Computational Complexity of Synchronization Under Sparse Regular Constraints
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture Notes in Computer Science 2929 Edited by G
Lecture Notes in Computer Science 2929 Edited by G. Goos, J. Hartmanis, and J. van Leeuwen 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Harrie de Swart Ewa Orłowska Gunther Schmidt Marc Roubens (Eds.) Theory and Applications of Relational Structures as Knowledge Instruments COST Action 274, TARSKI Revised Papers 13 Series Editors Gerhard Goos, Karlsruhe University, Germany Juris Hartmanis, Cornell University, NY, USA Jan van Leeuwen, Utrecht University, The Netherlands Volume Editors Harrie de Swart Tilburg University, Faculty of Philosophy P.O. Box 90153, 5000 LE Tilburg, The Netherlands E-mail: [email protected] Ewa Orłowska National Institute of Telecommunications ul. Szachowa 1, 04-894 Warsaw, Poland E-mail: [email protected] Gunther Schmidt Universit¨atder Bundeswehr M¨unchen Fakult¨atf¨urInformatik, Institut f¨urSoftwaretechnologie 85577 Neubiberg, Germany E-mail: [email protected] Marc Roubens University of Li`ege,Department of Mathematics Sart Tilman Building, 14 Grande Traverse, B37, 4000 Li`ege1, Belgium E-mail: [email protected] Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. CR Subject Classification (1998): I.1, I.2, F.4, H.2.8 ISSN 0302-9743 ISBN 3-540-20780-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. -
Sparse Complete Sets for Conp: Solution of the P Versus NP Problem
Sparse complete sets for coNP: Solution of the P versus NP problem Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected] https://orcid.org/0000-0001-8210-4126 Abstract P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP- complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP. 2012 ACM Subject Classification Theory of computation → Complexity classes, Theory of computation → Problems, reductions and completeness Keywords and phrases Complexity Classes, Sparse, Complement Language, Completeness, Poly- nomial Time 1 Issues The P versus NP problem is a major unsolved problem in computer science [6]. This is considered by many to be the most important open problem in the field [6]. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution [6]. It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency [1]. -
Sparse Complete Sets for Conp: Solution of the P Versus NP Problem
FRANK VEGA Computational Complextiy Sparse complete sets for coNP: Solution of the P versus NP problem Frank Vega March 4, 2019 Abstract: P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP. 1 Introduction The P versus NP problem is a major unsolved problem in computer science [6]. This is considered by many to be the most important open problem in the field [6]. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution [6]. It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency [1]. However, the precise statement of the P = NP problem was introduced in 1971 by Stephen Cook in a seminal paper [6]. ACM Classification: F.1.3.3, F.1.3.2 AMS Classification: 68Q15, 68Q17 Key words and phrases: complexity classes, complement language, sparse, completeness, polynomial time FRANK VEGA In 1936, Turing developed his theoretical computational model [19]. -
Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors' permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] | Thank You!! ii Chapter 6 Boolean Circuits \One might imagine that P 6= NP, but SAT is tractable in the following sense: for every ` there is a very short program that runs in time `2 and correctly treats all instances of size `." Karp and Lipton, 1982 This chapter investigates a model of computation called the Boolean circuit, that is a general- ization of Boolean formulae and a formalization of the silicon chips used by electronic computers. It is a natural model for non-uniform computation, which means a computational model that allows a different algorithm to be used for each input size, in contrast to the standard (or uniform) TM model where the same TM is used on all the infinitely many input sizes. Non-uniform computation crops up often in complexity theory (e.g., see chapters 18 and 19). Another motivation for studying Boolean circuits is that they seem mathematically simpler than Turing machines, and hence it might be easier to prove upper bounds on their computational power. Furthermore, as we shall see in Section 6.1, sufficiently strong results of this type for circuits will resolve some important questions about Turing machines such as P versus NP. -
Ladner's Theorem, Sparse and Dense Languages
