Lecture 16: Relativization Umans Complexity Theory Lecturess.

Slides:



Advertisements
Similar presentations
An Invitation to Quantum Complexity Theory The Study of What We Cant Do With Computers We Dont Have Scott Aaronson (MIT) QIP08, New Delhi BQP NP- complete.
Advertisements

Lower Bounds for Non-Black-Box Zero Knowledge Boaz Barak (IAS*) Yehuda Lindell (IBM) Salil Vadhan (Harvard) *Work done while in Weizmann Institute. Short.
COMPLEXITY THEORY CSci 5403 LECTURE VII: DIAGONALIZATION.
Complexity Theory Lecture 6
Resource-Bounded Computation
Complexity Classes: P and NP
Complexity Theory Lecture 1 Lecturer: Moni Naor. Computational Complexity Theory Study the resources needed to solve computational problems –Computer.
CS151 Complexity Theory Lecture 17 May 27, CS151 Lecture 172 Outline elements of the proof of the PCP Theorem counting problems –#P and its relation.
CSCI 4325 / 6339 Theory of Computation Zhixiang Chen.
Umans Complexity Theory Lectures Lecture 2a: Reductions & Completeness.
Complexity 11-1 Complexity Andrei Bulatov Space Complexity.
Computability and Complexity 22-1 Computability and Complexity Andrei Bulatov Hierarchy Theorem.
Complexity 18-1 Complexity Andrei Bulatov Probabilistic Algorithms.
1 Introduction to Computability Theory Lecture15: Reductions Prof. Amos Israeli.
1 Introduction to Computability Theory Lecture12: Reductions Prof. Amos Israeli.
CS151 Complexity Theory Lecture 7 April 20, 2004.
CS151 Complexity Theory Lecture 1 March 30, 2015.
Arithmetic Hardness vs. Randomness Valentine Kabanets SFU.
CS151 Complexity Theory Lecture 12 May 6, CS151 Lecture 122 Outline The Polynomial-Time Hierarachy (PH) Complete problems for classes in PH, PSPACE.
1 Slides by Golan Weisz, Omer Ben Shalom Nir Ailon & Tal Moran Adapted from Oded Goldreich’s course lecture notes by Moshe Lewenstien, Yehuda Lindell.
Submitted by : Estrella Eisenberg Yair Kaufman Ohad Lipsky Riva Gonen Shalom.
RELATIVIZATION CSE860 Vaishali Athale. Overview Introduction Idea behind “Relativization” Concept of “Oracle” Review of Diagonalization Proof Limits of.
CS151 Complexity Theory Lecture 1 March 30, 2004.
CS151 Complexity Theory Lecture 15 May 18, CS151 Lecture 152 Outline IP = PSPACE Arthur-Merlin games –classes MA, AM Optimization, Approximation,
Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.
Umans Complexity Theory Lectures Lecture 2c: EXP Complete Problem: Padding and succinctness.
February 20, 2015CS21 Lecture 191 CS21 Decidability and Tractability Lecture 19 February 20, 2015.
The Polynomial Hierarchy By Moti Meir And Yitzhak Sapir Based on notes from lectures by Oded Goldreich taken by Ronen Mizrahi, and lectures by Ely Porat.
Definition: Let M be a deterministic Turing Machine that halts on all inputs. Space Complexity of M is the function f:N  N, where f(n) is the maximum.
CSCI 4325 / 6339 Theory of Computation Zhixiang Chen.
Review Byron Gao. Overview Theory of computation: central areas: Automata, Computability, Complexity Computability: Is the problem solvable? –solvable.
Optimal Proof Systems and Sparse Sets Harry Buhrman, CWI Steve Fenner, South Carolina Lance Fortnow, NEC/Chicago Dieter van Melkebeek, DIMACS/Chicago.
Theory of Computing Lecture 17 MAS 714 Hartmut Klauck.
CS 345: Chapter 8 Noncomputability and Undecidability Or Sometimes You Can’t Get It Done At All.
Theory of Computing Lecture 21 MAS 714 Hartmut Klauck.
CS151 Complexity Theory Lecture 12 May 6, QSAT is PSPACE-complete Theorem: QSAT is PSPACE-complete. Proof: 8 x 1 9 x 2 8 x 3 … Qx n φ(x 1, x 2,
1 Design and Analysis of Algorithms Yoram Moses Lecture 11 June 3, 2010
Umans Complexity Theory Lectures Lecture 1a: Problems and Languages.
The Computational Complexity of Satisfiability Lance Fortnow NEC Laboratories America.
A Problem That Is Complete for PSPACE (Polynomial Space) BY TEJA SUDHA GARIGANTI.
Umans Complexity Theory Lectures Lecture 17: Natural Proofs.
CS151 Complexity Theory Lecture 16 May 20, The outer verifier Theorem: NP  PCP[log n, polylog n] Proof (first steps): –define: Polynomial Constraint.
Lecture 21 Reducibility. Many-one reducibility For two sets A c Σ* and B c Γ*, A ≤ m B if there exists a Turing-computable function f: Σ* → Γ* such that.
Umans Complexity Theory Lectures Lecture 1c: Robust Time & Space Classes.
Complexity 24-1 Complexity Andrei Bulatov Interactive Proofs.
Comparing Notions of Full Derandomization Lance Fortnow NEC Research Institute With thanks to Dieter van Melkebeek.
Eric Allender Rutgers University Curiouser and Curiouser: The Link between Incompressibility and Complexity CiE Special Session, June 19, 2012.
Homework 8 Solutions Problem 1. Draw a diagram showing the various classes of languages that we have discussed and alluded to in terms of which class.
Theory of Computational Complexity Yuji Ishikawa Avis lab. M1.
Recall last lecture and Nondeterministic TMs Ola Svensson.
Complexity 27-1 Complexity Andrei Bulatov Interactive Proofs (continued)
Umans Complexity Theory Lectures Lecture 3c: Nondeterminism: Ladner’s Theorem: If P≠ NP then ∃ NP Languages neither NP Complete nor in P.
Umans Complexity Theory Lectures
Probabilistic Algorithms
Umans Complexity Theory Lectures
Scott Aaronson (MIT) QIP08, New Delhi
Computational Complexity Theory
Umans Complexity Theory Lectures
Theory of Computability
Perspective on Lower Bounds: Diagonalization
Umans Complexity Theory Lectures
CS151 Complexity Theory Lecture 17 May 31, 2017.
Umans Complexity Theory Lectures
Introduction to Oracles in Complexity Theory
CS21 Decidability and Tractability
Our First NP-Complete Problem
The Polynomial Hierarchy
CS151 Complexity Theory Lecture 1 April 2, 2019.
Instructor: Aaron Roth
Intro to Theory of Computation
Presentation transcript:

