The Complexity of Class Polyl
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CS601 DTIME and DSPACE Lecture 5 Time and Space Functions: T, S
CS601 DTIME and DSPACE Lecture 5 Time and Space functions: t, s : N → N+ Definition 5.1 A set A ⊆ U is in DTIME[t(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M) ≡ w ∈ U M(w)=1 , and 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps. Definition 5.2 A set A ⊆ U is in DSPACE[s(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M), and 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells. (Input tape is “read-only” and not counted as space used.) Example: PALINDROMES ∈ DTIME[n], DSPACE[n]. In fact, PALINDROMES ∈ DSPACE[log n]. [Exercise] 1 CS601 F(DTIME) and F(DSPACE) Lecture 5 Definition 5.3 f : U → U is in F (DTIME[t(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. Definition 5.4 f : U → U is in F (DSPACE[s(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. (Input tape is “read-only”; Output tape is “write-only”. -
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. -
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/. -
Lecture 10: Space Complexity III
Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Lecture 10: Space Complexity III Arijit Bishnu 27.03.2010 Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). -
User's Guide for Complexity: a LATEX Package, Version 0.80
User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required). -
Lecture 4: Space Complexity II: NL=Conl, Savitch's Theorem
COM S 6810 Theory of Computing January 29, 2009 Lecture 4: Space Complexity II Instructor: Rafael Pass Scribe: Shuang Zhao 1 Recall from last lecture Definition 1 (SPACE, NSPACE) SPACE(S(n)) := Languages decidable by some TM using S(n) space; NSPACE(S(n)) := Languages decidable by some NTM using S(n) space. Definition 2 (PSPACE, NPSPACE) [ [ PSPACE := SPACE(nc), NPSPACE := NSPACE(nc). c>1 c>1 Definition 3 (L, NL) L := SPACE(log n), NL := NSPACE(log n). Theorem 1 SPACE(S(n)) ⊆ NSPACE(S(n)) ⊆ DTIME 2 O(S(n)) . This immediately gives that N L ⊆ P. Theorem 2 STCONN is NL-complete. 2 Today’s lecture Theorem 3 N L ⊆ SPACE(log2 n), namely STCONN can be solved in O(log2 n) space. Proof. Recall the STCONN problem: given digraph (V, E) and s, t ∈ V , determine if there exists a path from s to t. 4-1 Define boolean function Reach(u, v, k) with u, v ∈ V and k ∈ Z+ as follows: if there exists a path from u to v with length smaller than or equal to k, then Reach(u, v, k) = 1; otherwise Reach(u, v, k) = 0. It is easy to verify that there exists a path from s to t iff Reach(s, t, |V |) = 1. Next we show that Reach(s, t, |V |) can be recursively computed in O(log2 n) space. For all u, v ∈ V and k ∈ Z+: • k = 1: Reach(u, v, k) = 1 iff (u, v) ∈ E; • k > 1: Reach(u, v, k) = 1 iff there exists w ∈ V such that Reach(u, w, dk/2e) = 1 and Reach(w, v, bk/2c) = 1. -
Supercuspidal Part of the Mod L Cohomology of GU(1,N - 1)-Shimura Varieties
Supercuspidal part of the mod l cohomology of GU(1,n - 1)-Shimura varieties The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Shin, Sug Woo. “Supercuspidal Part of the Mod L Cohomology of GU(1,n - 1)-Shimura Varieties.” Journal für die reine und angewandte Mathematik (Crelles Journal) 2015.705 (2015): 1–21. © 2015 De Gruyter As Published http://dx.doi.org/10.1515/crelle-2013-0057 Publisher Walter de Gruyter Version Final published version Citable link http://hdl.handle.net/1721.1/108760 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. J. reine angew. Math. 705 (2015), 1–21 Journal für die reine und angewandte Mathematik DOI 10.1515/crelle-2013-0057 © De Gruyter 2015 Supercuspidal part of the mod l cohomology of GU.1; n 1/-Shimura varieties By Sug Woo Shin at Cambridge, MA Abstract. Let l be a prime. In this paper we are concerned with GU.1; n 1/-type Shimura varieties with arbitrary level structure at l and investigate the part of the cohomology on which G.Qp/ acts through mod l supercuspidal representations, where p l is any prime ¤ such that G.Qp/ is a general linear group. The main theorem establishes the mod l analogue of the local-global compatibility. Our theorem also encodes a global mod l Jacquet–Langlands correspondence in that the cohomology is described in terms of mod l automorphic forms on some compact inner form of G. -