Ladner's theorem Density Ladner's Theorem, Sparse and Dense Languages Vasiliki Velona µ Q λ8 November 2014 Ladner's theorem Density Ladner's theorem Part 1, Ladner's Theorem Ladner's theorem Density Ladner's theorem Part 1: Ladner's theorem (Ladner, 1975): If P 6= NP, then there is a language in P which is neither in P or is NP complete The second scenario is impossible. Ladner's theorem Density Ladner's theorem Ladner's theorem proof Preliminaries We can compute an enumeration of all polynomialy bounded TMs (M1; M2; M3; :::) and all logarithmic space reductions (R1; R2; R3; :::) Why? One way is to use a polynomial "clock" on every Mi that will allow it to run for no more than jxji steps for input x. Similarly, we can do this for logarithmic space reductions. Ladner's theorem Density Ladner's theorem Ladner's theorem proof The wanted language is described in terms of the machine K that decides it: L(K) = fxjx 2 SAT and f (jxj) is eveng f (n) will be described later. The demands for this language are the following: 1 8iL(K) 6= L(Mi ) (out of P) 2 8i9w : K(Ri (w)) 6= S(w) (out of NP-complete) and we'll prove that they're met. Ladner's theorem Density Ladner's theorem Ladner's theorem proof We want to check our conditions one after the other: M1; R1; M2; R2; ::: Let F be the turing machine that computes f. For n = 1 F makes two steps and outputs 2. -
Signature Redacted
Connections between Circuit Analysis Problems and Circuit Lower Bounds by Cody Murray Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2018 Massachusetts Institute of Technology 2018. All rights reserved. Signature redacted- Author .... ......................... Department of Ele ngineering and Computer Science June 22, 2018 redacted. Certified by. Signature Ryan Williams Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Signature redacted Accepted by.......... Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students MASSACHUSMlS INSTITUTE c Of TECHNOLOGY- OCT 10 2018 L_BRARIES 2 Connections between Circuit Analysis Problems and Circuit Lower Bounds by Cody Murray Submitted to the Department of Electrical Engineering and Computer Science on June 22, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science and Engineering Abstract A circuit analysis problem takes a Boolean function f as input (where f is repre- sented either as a logical circuit, or as a truth table) and determines some interesting property of f. Examples of circuit analysis problems include Circuit Satisfiability, Circuit Composition, and the Minimum Size Circuit Problem (MCSP). A circuit lower bound presents an interesting function f and shows that no "easy" family of logical circuits can compute f correctly on all inputs, for some definition of "easy". Lower bounds are infamously hard to prove, but are of significant interest for under- standing computation. In this thesis, we derive new connections between circuit analysis problems and cir- cuit lower bounds, to prove new lower bounds for various well-studied circuit classes. -
Sparse Complete Sets for Conp: Solution of the P Versus NP Problem
1 Sparse complete sets for coNP: Solution of the P versus NP problem Frank Vega Abstract—P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP. Keywords—Complexity Classes, Sparse, Complement Language, Completeness, Polynomial Time. F 1 INTRODUCTION The P versus NP problem is a major unsolved problem possible positive constant k [5]. Whether P = NP or not in computer science [1]. This is considered by many to be is still a controversial and unsolved problem [2]. In this the most important open problem in the field [1]. It is one work, we proved the complexity class P is equal to NP . of the seven Millennium Prize Problems selected by the Hence, we solved one of the most important open problems Clay Mathematics Institute to carry a US$1,000,000 prize for in computer science. the first correct solution [1]. It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency [2]. -
Sparse Complete Sets for Conp: Solution of the P Versus NP Problem Frank Vega
Sparse complete sets for coNP: Solution of the P versus NP problem Frank Vega To cite this version: Frank Vega. Sparse complete sets for coNP: Solution of the P versus NP problem. 2021. hal-03320440 HAL Id: hal-03320440 https://hal.archives-ouvertes.fr/hal-03320440 Preprint submitted on 16 Aug 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. FRANK VEGA Computational Complextiy Sparse complete sets for coNP: Solution of the P versus NP problem Frank Vega March 4, 2019 Abstract: P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete.