Computational Complexity: a Modern Approach PDF Book
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Advances in Theoretical & Computational Physics
ISSN: 2639-0108 Review Article Advances in Theoretical & Computational Physics Cosmology: Preprogrammed Evolution (The problem of Providence) Besud Chu Erdeni Unified Theory Lab, Bayangol disrict, Ulan-Bator, Mongolia *Corresponding author Besud Chu Erdeni, Unified Theory Lab, Bayangol disrict, Ulan-Bator, Mongolia. Submitted: 25 March 2021; Accepted: 08 Apr 2021; Published: 18 Apr 2021 Citation: Besud Chu Erdeni (2021) Cosmology: Preprogrammed Evolution (The problem of Providence). Adv Theo Comp Phy 4(2): 113-120. Abstract This is continued from the article Superunification: Pure Mathematics and Theoretical Physics published in this journal and intended to discuss the general logical and philosophical consequences of the universal mathematical machine described by the superunified field theory. At first was mathematical continuum, that is, uncountably infinite set of real numbers. The continuum is self-exited and self- organized into the universal system of mathematical harmony observed by the intelligent beings in the Cosmos as the physical Universe. Consequently, cosmology as a science of evolution in the Uni- verse can be thought as a preprogrammed natural phenomenon beginning with the Big Bang event, or else, the Creation act pro- (6) cess. The reader is supposed to be acquinted with the Pythagoras’ (Arithmetization) and Plato’s (Geometrization) concepts. Then, With this we can derive even the human gene-chromosome topol- the numeric, Pythagorean, form of 4-dim space-time shall be ogy. (1) The operator that images the organic growth process from nothing is where time is an agorithmic (both geometric and algebraic) bifur- cation of Newton’s absolute space denoted by the golden section exp exp e. -
Journal of Computational Physics
JOURNAL OF COMPUTATIONAL PHYSICS AUTHOR INFORMATION PACK TABLE OF CONTENTS XXX . • Description p.1 • Audience p.2 • Impact Factor p.2 • Abstracting and Indexing p.2 • Editorial Board p.2 • Guide for Authors p.6 ISSN: 0021-9991 DESCRIPTION . Journal of Computational Physics has an open access mirror journal Journal of Computational Physics: X which has the same aims and scope, editorial board and peer-review process. To submit to Journal of Computational Physics: X visit https://www.editorialmanager.com/JCPX/default.aspx. The Journal of Computational Physics focuses on the computational aspects of physical problems. JCP encourages original scientific contributions in advanced mathematical and numerical modeling reflecting a combination of concepts, methods and principles which are often interdisciplinary in nature and span several areas of physics, mechanics, applied mathematics, statistics, applied geometry, computer science, chemistry and other scientific disciplines as well: the Journal's editors seek to emphasize methods that cross disciplinary boundaries. The Journal of Computational Physics also publishes short notes of 4 pages or less (including figures, tables, and references but excluding title pages). Letters to the Editor commenting on articles already published in this Journal will also be considered. Neither notes nor letters should have an abstract. Review articles providing a survey of particular fields are particularly encouraged. Full text articles have a recommended length of 35 pages. In order to estimate the page limit, please use our template. Published conference papers are welcome provided the submitted manuscript is a significant enhancement of the conference paper with substantial additions. Reproducibility, that is the ability to reproduce results obtained by others, is a core principle of the scientific method. -
On Physical Problems That Are Slightly More Difficult Than
On physical problems that are slightly more difficult than QMA Andris Ambainis University of Latvia and IAS, Princeton Email: [email protected] Abstract We study the complexity of computational problems from quantum physics. Typi- cally, they are studied using the complexity class QMA (quantum counterpart of NP ) but some natural computational problems appear to be slightly harder than QMA. We introduce new complexity classes consisting of problems that are solvable with a small number of queries to a QMA oracle and use these complexity classes to quantify the complexity of several natural computational problems (for example, the complexity of estimating the spectral gap of a Hamiltonian). 1 Introduction Quantum Hamiltonian complexity [30] is a new field that combines quantum physics with computer science, by using the notions from computational complexity to study the com- plexity of problems that appear in quantum physics. One of central notions of Hamiltonian complexity is the complexity class QMA [25, 24, 40, 21, 3] which is the quantum counterpart of NP. QMA consists of all computational problems whose solutions can be verified in polynomial time on a quantum computer, given a quantum witness (a quantum state on a polynomial number of qubits). QMA captures the complexity of several interesting physical problems. For example, estimating the ground state energy of a physical system (described by a Hamiltonian) is arXiv:1312.4758v2 [quant-ph] 10 Apr 2014 a very important task in quantum physics. We can characterize the complexity of this problem by showing that it is QMA-complete, even if we restrict it to natural classes of Hamiltonians. -
