Fixed-Polynomial Size Circuit Bounds
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Interactive Proof Systems and Alternating Time-Space Complexity
Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement. -
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/. -
Delegating Computation: Interactive Proofs for Muggles∗
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 108 (2017) Delegating Computation: Interactive Proofs for Muggles∗ Shafi Goldwasser Yael Tauman Kalai Guy N. Rothblum MIT and Weizmann Institute Microsoft Research Weizmann Institute [email protected] [email protected] [email protected] Abstract In this work we study interactive proofs for tractable languages. The (honest) prover should be efficient and run in polynomial time, or in other words a \muggle".1 The verifier should be super-efficient and run in nearly-linear time. These proof systems can be used for delegating computation: a server can run a computation for a client and interactively prove the correctness of the result. The client can verify the result's correctness in nearly-linear time (instead of running the entire computation itself). Previously, related questions were considered in the Holographic Proof setting by Babai, Fortnow, Levin and Szegedy, in the argument setting under computational assumptions by Kilian, and in the random oracle model by Micali. Our focus, however, is on the original inter- active proof model where no assumptions are made on the computational power or adaptiveness of dishonest provers. Our main technical theorem gives a public coin interactive proof for any language computable by a log-space uniform boolean circuit with depth d and input length n. The verifier runs in time n · poly(d; log(n)) and space O(log(n)), the communication complexity is poly(d; log(n)), and the prover runs in time poly(n). In particular, for languages computable by log-space uniform NC (circuits of polylog(n) depth), the prover is efficient, the verifier runs in time n · polylog(n) and space O(log(n)), and the communication complexity is polylog(n). -
Interactive Proofs
Interactive proofs April 12, 2014 [72] L´aszl´oBabai. Trading group theory for randomness. In Proc. 17th STOC, pages 421{429. ACM Press, 1985. doi:10.1145/22145.22192. [89] L´aszl´oBabai and Shlomo Moran. Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. J. Comput. System Sci., 36(2):254{276, 1988. doi:10.1016/0022-0000(88)90028-1. [99] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic ex- ponential time has two-prover interactive protocols. In Proc. 31st FOCS, pages 16{25. IEEE Comp. Soc. Press, 1990. doi:10.1109/FSCS.1990.89520. See item 1991.108. [108] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic expo- nential time has two-prover interactive protocols. Comput. Complexity, 1 (1):3{40, 1991. doi:10.1007/BF01200056. Full version of 1990.99. [136] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. In Proc. 34th FOCS, pages 724{733, Palo Alto CA, 1993. IEEE Comp. Soc. Press. doi:10.1109/SFCS.1993.366815. Conference version of item 1997:160. [160] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. System Sci., 54(2):317{331, 1997. doi:10.1006/jcss.1997.1472. Full version of 1993.136. [111] L´aszl´oBabai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. In Proc. -
Simple Doubly-Efficient Interactive Proof Systems for Locally
Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 18 (2017) Simple doubly-efficient interactive proof systems for locally-characterizable sets Oded Goldreich∗ Guy N. Rothblumy September 8, 2017 Abstract A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier’s strategy can be implemented in almost-linear-time. We present direct constructions of doubly-efficient interactive proof systems for problems in P that are believed to have relatively high complexity. Specifically, such constructions are presented for t-CLIQUE and t-SUM. In addition, we present a generic construction of such proof systems for a natural class that contains both problems and is in NC (and also in SC). The proof systems presented by us are significantly simpler than the proof systems presented by Goldwasser, Kalai and Rothblum (JACM, 2015), let alone those presented by Reingold, Roth- blum, and Rothblum (STOC, 2016), and can be implemented using a smaller number of rounds. Contents 1 Introduction 1 1.1 The current work . 1 1.2 Relation to prior work . 3 1.3 Organization and conventions . 4 2 Preliminaries: The sum-check protocol 5 3 The case of t-CLIQUE 5 4 The general result 7 4.1 A natural class: locally-characterizable sets . 7 4.2 Proof of Theorem 1 . 8 4.3 Generalization: round versus computation trade-off . 9 4.4 Extension to a wider class . 10 5 The case of t-SUM 13 References 15 Appendix: An MA proof system for locally-chracterizable sets 18 ∗Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. -
A Study of the NEXP Vs. P/Poly Problem and Its Variants by Barıs