Lecture 16: Relativization Umans Complexity Theory Lecturess

2 Approaches to open problems Almost all major open problems we have seen entail proving lower bounds –P ≠ NP- P = BPP * –L ≠ P- NP = AM * –P ≠ PSPACE –NC proper –BPP ≠ EXP –PH proper –EXP * P/poly we know circuit lower bounds imply derandomization more difficult (and recent): derandomization implies circuit lower bounds!

3 Approaches to open problems two natural approaches –simulation + diagonalization (uniform) –circuit lower bounds (non-uniform) no success for either approach as applied to date Why ?

4 Approaches to open problems in a precise, formal sense these approaches are too powerful ! if they could be used to resolve major open problems, a side effect would be: –proving something that is false, or –proving something that is believed to be false

5 Relativization Many proofs and techniques we have seen relativize: –they hold after replacing all TMs with oracle TMs that have access to an oracle A –e.g. L A  P A for all oracles A –e.g. P A ≠ EXP A for all oracles A

6 Relativization Idea: design an oracle A relative to which some statement is false –implies there can be no relativizing proof of that statement –e.g. design A for which P A = NP A Better: also design an oracle B relative to which statement is true –e.g. also design B for which P B ≠ NP B –implies no relativizing proof can resolve truth of the statement either way !

7 Relativization Oracles are known that falsify almost every major conjecture concerning complexity classes –for these conjectures, non-relativizing proofs are required –almost all known proofs in Complexity relativize (sometimes after some reformulation) – notable exceptions: The PCP Theorem IP = PSPACE most circuit lower bounds (more on these later)

8 Oracles for P vs. NP Goal: –oracle A for which P A = NP A –oracle B for which P B ≠ NP B conclusion: resolving P vs. NP requires a non-relativizing proof

9 Oracles for P vs. NP for P A = NP A need A to be powerful –warning: intend to make P more powerful, but also make NP more powerful. –e.g. A = SAT doesn’t work –however A = QSAT works: (QSAT = satisfiability of quantified Boolean formulas) PSPACE  P QSAT  NP QSAT  NPSPACE and we know NPSPACE  PSPACE

10 Oracles for P vs. NP Theorem: there exists an oracle B for which P B ≠ NP B. Proof: –define L = {1 k :  x  B s.t. |x| = k} –we will show L  NP B – P B. –easy: L  NP B (no matter what B is)

11 Oracles for P vs. NP –design B by diagonalizing against all “P B machines” –M 1, M 2, M 3, … is an enumeration of deterministic OTMs –each machine appears infinitely often –B i will be those strings of length  i in B –we build B i after simulating machine M i

12 Oracles for P vs. NP L = {1 k :  x  B s.t. |x| = k} Proof (continued): –maintain “exceptions” X that must not go in B –initially X = { }, B 0 = { } Stage i: –simulate M i (1 i ) for i log i steps –when M i makes an oracle query q: if |q| < i, answer using B i-1 if |q|  i, answer “no”; add q to X –if simulated M i accepts 1 i then B i = B i-1 –if simulated M i rejects 1 i, B i = B i-1  {x  {0,1} i : x  X}

13 Oracles for P vs. NP L = {1 k :  x  B s.t. |x| = k} Proof (continued): –if M i accepts, we ensure no strings of length i in B –therefore 1 i  L, and so M i does not decide L –if M i rejects, we ensure some string of length i in B –Why? B i = B i-1  {x  {0,1} i : x  X} and |X| is at most Σ j  i j log j << 2 i –therefore 1 i  L, and so M i does not decide L –Conclude: L  P B