On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Computational Science and Engineering
Computational Science and Engineering, PhD College of Engineering Graduate Coordinator: TBA Email: Phone: Department Chair: Marwan Bikdash Email: [email protected] Phone: 336-334-7437 The PhD in Computational Science and Engineering (CSE) is an interdisciplinary graduate program designed for students who seek to use advanced computational methods to solve large problems in diverse fields ranging from the basic sciences (physics, chemistry, mathematics, etc.) to sociology, biology, engineering, and economics. The mission of Computational Science and Engineering is to graduate professionals who (a) have expertise in developing novel computational methodologies and products, and/or (b) have extended their expertise in specific disciplines (in science, technology, engineering, and socioeconomics) with computational tools. The Ph.D. program is designed for students with graduate and undergraduate degrees in a variety of fields including engineering, chemistry, physics, mathematics, computer science, and economics who will be trained to develop problem-solving methodologies and computational tools for solving challenging problems. Research in Computational Science and Engineering includes: computational system theory, big data and computational statistics, high-performance computing and scientific visualization, multi-scale and multi-physics modeling, computational solid, fluid and nonlinear dynamics, computational geometry, fast and scalable algorithms, computational civil engineering, bioinformatics and computational biology, and computational physics. Additional Admission Requirements Master of Science or Engineering degree in Computational Science and Engineering (CSE) or in science, engineering, business, economics, technology or in a field allied to computational science or computational engineering field. GRE scores Program Outcomes: Students will demonstrate critical thinking and ability in conducting research in engineering, science and mathematics through computational modeling and simulations. -
Foundations of Cryptography – a Primer Oded Goldreich
Foundations of Cryptography –APrimer Foundations of Cryptography –APrimer Oded Goldreich Department of Computer Science Weizmann Institute of Science Rehovot Israel [email protected] Boston – Delft Foundations and TrendsR in Theoretical Computer Science Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1 781 871 0245 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 A Cataloging-in-Publication record is available from the Library of Congress Printed on acid-free paper ISBN: 1-933019-02-6; ISSNs: Paper version 1551-305X; Electronic version 1551-3068 c 2005 O. Goldreich All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. now Publishers Inc. has an exclusive license to publish this mate- rial worldwide. Permission to use this content must be obtained from the copyright license holder. Please apply to now Publishers, PO Box 179, 2600 AD Delft, The Netherlands, www.nowpublishers.com; e-mail: [email protected] Contents 1 Introduction and Preliminaries 1 1.1 Introduction 1 1.2 Preliminaries 7 IBasicTools 10 2 Computational Difficulty and One-way Functions 13 2.1 One-way functions 14 2.2 Hard-core predicates 18 3 Pseudorandomness 23 3.1 Computational indistinguishability 24 3.2 Pseudorandom generators -
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/. -
Boolean Hierarchies
Bo olean Hierarchies On Collapse Prop erties and Query Order Dissertation zur Erlangung des akademischen Grades do ctor rerum naturalium Dr rer nat vorgelegt dem Rat der Fakultat fur Mathematik und Informatik der FriedrichSchillerUniversitat Jena von DiplomMathematiker Harald Hemp el geb oren am August in Jena Gutachter Prof Dr Gerd Wechsung Prof Edith Hemaspaandra Prof Dr Klaus Wagner Tag des Rigorosums Tag der oentlichen Verteidigung To my family Acknowledgements Words can not express my deep gratitude to my advisor Professor Gerd Wechsung Gen erously he oered supp ort guidance and encouragement throughout the past four years Learning from him and working with him was and still is a pleasure and privilege I much ap preciate Through all the ups and downs of my research his optimism and humane warmth have made the downs less frustrating and the ups more encouraging I want to express my deep gratitude to Professor Lane Hemaspaandra and Professor Edith Hemaspaandra Allowing me to become part of so many joint pro jects has been a wonderful learning exp erience and I much b eneted from their scien tic exp ertise Their generous help and advice help ed me to gain insights into how research is done and made this thesis p ossible For serving as referees for this thesis I am grateful to Professor Edith Hemaspaandra and Professor Klaus Wagner Iwant to thank all my colleagues at Jena esp ecially HaikoMuller Dieter Kratsch Jorg Rothe Johannes Waldmann and Maren Hinrichs for generously oering help and supp ort A regarding the many little things -
The Polynomial Hierarchy
ij 'I '""T', :J[_ ';(" THE POLYNOMIAL HIERARCHY Although the complexity classes we shall study now are in one sense byproducts of our definition of NP, they have a remarkable life of their own. 17.1 OPTIMIZATION PROBLEMS Optimization problems have not been classified in a satisfactory way within the theory of P and NP; it is these problems that motivate the immediate extensions of this theory beyond NP. Let us take the traveling salesman problem as our working example. In the problem TSP we are given the distance matrix of a set of cities; we want to find the shortest tour of the cities. We have studied the complexity of the TSP within the framework of P and NP only indirectly: We defined the decision version TSP (D), and proved it NP-complete (corollary to Theorem 9.7). For the purpose of understanding better the complexity of the traveling salesman problem, we now introduce two more variants. EXACT TSP: Given a distance matrix and an integer B, is the length of the shortest tour equal to B? Also, TSP COST: Given a distance matrix, compute the length of the shortest tour. The four variants can be ordered in "increasing complexity" as follows: TSP (D); EXACTTSP; TSP COST; TSP. Each problem in this progression can be reduced to the next. For the last three problems this is trivial; for the first two one has to notice that the reduction in 411 j ;1 17.1 Optimization Problems 413 I 412 Chapter 17: THE POLYNOMIALHIERARCHY the corollary to Theorem 9.7 proving that TSP (D) is NP-complete can be used with DP. -
Complexity Theory
Complexity Theory Course Notes Sebastiaan A. Terwijn Radboud University Nijmegen Department of Mathematics P.O. Box 9010 6500 GL Nijmegen the Netherlands [email protected] Copyright c 2010 by Sebastiaan A. Terwijn Version: December 2017 ii Contents 1 Introduction 1 1.1 Complexity theory . .1 1.2 Preliminaries . .1 1.3 Turing machines . .2 1.4 Big O and small o .........................3 1.5 Logic . .3 1.6 Number theory . .4 1.7 Exercises . .5 2 Basics 6 2.1 Time and space bounds . .6 2.2 Inclusions between classes . .7 2.3 Hierarchy theorems . .8 2.4 Central complexity classes . 10 2.5 Problems from logic, algebra, and graph theory . 11 2.6 The Immerman-Szelepcs´enyi Theorem . 12 2.7 Exercises . 14 3 Reductions and completeness 16 3.1 Many-one reductions . 16 3.2 NP-complete problems . 18 3.3 More decision problems from logic . 19 3.4 Completeness of Hamilton path and TSP . 22 3.5 Exercises . 24 4 Relativized computation and the polynomial hierarchy 27 4.1 Relativized computation . 27 4.2 The Polynomial Hierarchy . 28 4.3 Relativization . 31 4.4 Exercises . 32 iii 5 Diagonalization 34 5.1 The Halting Problem . 34 5.2 Intermediate sets . 34 5.3 Oracle separations . 36 5.4 Many-one versus Turing reductions . 38 5.5 Sparse sets . 38 5.6 The Gap Theorem . 40 5.7 The Speed-Up Theorem . 41 5.8 Exercises . 43 6 Randomized computation 45 6.1 Probabilistic classes . 45 6.2 More about BPP . 48 6.3 The classes RP and ZPP . -
Collapsing Exact Arithmetic Hierarchies
Electronic Colloquium on Computational Complexity, Report No. 131 (2013) Collapsing Exact Arithmetic Hierarchies Nikhil Balaji and Samir Datta Chennai Mathematical Institute fnikhil,[email protected] Abstract. We provide a uniform framework for proving the collapse of the hierarchy, NC1(C) for an exact arith- metic class C of polynomial degree. These hierarchies collapses all the way down to the third level of the AC0- 0 hierarchy, AC3(C). Our main collapsing exhibits are the classes 1 1 C 2 fC=NC ; C=L; C=SAC ; C=Pg: 1 1 NC (C=L) and NC (C=P) are already known to collapse [1,18,19]. We reiterate that our contribution is a framework that works for all these hierarchies. Our proof generalizes a proof 0 1 from [8] where it is used to prove the collapse of the AC (C=NC ) hierarchy. It is essentially based on a polynomial degree characterization of each of the base classes. 1 Introduction Collapsing hierarchies has been an important activity for structural complexity theorists through the years [12,21,14,23,18,17,4,11]. We provide a uniform framework for proving the collapse of the NC1 hierarchy over an exact arithmetic class. Using 0 our method, such a hierarchy collapses all the way down to the AC3 closure of the class. 1 1 1 Our main collapsing exhibits are the NC hierarchies over the classes C=NC , C=L, C=SAC , C=P. Two of these 1 1 hierarchies, viz. NC (C=L); NC (C=P), are already known to collapse ([1,19,18]) while a weaker collapse is known 0 1 for a third one viz.