Comparing Complexity Classes*
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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 of Space Bounded Interactive Proof Systems
The Complexity of Space Bounded Interactive Proof Systems ANNE CONDON Computer Science Department, University of Wisconsin-Madison 1 INTRODUCTION Some of the most exciting developments in complexity theory in recent years concern the complexity of interactive proof systems, defined by Goldwasser, Micali and Rackoff (1985) and independently by Babai (1985). In this paper, we survey results on the complexity of space bounded interactive proof systems and their applications. An early motivation for the study of interactive proof systems was to extend the notion of NP as the class of problems with efficient \proofs of membership". Informally, a prover can convince a verifier in polynomial time that a string is in an NP language, by presenting a witness of that fact to the verifier. Suppose that the power of the verifier is extended so that it can flip coins and can interact with the prover during the course of a proof. In this way, a verifier can gather statistical evidence that an input is in a language. As we will see, the interactive proof system model precisely captures this in- teraction between a prover P and a verifier V . In the model, the computation of V is probabilistic, but is typically restricted in time or space. A language is accepted by the interactive proof system if, for all inputs in the language, V accepts with high probability, based on the communication with the \honest" prover P . However, on inputs not in the language, V rejects with high prob- ability, even when communicating with a \dishonest" prover. In the general model, V can keep its coin flips secret from the prover. -
Lecture 14 (Feb 28): Probabilistically Checkable Proofs (PCP) 14.1 Motivation: Some Problems and Their Approximation Algorithms
CMPUT 675: Computational Complexity Theory Winter 2019 Lecture 14 (Feb 28): Probabilistically Checkable Proofs (PCP) Lecturer: Zachary Friggstad Scribe: Haozhou Pang 14.1 Motivation: Some problems and their approximation algorithms Many optimization problems are NP-hard, and therefore it is unlikely to efficiently find optimal solutions. However, in some situations, finding a provably good-enough solution (approximation of the optimal) is also acceptable. We say an algorithm is an α-approximation algorithm for an optimization problem iff for every instance of the problem it can find a solution whose cost is a factor of α of the optimum solution cost. Here are some selected problems and their approximation algorithms: Example: Min Vertex Cover is the problem that given a graph G = (V; E), the goal is to find a vertex cover C ⊆ V of minimum size. The following algorithm gives a 2-approximation for Min Vertex Cover: Algorithm 1 Min Vertex Cover Algorithm Input: Graph G = (V; E) Output: a vertex cover C of G. C ; while some edge (u; v) has u; v2 = C do C C [ fu; vg end while return C Claim 1 Let C∗ be an optimal solution, the returned solution C satisfies jCj ≤ 2jC∗j. Proof. Let M be the edges considered by the algorithm in the loop, we have jCj = 2jMj. Also, jC∗j covers M and no two edges in M share an endpoint (M is a matching), so jC∗j ≥ jMj. Therefore, jCj = 2jMj ≤ 2jC∗j. Example: Max SAT is the problem that given a CNF formula, the goal is to find a assignment that satisfies as many clauses as possible. -
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 ). -
Probabilistic Proof Systems: a Primer
Probabilistic Proof Systems: A Primer Oded Goldreich Department of Computer Science and Applied Mathematics Weizmann Institute of Science, Rehovot, Israel. June 30, 2008 Contents Preface 1 Conventions and Organization 3 1 Interactive Proof Systems 4 1.1 Motivation and Perspective ::::::::::::::::::::::: 4 1.1.1 A static object versus an interactive process :::::::::: 5 1.1.2 Prover and Veri¯er :::::::::::::::::::::::: 6 1.1.3 Completeness and Soundness :::::::::::::::::: 6 1.2 De¯nition ::::::::::::::::::::::::::::::::: 7 1.3 The Power of Interactive Proofs ::::::::::::::::::::: 9 1.3.1 A simple example :::::::::::::::::::::::: 9 1.3.2 The full power of interactive proofs ::::::::::::::: 11 1.4 Variants and ¯ner structure: an overview ::::::::::::::: 16 1.4.1 Arthur-Merlin games a.k.a public-coin proof systems ::::: 16 1.4.2 Interactive proof systems with two-sided error ::::::::: 16 1.4.3 A hierarchy of interactive proof systems :::::::::::: 17 1.4.4 Something completely di®erent ::::::::::::::::: 18 1.5 On computationally bounded provers: an overview :::::::::: 18 1.5.1 How powerful should the prover be? :::::::::::::: 19 1.5.2 Computational Soundness :::::::::::::::::::: 20 2 Zero-Knowledge Proof Systems 22 2.1 De¯nitional Issues :::::::::::::::::::::::::::: 23 2.1.1 A wider perspective: the simulation paradigm ::::::::: 23 2.1.2 The basic de¯nitions ::::::::::::::::::::::: 24 2.2 The Power of Zero-Knowledge :::::::::::::::::::::: 26 2.2.1 A simple example :::::::::::::::::::::::: 26 2.2.2 The full power of zero-knowledge proofs :::::::::::: -
