Umans Complexity Theory Lectures
|
|
- Irene O’Neal’
- 5 years ago
- Views:
Transcription
1 Umans Complexity Theory Lectures Lecture 12: The Polynomial-Time Hierarchy Oracle Turing Machines Oracle Turing Machine (OTM): Deterministic multitape TM M with special query tape special states q?, q yes, q no on input x, with oracle language A M A runs as usual, except when M A enters state q? : y = contents of query tape y A transition to q yes y A transition to q no 2 Oracle Turing Machines Nondeterministic Oracle TM (NOTM) defined in the same way (transition relation, rather than function) oracle is like a subroutine, or function in your favorite programming language but each call counts as single step e.g.: given φ 1, φ 2,, φ n are even # satisfiable? poly-time OTM solves with SAT oracle Oracle Turing Machines Shorthand #1: applying oracles to entire complexity classes: complexity class C language A C A = {L decided by OTM M with oracle A with M in C} example: P SAT 3 4
2 Oracle Turing Machines Shorthand #2: using complexity classes as oracles: OTM M complexity class C M C decides language L if for some language A C, M A decides L Both together: C D = languages decided by OTM in C with oracle language from D exercise: show P SAT = P NP The Polynomial-Time Hierarchy can define lots of complexity classes using oracles the classes on the next slide stand out they have natural complete problems they have a natural interpretation in terms of alternating quantifiers they help us state certain consequences and containments (more later) 5 6 The Polynomial-Time Hierarchy Σ 0 = Π 0 = P Δ 1 =P P Σ 1 =NP Π 1 =conp Δ 2 =P NP Σ 2 =NP NP Π 2 =conp NP Δ i+1 =P Σ i Σ i+i =NP Σ i Π i+1 =conp Σ i Polynomial Hierarchy PH = i Σ i The Polynomial-Time Hierarchy Σ 0 = Π 0 = P Δ i+1 =P Σ i Σ i+i =NP Σ i Π i+1 =conp Σ i Example: MN CRCUT: given Boolean circuit C, integer k; is there a circuit C of size at most k that computes the same function C does? MN CRCUT Σ 2 7 8
3 The Polynomial-Time Hierarchy Σ 0 = Π 0 = P Δ i+1 =P Σ i Σ i+i =NP Σ i Π i+1 =conp Σ i Example: EXACT TSP: given a weighted graph G, and an integer k; is the k-th bit of the length of the shortest TSP tour in G a 1? EXACT TSP Δ 2 The PH PSPACE: generalized geography, 2-person games 3rd level: V-C dimension 2nd level: MN CRCUT, BPP 1st level: SAT, UNSAT, factoring, etc EXP PSPACE PH Σ 3 Π 3 Δ 3 Σ 2 Π 2 Δ 2 NP conp 9 P 10 Useful characterization Recall: L NP iff expressible as L = { x y, y x k, (x, y) R } where R P. Useful characterization Theorem: L Σ i iff expressible as L = { x y, y x k, (x, y) R } where R Π i-1. Corollary: L conp iff expressible as L = { x y, y x k, (x, y) R } where R P. Corollary: L Π i iff expressible as L = { x y, y x k, (x, y) R } where R Σ i
