese library; v. 137) Bibliography: p. Includes indexes. 1. Metamathematics. I. Title. QA9.8.w4 510: 1 79-27819 ISBN 978-90-481-8357-9 ISBN 978-94-015-7653-6 (eBook) DOl 10.1007/978-94-015-7653-6 All Rights Reserved Copyright © 1980 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1980. Softcover reprint of the hardcover 1st edition 1980 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE vii INTRODUCTION ix CHAPTER I / MECHANISM: SOME HISTORICAL NOTES I. Machines and Demons 2. Machines and Men 17 3. Machines, Arithmetic, and Logic 22 CHAPTER II / MIND, NUMBER, AND THE INFINITE 33 I. The Obligations of Infinity 33 2. Mind and Philosophy of Number 40 3. Dedekind's Theory of Arithmetic 46 4.