Space Complexity
Chapter 3 Space complexity “(our) construction... also suggests that what makes “games” harder than “puzzles” (e.g. NP-complete problems) is the fact that the initiative (“the move”) can shift back and forth between the players.” Shimon Even and Robert Tarjan, 1976 In this chapter we will study the memory requirements of computational tasks. To do this we define space-bounded computation, which has to be per- formed by the TM using a restricted number of tape cells, the number being a function of the input size. We also study nondeterministic space-bounded TMs. As in the chapter on NP, our goal in introducing a complexity class is to “capture” interesting computational phenomena— in other words, iden- tify an interesting set of computational problems that lie in the complexity class and are complete for it. One phenomenon we will “capture” this way (see Section 3.3.2) concerns computation of winning strategies in 2-person games, which seems inherently different from (and possibly more difficult than) solving NP problems such as SAT, as alluded to in the above quote. The formal definition of deterministic and non-deterministic space bounded computation is as follows (see also Figure 3.1): Definition 3.1 (Space-bounded computation.) ∗ Let S : N → N and L ⊆ {0, 1} . We say that L ∈ SPACE(s(n)) (resp. L ∈ NSPACE(s(n))) if there is a constant c and TM (resp. NDTM) M deciding L such that on every input x ∈ {0, 1}∗, the total number of locations that are at some point non-blank during M’s execution on x is at most c·s(|x|). -
Complexity Theory Lectures 1–6
Complexity Theory 1 Complexity Theory Lectures 1–6 Lecturer: Dr. Timothy G. Griffin Slides by Anuj Dawar Computer Laboratory University of Cambridge Easter Term 2009 http://www.cl.cam.ac.uk/teaching/0809/Complexity/ Cambridge Easter 2009 Complexity Theory 2 Texts The main texts for the course are: Computational Complexity. Christos H. Papadimitriou. Introduction to the Theory of Computation. Michael Sipser. Other useful references include: Computers and Intractability: A guide to the theory of NP-completeness. Michael R. Garey and David S. Johnson. Structural Complexity. Vols I and II. J.L. Balc´azar, J. D´ıaz and J. Gabarr´o. Computability and Complexity from a Programming Perspective. Neil Jones. Cambridge Easter 2009 Complexity Theory 3 Outline A rough lecture-by-lecture guide, with relevant sections from the text by Papadimitriou (or Sipser, where marked with an S). Algorithms and problems. 1.1–1.3. • Time and space. 2.1–2.5, 2.7. • Time Complexity classes. 7.1, S7.2. • Nondeterminism. 2.7, 9.1, S7.3. • NP-completeness. 8.1–8.2, 9.2. • Graph-theoretic problems. 9.3 • Cambridge Easter 2009 Complexity Theory 4 Outline - contd. Sets, numbers and scheduling. 9.4 • coNP. 10.1–10.2. • Cryptographic complexity. 12.1–12.2. • Space Complexity 7.1, 7.3, S8.1. • Hierarchy 7.2, S9.1. • Descriptive Complexity 5.6, 5.7. • Cambridge Easter 2009 Complexity Theory 5 Complexity Theory Complexity Theory seeks to understand what makes certain problems algorithmically difficult to solve. In Data Structures and Algorithms, we saw how to measure the complexity of specific algorithms, by asymptotic measures of number of steps. -
An Entropy Proof of the Switching Lemma and Tight Bounds on the Decision-Tree Size of AC0