4 Useful characterization Theorem: L Σ i iff expressible as L = { x y, y x k, (x, y) R }, where R Π i-1. Proof of Theorem: induction on i base case (i =1) on previous slide ( ) we know Σ i = NP Σ i-1 = NP Π i-1 guess y, ask oracle if (x, y) R Useful characterization Theorem: L Σ i iff expressible as L = { x y, y x k, (x, y) R }, where R Π i-1. ( ) given L Σ i = NP Σ i-1 decided by ONTM M running in time n k try: R = { (x, y) : y describes valid path of M s computation leading to q accept } but how to recognize valid computation path when it depends on result of oracle queries? Useful characterization Theorem: L Σ i iff expressible as L = { x y, y x k, (x, y) R }, where R Π i-1. try: R = { (x, y) : y describes valid path of M s computation leading to q accept } valid path = step-by-step description including correct yes/no answer for each A-oracle query z j (A Σ i-1 ) verify no queries in Π i-1 : e.g: z 1 A z 3 A z 8 A for each yes query z j : w j, w j z j k with (z j, w j ) R for some R Π i-2 by induction. for each yes query z j put w j in description of path y 15 Useful characterization Theorem: L Σ i iff expressible as L = { x y, y x k, (x, y) R }, where R Π i-1. single language R in Π i-1 : (x, y) R all no z j are not in A and all yes z j have (z j, w j ) R and y is a path leading to q accept. Note: AND of polynomially-many Π i-1 predicates is in Π i-1. 16
5 Alternating quantifiers Nicer, more usable version: L Σ i iff expressible as L = { x y 1 y 2 y 3 Qy i (x, y 1,y 2,,y i ) R } where Q= / if i even/odd, and R P L Π i iff expressible as L = { x y 1 y 2 y 3 Qy i (x, y 1,y 2,,y i ) R } where Q= / if i even/odd, and R P 17 Alternating quantifiers Proof: ( ) induction on i base case: true for Σ 1 =NP and Π 1 =conp consider L Σ i : L = {x y 1 (x, y 1 ) R }, for R Π i-1 L = {x y 1 y 2 y 3 Qy i ((x, y 1 ), y 2,,y i ) R} L = {x y 1 y 2 y 3 Qy i (x, y 1,y 2,,y i ) R} where Q= / if i even/odd, and R P same argument for L Π i ( ) exercise. 18 Alternating quantifiers Complete problems Pleasing viewpoint: const. # of alternations NP Δ 3 Σ 2 Π 2 Δ 2 P conp poly(n) alternations PSPACE PH Σ i Π i Σ 3 Π 3 19 three variants of SAT: QSAT i (i odd) = {3-CNFs φ(x 1, x 2,, x i ) for which x 1 x 2 x 3 x i φ(x 1, x 2,, x i ) = 1} QSAT i (i even) = {3-DNFs φ(x 1, x 2,, x i ) for which x 1 x 2 x 3 x i φ(x 1, x 2,, x i ) = 1} QSAT = {3-CNFs φ for which x 1 x 2 x 3 Qx n φ(x 1, x 2,, x n ) = 1} 20
6 QSAT i is Σ i -complete Theorem: QSAT i is Σ i -complete. Proof: (clearly in Σ i ) assume i odd; given L Σ i in form { x y 1 y 2 y 3 y i (x, y 1,y 2,,y i ) R } x y 1 y 2 y 3 y i Boolean Circuit C 1 iff (x, y 1,y 2,,y i ) R CVAL reduction for R QSAT i is Σ i -complete x y 1 y 2 y 3 y i 1 iff (x, y 1,y 2,,y i ) R Boolean Circuit C CVAL reduction for R Problem set: can construct 3-CNF φ from C: z φ(x,y 1,,y i, z) = 1 C(x,y 1,,y i ) = 1 we get: y 1 y 2 y i z φ(x,y 1,,y i, z) = 1 y 1 y 2 y i C(x,y 1,,y i ) = 1 x L Where CVAL = Boolean Circuit evaluation problem QSAT i is Σ i -complete Proof (continued) assume i even; given L Σ i in form { x y 1 y 2 y 3 y i (x, y 1,y 2,,y i ) R } x y 1 y 2 y 3 y i Boolean Circuit C 1 iff (x, y 1,y 2,,y i ) R CVAL reduction for R QSAT i is Σ i -complete x y 1 y 2 y 3 y i 1 iff (x, y 1,y 2,,y i ) R Boolean Circuit C CVAL reduction for R Problem set: can construct 3-DNF φ from C: z φ(x,y 1,,y i, z) = 1 C(x,y 1,,y i ) = 1 we get: y 1 y 2 y i z φ(x,y 1,y 2,,y i, z) = 1 y 1 y 2 y i C(x,y 1,y 2,,y i ) = 1 x L 23 24