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. -
Introduction to Complexity Theory Big O Notation Review Linear Function: R(N)=O(N)
GS019 - Lecture 1 on Complexity Theory Jarod Alper (jalper) Introduction to Complexity Theory Big O Notation Review Linear function: r(n)=O(n). Polynomial function: r(n)=2O(1) Exponential function: r(n)=2nO(1) Logarithmic function: r(n)=O(log n) Poly-log function: r(n)=logO(1) n Definition 1 (TIME) Let t : . Define the time complexity class, TIME(t(n)) to be ℵ−→ℵ TIME(t(n)) = L DTM M which decides L in time O(t(n)) . { |∃ } Definition 2 (NTIME) Let t : . Define the time complexity class, NTIME(t(n)) to be ℵ−→ℵ NTIME(t(n)) = L NDTM M which decides L in time O(t(n)) . { |∃ } Example 1 Consider the language A = 0k1k k 0 . { | ≥ } Is A TIME(n2)? Is A ∈ TIME(n log n)? Is A ∈ TIME(n)? Could∈ we potentially place A in a smaller complexity class if we consider other computational models? Theorem 1 If t(n) n, then every t(n) time multitape Turing machine has an equivalent O(t2(n)) time single-tape turing≥ machine. Proof: see Theorem 7:8 in Sipser (pg. 232) Theorem 2 If t(n) n, then every t(n) time RAM machine has an equivalent O(t3(n)) time multi-tape turing machine.≥ Proof: optional exercise Conclusion: Linear time is model specific; polynomical time is model indepedent. Definition 3 (The Class P ) k P = [ TIME(n ) k Definition 4 (The Class NP) GS019 - Lecture 1 on Complexity Theory Jarod Alper (jalper) k NP = [ NTIME(n ) k Equivalent Definition of NP NP = L L has a polynomial time verifier . -
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. -
Introduction to Complexity Classes
Introduction to Complexity Classes Marcin Sydow Introduction to Complexity Classes Marcin Sydow Introduction Denition to Complexity Classes TIME(f(n)) TIME(f(n)) denotes the set of languages decided by Marcin deterministic TM of TIME complexity f(n) Sydow Denition SPACE(f(n)) denotes the set of languages decided by deterministic TM of SPACE complexity f(n) Denition NTIME(f(n)) denotes the set of languages decided by non-deterministic TM of TIME complexity f(n) Denition NSPACE(f(n)) denotes the set of languages decided by non-deterministic TM of SPACE complexity f(n) Linear Speedup Theorem Introduction to Complexity Classes Marcin Sydow Theorem If L is recognised by machine M in time complexity f(n) then it can be recognised by a machine M' in time complexity f 0(n) = f (n) + (1 + )n, where > 0. Blum's theorem Introduction to Complexity Classes Marcin Sydow There exists a language for which there is no fastest algorithm! (Blum - a Turing Award laureate, 1995) Theorem There exists a language L such that if it is accepted by TM of time complexity f(n) then it is also accepted by some TM in time complexity log(f (n)). Basic complexity classes Introduction to Complexity Classes Marcin (the functions are asymptotic) Sydow P S TIME nj , the class of languages decided in = j>0 ( ) deterministic polynomial time NP S NTIME nj , the class of languages decided in = j>0 ( ) non-deterministic polynomial time EXP S TIME 2nj , the class of languages decided in = j>0 ( ) deterministic exponential time NEXP S NTIME 2nj , the class of languages decided -
Computational Complexity
Computational complexity Plan Complexity of a computation. Complexity classes DTIME(T (n)). Relations between complexity classes. Complete problems. Domino problems. NP-completeness. Complete problems for other classes Alternating machines. – p.1/17 Complexity of a computation A machine M is T (n) time bounded if for every n and word w of size n, every computation of M has length at most T (n). A machine M is S(n) space bounded if for every n and word w of size n, every computation of M uses at most S(n) cells of the working tape. Fact: If M is time or space bounded then L(M) is recursive. If L is recursive then there is a time and space bounded machine recognizing L. DTIME(T (n)) = fL(M) : M is det. and T (n) time boundedg NTIME(T (n)) = fL(M) : M is T (n) time boundedg DSPACE(S(n)) = fL(M) : M is det. and S(n) space boundedg NSPACE(S(n)) = fL(M) : M is S(n) space boundedg . – p.2/17 Hierarchy theorems A function S(n) is space constructible iff there is S(n)-bounded machine that given w writes aS(jwj) on the tape and stops. A function T (n) is time constructible iff there is a machine that on a word of size n makes exactly T (n) steps. Thm: Let S2(n) be a space-constructible function and let S1(n) ≥ log(n). If S1(n) lim infn!1 = 0 S2(n) then DSPACE(S2(n)) − DSPACE(S1(n)) is nonempty. Thm: Let T2(n) be a time-constructible function and let T1(n) log(T1(n)) lim infn!1 = 0 T2(n) then DTIME(T2(n)) − DTIME(T1(n)) is nonempty.