An entropy proof of the switching lemma and tight bounds on the decision-tree size of AC0 Benjamin Rossman∗ University of Toronto November 3, 2017 Abstract We first give a simple entropy argument showing that every m-clause DNF with expected value λ 2 [0; 1] under the uniform distribution has average sensitivity (a.k.a. total influence) at most 2λ log(m/λ). Using a similar idea, we then show the following switching lemma for an m-clause DNF (or CNF) formula F : P t (1) [ DTdepth(F Rp) ≥ t ] ≤ O(p log(m + 1)) : for all p 2 [0; 1] and t 2 N where Rp is the p-random restriction and DTdepth(·) denotes decision- tree depth. Our proof replaces the counting arguments in previous proofs of H˚astad's O(pw)t switching lemma for width-w DNFs [5,9,2] with an entropy argument that naturally applies to unbounded-width DNFs with a bounded number of clauses. With respect to AC0 circuits, our m-clause switching lemma has similar applications as H˚astad'swidth-w switching lemma, 1=(d−1) including a 2Ω(n ) lower bound for PARITY. An additional result of this paper extends inequality (1) to AC0 circuits via a combination of H˚astad'sswitching and multi-switching lemmas [5,6]. For boolean functions f : f0; 1gn ! f0; 1g computable by AC0 circuits of depth d and size s, we show that P d−1 t (2) [ DTdepth(fRp) ≥ t ] ≤ (p · O(log s) ) for all p 2 [0; 1] and t 2 N. -
18.405J S16 Lecture 3: Circuits and Karp-Lipton
18.405J/6.841J: Advanced Complexity Theory Spring 2016 Lecture 3: Circuits and Karp-Lipton Scribe: Anonymous Student Prof. Dana Moshkovitz Scribe Date: Fall 2012 1 Overview In the last lecture we examined relativization. We found that a common method (diagonalization) for proving non-inclusions between classes of problems was, alone, insufficient to settle the P versus NP problem. This was due to the existence of two orcales: one for which P = NP and another for which P =6 NP . Therefore, in our search for a solution, we must look into the inner workings of computation. In this lecture we will examine the role of Boolean Circuits in complexity theory. We will define the class P=poly based on the circuit model of computation. We will go on to give a weak circuit lower bound and prove the Karp-Lipton Theorem. 2 Boolean Circuits Turing Machines are, strange as it may sound, a relatively high level programming language. Because we found that diagonalization alone is not sufficient to settle the P versus NP problem, we want to look at a lower level model of computation. It may, for example, be easier to work at the bit level. To this end, we examine boolean circuits. Note that any function f : f0; 1gn ! f0; 1g is expressible as a boolean circuit. 2.1 Definitions Definition 1. Let T : N ! N.A T -size circuit family is a sequence fCng such that for all n, Cn is a boolean circuit of size (in, say, gates) at most T (n). SIZE(T ) is the class of languages with T -size circuits. -
PSPACE-Complete Languages
CSCI 1590 Intro to Computational Complexity PSPACE-Complete Languages John E. Savage Brown University February 11, 2009 John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 1 / 10 Summary 1 Complexity Class Containment 2 Polynomial Time Hierarchy 3 PH-Complete Languages 4 Games and tqbf 5 tqbf is PSPACE-Complete John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 2 / 10 Complexity Classes from Last Lecture P, coNP, p p Πi and Σi PH PSPACE NP ∪ coNP PSPACE = NPSPACE NP ∩ coNP NP coNP L2 P NL L John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 3 / 10 Polynomial Time Hierarchy A language is in NP(coNP) if and only if it can be reduced in polynomial time to a statement of the form ∃x b(x) (∀x b(x)) By adding additional levels of quantification, as shown below, potentially new complexity classes are added. ∀x1 ∃x2 b(x1, x2) ∃x1 ∀x2 b(x1, x2) The sets of languages PTIME reducible to statements of this form are p p denoted Πi and Σi respectively, when there are i alternations of existential and universal quantifiers and the outermost quantifier is ∀ and ∃, respectively. Definition The Polynomial Hierarchy (PH) is defined as p PH = [ Σi i John E. Savage (Brown University) CSCI 1590 Intro to Computational Complexity February 11, 2009 4 / 10 PH and PSPACE Definition A language L is PH-complete if a) L ∈ PH and b) all languages in PH are ptime reducible to L. It is not hard to see that PH ⊆ PSPACE.