7 QSAT is PSPACE-complete Theorem: QSAT is PSPACE-complete. Proof: x 1 x 2 x 3 Qx n φ(x 1, x 2,, x n )? in PSPACE: x 1 x 2 x 3 Qx n φ(x 1, x 2,, x n )? x 1 : for each x 1, recursively solve x 2 x 3 Qx n φ(x 1, x 2,, x n )? if encounter yes, return yes x 1 : for each x 1, recursively solve x 2 x 3 Qx n φ(x 1, x 2,, x n )? if encounter no, return no base case: evaluating a 3-CNF expression poly(n) recursion depth poly(n) bits of state at each level QSAT is PSPACE-complete given TM M deciding L PSPACE; input x 2 nk possible configurations single START configuration assume single ACCEPT configuration define: REACH(X, Y, i) configuration Y reachable from configuration X in at most 2 i steps QSAT is PSPACE-complete QSAT is PSPACE-complete REACH(X, Y, i) configuration Y reachable from configuration X in at most 2 i steps. Goal: produce 3-CNF φ(w 1,w 2,w 3,,w m ) such that w 1 w 2 Qw m φ(w 1,,w m ) REACH(START, ACCEPT, n k ) for i = 0, 1, n k produce quantified Boolean expressions ψ i (A, B, W) w 1 w 2 ψ i (A, B, W) REACH(A, B, i) convert ψ n k to 3-CNF φ add variables V w 1 w 2 V φ(a, B, W, V) REACH(A, B, n k ) hardwire A = START, B = ACCEPT w 1 w 2 V φ(w, V) x L 27 28
8 QSAT is PSPACE-complete QSAT is PSPACE-complete ψ o (A, B) = true iff A = B or A yields B in one step of M Boolean expression of size O(n k ) Key observation #1: REACH(A, B, i+1) Z [REACH(A, Z, i) REACH(Z, B, i)] STEP STEP STEP STEP config. A config. B cannot define ψ i+1 (A, B) to be Z [ψ i (A, Z) ψ i (Z, B)] why: then formula grows exponentially large) QSAT is PSPACE-complete QSAT is PSPACE-complete Key idea #2: use quantifiers Couldn t do ψ i+1 (A, B) = Z [ψ i (A, Z) ψ i (Z, B)] Key dea: define ψ i+1 (A, B) to be ψ o (A, B) = true iff A = B or A yields B in 1 step ψ i+1 (A, B) = Z X Y[((X=A Y=Z) (X=Z Y=B)) ψ i (X, Y)] Z X Y [((X=A Y=Z) (X=Z Y=B)) ψ i (X, Y)] ψ 0 = O(n k ) ψ i+1 = O(n k ) + ψ i ψ i (X, Y) is preceded by quantifiers move to front (they don t involve X,Y,Z,A,B) 31 total size of ψ n k is O(n k ) 2 = poly(n) logspace reduction 32
9 PH collapse PH Σ 3 Π 3 PH collapse Theorem: if Σ i = Π i then for all j > i Σ j = Π j = Δ j = Σ i the polynomial hierarchy collapses to the i-th level Proof: sufficient to show Σ i = Σ i+1 then Σ i+1 = Σ i = Π i = Π i+1 ; apply theorem again NP Δ 3 Σ 2 Π 2 Δ 2 P conp recall: L Σ i+1 iff expressible as L = { x y (x, y) R } where R Π i since Π i = Σ i, R expressible as R = { (x,y) z ((x, y), z) R } where R Π i-1 together: L = { x (y, z) (x, (y, z)) R } conclude L Σ i Oracles vs. Algorithms Natural complete problems A point to ponder: given poly-time algorithm for SAT can you solve MN CRCUT efficiently? what other problems? Entire complexity classes? given SAT oracle same input/output behavior can you solve MN CRCUT efficiently? 35 We now have quantified versions of SAT complete for levels in PH, PSPACE Natural complete problems? PSPACE: QSAT and games PH: bounded quantifier QSAT & games with bounded alternations almost all other natural problems lie in the second and third level of PH 36