A Study of the NEXP vs. P/poly Problem and Its Variants by Barı¸sAydınlıoglu˘ A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Sciences) at the UNIVERSITY OF WISCONSIN–MADISON 2017 Date of final oral examination: August 15, 2017 This dissertation is approved by the following members of the Final Oral Committee: Eric Bach, Professor, Computer Sciences Jin-Yi Cai, Professor, Computer Sciences Shuchi Chawla, Associate Professor, Computer Sciences Loris D’Antoni, Asssistant Professor, Computer Sciences Joseph S. Miller, Professor, Mathematics © Copyright by Barı¸sAydınlıoglu˘ 2017 All Rights Reserved i To Azadeh ii acknowledgments I am grateful to my advisor Eric Bach, for taking me on as his student, for being a constant source of inspiration and guidance, for his patience, time, and for our collaboration in [9]. I have a story to tell about that last one, the paper [9]. It was a late Monday night, 9:46 PM to be exact, when I e-mailed Eric this: Subject: question Eric, I am attaching two lemmas. They seem simple enough. Do they seem plausible to you? Do you see a proof/counterexample? Five minutes past midnight, Eric responded, Subject: one down, one to go. I think the first result is just linear algebra. and proceeded to give a proof from The Book. I was ecstatic, though only for fifteen minutes because then he sent a counterexample refuting the other lemma. But a third lemma, inspired by his counterexample, tied everything together. All within three hours. On a Monday midnight. I only wish that I had asked to work with him sooner. -
NP-Complete Problems and Physical Reality
NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia- batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing.” The section on soap bubbles even includes some “experimental” re- sults. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1 Introduction “Let a computer smear—with the right kind of quantum randomness—and you create, in effect, a ‘parallel’ machine with an astronomical number of processors . All you have to do is be sure that when you collapse the system, you choose the version that happened to find the needle in the mathematical haystack.” —From Quarantine [31], a 1992 science-fiction novel by Greg Egan If I had to debate the science writer John Horgan’s claim that basic science is coming to an end [48], my argument would lean heavily on one fact: it has been only a decade since we learned that quantum computers could factor integers in polynomial time. In my (unbiased) opinion, the showdown that quantum computing has forced—between our deepest intuitions about computers on the one hand, and our best-confirmed theory of the physical world on the other—constitutes one of the most exciting scientific dramas of our time. -
A Note on NP ∩ Conp/Poly Copyright C 2000, Vinodchandran N
BRICS Basic Research in Computer Science BRICS RS-00-19 V. N. Variyam: A Note on A Note on NP \ coNP=poly NP \ coNP = Vinodchandran N. Variyam poly BRICS Report Series RS-00-19 ISSN 0909-0878 August 2000 Copyright c 2000, Vinodchandran N. Variyam. BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory RS/00/19/ A Note on NP ∩ coNP/poly N. V. Vinodchandran BRICS, Department of Computer Science, University of Aarhus, Denmark. [email protected] August, 2000 Abstract In this note we show that AMexp 6⊆ NP ∩ coNP=poly, where AMexp denotes the exponential version of the class AM.Themain part of the proof is a collapse of EXP to AM under the assumption that EXP ⊆ NP ∩ coNP=poly 1 Introduction The issue of how powerful circuit based computation is, in comparison with Turing machine based computation has considerable importance in complex- ity theory. There are a large number of important open problems in this area. -
April 2017 Table of Contents
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society April 2017 Volume 64, Number 4 AMS Prize Announcements page 311 Spring Sectional Sampler page 333 AWM Research Symposium 2017 Lecture Sampler page 341 Mathematics and Statistics Awareness Month page 362 About the Cover: How Minimal Surfaces Converge to a Foliation (see page 307) MATHEMATICAL CONGRESS OF THE AMERICAS MCA 2017 JULY 2428, 2017 | MONTREAL CANADA MCA2017 will take place in the beautiful city of Montreal on July 24–28, 2017. The many exciting activities planned include 25 invited lectures by very distinguished mathematicians from across the Americas, 72 special sessions covering a broad spectrum of mathematics, public lectures by Étienne Ghys and Erik Demaine, a concert by the Cecilia String Quartet, presentation of the MCA Prizes and much more. SPONSORS AND PARTNERS INCLUDE Canadian Mathematical Society American Mathematical Society Pacifi c Institute for the Mathematical Sciences Society for Industrial and Applied Mathematics The Fields Institute for Research in Mathematical Sciences National Science Foundation Centre de Recherches Mathématiques Conacyt, Mexico Atlantic Association for Research in Mathematical Sciences Instituto de Matemática Pura e Aplicada Tourisme Montréal Sociedade Brasileira de Matemática FRQNT Quebec Unión Matemática Argentina Centro de Modelamiento Matemático For detailed information please see the web site at www.mca2017.org. AMERICAN MATHEMATICAL SOCIETY PUSHING LIMITS From West Point to Berkeley & Beyond PUSHING LIMITS FROM WEST POINT TO BERKELEY & BEYOND Ted Hill, Georgia Tech, Atlanta, GA, and Cal Poly, San Luis Obispo, CA Recounting the unique odyssey of a noted mathematician who overcame military hurdles at West Point, Army Ranger School, and the Vietnam War, this is the tale of an academic career as noteworthy for its o beat adventures as for its teaching and research accomplishments. -