10 Natural complete problems in PH MN CRCUT good candidate to be Σ 2 -complete, still open MN DNF: given DNF φ, integer k; is there a DNF φ of size at most k computing same function φ does? Theorem (U): MN DNF is Σ 2 -complete. Natural complete problems in PSPACE General phenomenon: many 2-player games are PSPACE-complete. 2 players, alternate picking edges lose when no unvisited choice auckland san francisco pasadena athens davis oakland GEOGRAPHY = {(G, s) : G is a directed graph and player can win from node s} Natural complete problems in PSPACE Theorem: GEOGRAPHY is PSPACEcomplete. Proof: in PSPACE easily expressed with alternating quantifiers PSPACE-hard reduction from QSAT 39 Proof Sketch: GEOGRAPHY is PSPACE-complete. x 1 x 2 x 3 ( x 1 x 2 x 3 ) ( x 3 x 1 ) (x 1 x 2 ) true true true x 1 x 2 x 3 false false false alternately pick truth assignment pick a true literal? pick a clause 40
11 Karp-Lipton Theorem we know that P = NP implies SAT has polynomial-size circuits. (showing SAT does not have poly-size circuits is one route to proving P NP) P/poly = Family of languages recognized by polynomial size boolean circuits suppose SAT has poly-size circuits, so: SAT P/poly any consequences? might hope: SAT P/poly PH collapses to Karp-Lipton Theorem Theorem (KL): if SAT has poly-size circuits then PH collapses to the second level. Proof: suffices to show Π 2 Σ 2 L Π 2 implies L expressible as: L = { x : y z (x, y, z) R} with R P. P, same as if SAT P Proof of Karp-Lipton Theorem L = { x : y z (x, y, z) R} given (x, y), z (x, y, z) R? is in NP pretend C solves SAT, use self-reducibility Claim: if SAT P/poly, then L = poly time { x : C y [use C repeatedly to find some z for which (x, y, z) R; accept iff (x, y, z) R] } Proof of Karp-Lipton Theorem L = { x : y z (x, y, z) R} {x : C y [use C repeatedly to find some z for which (x,y,z) R; accept iff (x,y,z) R] } x L: some C decides SAT C y [ ] accepts x L: y z (x, y, z) R C y [ ] rejects 43 44
12 Theorem BPP PH BPP language L: probabilistic TM M: x L Pr y [M(x,y) accepts] 2/3 x L Pr y [M(x,y) rejects] 2/3 Don t know BPP different from EXP Also don t know if Π 2 Σ 2 is different from EXP but believe much weaker Theorem (S,L,GZ): BPP (Π 2 Σ 2 ) 45 Proof of Theorem BPP PH Proof: BPP language L: p.p.t. TM M: x L Pr y [M(x,y) accepts] 2/3 x L Pr y [M(x,y) rejects] 2/3 Known Strong error reduction: There exists a probabilistic TM M : use n random bits ( y = n) # strings y for which M (x, y ) incorrect is at most 2 n/3 (can t achieve with naïve amplification) 46 Proof of Theorem BPP PH view y = (w, z), each of length n/2 consider output of M (x, (w, z)): w = x L x L so so many few ones, ones, not some enough disk fo is whole all ones disk Proof of Theorem BPP PH proof (continued): strong error reduction: # bad y < 2 n/3 y = (w, z) with w = z = n/2 Claim: L = {x : w z M (x, (w, z)) = 1 } x L: suppose w z M (x, (w, z)) = 0 implies 2 n/2 number of 0 s in output; contradiction x L: suppose w z M (x, (w, z)) = 1 implies 2 n/2 number of 1 s in output; contradiction 48
13 Proof of Theorem BPP PH given BPP language L: p.p.t. TM M: x L Pr y [M(x,y) accepts] 2/3 x L Pr y [M(x,y) rejects] 2/3 showed L = {x : w z M (x, (w, z)) = 1} thus BPP Σ 2 BPP closed under complement BPP Π 2 conclude: BPP (Π 2 Σ 2 ) 49
Complete problems for classes in PH, The Polynomial-Time Hierarchy (PH) oracle is like a subroutine, or function in