Circuit Lower Bounds for Merlin-Arthur Classes
Electronic Colloquium on Computational Complexity, Report No. 5 (2007) Circuit Lower Bounds for Merlin-Arthur Classes Rahul Santhanam Simon Fraser University [email protected] January 16, 2007 Abstract We show that for each k > 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA AM ZPPNP , and k . We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the average-case setting, i.e., we show that MA/1 cannot be solved on more than 1/2+1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudo-random generators computable using O(1) bits of advice. 1 Introduction Proving circuit lower bounds within uniform complexity classes is one of the most fundamental and challenging tasks in complexity theory. Apart from clarifying our understanding of the power of non-uniformity, circuit lower bounds have direct relevance to some longstanding open questions. Proving super-polynomial circuit lower bounds for NP would separate P from NP. The weaker result that for each k there is a language in NP which doesn’t have circuits of size nk would separate BPP from NEXP, thus answering an important question in the theory of derandomization. -
Nondeterministic Direct Product Reducations and the Success
Nondeterministic Direct Product Reductions and the Success Probability of SAT Solvers [Preliminary version] Andrew Drucker∗ May 6, 2013 Last updated: May 16, 2013 Abstract We give two nondeterministic reductions which yield new direct product theorems (DPTs) for Boolean circuits. In these theorems one assumes that a target function f is mildly hard against nondeterministic circuits, and concludes that the direct product f ⊗t is extremely hard against (only polynomially smaller) probabilistic circuits. The main advantage of these results compared with previous DPTs is the strength of the size bound in our conclusion. As an application, we show that if NP is not in coNP=poly then, for every PPT algorithm attempting to produce satisfying assigments to Boolean formulas, there are infinitely many instances where the algorithm's success probability is nearly-exponentially small. This furthers a project of Paturi and Pudl´ak[STOC'10]. Contents 1 Introduction 2 1.1 Hardness amplification and direct product theorems..................2 1.2 Our new direct product theorems.............................4 1.3 Application to the success probability of PPT SAT solvers...............6 1.4 Our techniques.......................................7 2 Preliminaries 9 2.1 Deterministic and probabilistic Boolean circuits..................... 10 2.2 Nondeterministic circuits and nondeterministic mappings............... 10 2.3 Direct-product solvers................................... 12 2.4 Hash families and their properties............................ 13 2.5 A general nondeterministic circuit construction..................... 14 ∗School of Mathematics, IAS, Princeton, NJ. Email: [email protected]. This material is based upon work supported by the National Science Foundation under agreements Princeton University Prime Award No. CCF- 0832797 and Sub-contract No. 00001583. -
The Power of Algorithmic Approaches to the Graph Isomorphism Problem
The Power of Algorithmic Approaches to the Graph Isomorphism Problem Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Daniel Neuen, Master of Science aus Dormagen Berichter: Universit¨atsprofessorDr. Martin Grohe Universit¨atsprofessorDr. Pascal Schweitzer Universit¨atsprofessorDr. L´aszl´oBabai Tag der m¨undlichen Pr¨ufung:17. Dezember 2019 Diese Dissertation ist auf den Internetseiten der Universit¨atsbibliothek online verf¨ugbar. ii Abstract The Graph Isomorphism Problem asks, given two input graphs, whether they are structurally the same, that is, whether there is a renaming of the vertices of the first graph in order to transform it to the second graph. By a recent breakthrough result of Babai (STOC 2016), this problem can be solved in quasipolynomial time. However, despite extensive research efforts, it remains one of only few natural problems in NP that are neither known to be solvable in polynomial time nor known to be NP- complete. Over the past five decades several powerful techniques tackling the Graph Isomor- phism Problem have been investigated uncovering various surprising links between different approaches. Also, the situation has led to a number of algorithms solving the isomorphism problem on restricted classes of input graphs. In this thesis, we continue the investigation of various standard approaches to the Graph Isomorphism Problem to further broaden our understanding on the power and limits of such approaches. In particular, this leads to several improved algorithms solving the isomorphism problem for important restricted classes of graphs. One of the most fundamental methods in the context of graph isomorphism test- ing is the Weisfeiler-Leman algorithm, which iteratively computes an isomorphism- invariant coloring of vertex-tuples.