Oracle Turing Machines Nondeterministic OTM defined in the same way (transition relation, rather than function) oracle is like a subroutine, or function in your favorite PL but each call counts as single
More information1 PSPACE-Completeness
CS 6743 Lecture 14 1 Fall 2007 1 PSPACE-Completeness Recall the NP-complete problem SAT: Is a given Boolean formula φ(x 1,..., x n ) satisfiable? The same question can be stated equivalently as: Is the
More informationUmans Complexity Theory Lectures
Umans Complexity Theory Lectures Lecture 8: Introduction to Randomized Complexity: - Randomized complexity classes, - Error reduction, - in P/poly - Reingold s Undirected Graph Reachability in RL Randomized
More informationNotes on Complexity Theory Last updated: October, Lecture 6
Notes on Complexity Theory Last updated: October, 2015 Lecture 6 Notes by Jonathan Katz, lightly edited by Dov Gordon 1 PSPACE and PSPACE-Completeness As in our previous study of N P, it is useful to identify
More informationCS 151 Complexity Theory Spring Solution Set 5
CS 151 Complexity Theory Spring 2017 Solution Set 5 Posted: May 17 Chris Umans 1. We are given a Boolean circuit C on n variables x 1, x 2,..., x n with m, and gates. Our 3-CNF formula will have m auxiliary
More informationA.Antonopoulos 18/1/2010
Class DP Basic orems 18/1/2010 Class DP Basic orems 1 Class DP 2 Basic orems Class DP Basic orems TSP Versions 1 TSP (D) 2 EXACT TSP 3 TSP COST 4 TSP (1) P (2) P (3) P (4) DP Class Class DP Basic orems
More informationComplexity Theory VU , SS The Polynomial Hierarchy. Reinhard Pichler
Complexity Theory Complexity Theory VU 181.142, SS 2018 6. The Polynomial Hierarchy Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 15 May, 2018 Reinhard
More informationOutline. Complexity Theory EXACT TSP. The Class DP. Definition. Problem EXACT TSP. Complexity of EXACT TSP. Proposition VU 181.
Complexity Theory Complexity Theory Outline Complexity Theory VU 181.142, SS 2018 6. The Polynomial Hierarchy Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität
More informationThe Polynomial Hierarchy
The Polynomial Hierarchy Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Motivation..synthesizing circuits is exceedingly difficulty. It is even
More informationCSC 5170: Theory of Computational Complexity Lecture 9 The Chinese University of Hong Kong 15 March 2010
CSC 5170: Theory of Computational Complexity Lecture 9 The Chinese University of Hong Kong 15 March 2010 We now embark on a study of computational classes that are more general than NP. As these classes
More informationLecture 8. MINDNF = {(φ, k) φ is a CNF expression and DNF expression ψ s.t. ψ k and ψ is equivalent to φ}
6.841 Advanced Complexity Theory February 28, 2005 Lecture 8 Lecturer: Madhu Sudan Scribe: Arnab Bhattacharyya 1 A New Theme In the past few lectures, we have concentrated on non-uniform types of computation
More informationLecture 7: Polynomial time hierarchy
Computational Complexity Theory, Fall 2010 September 15 Lecture 7: Polynomial time hierarchy Lecturer: Kristoffer Arnsfelt Hansen Scribe: Mads Chr. Olesen Recall that adding the power of alternation gives
More informationLecture 20: conp and Friends, Oracles in Complexity Theory
6.045 Lecture 20: conp and Friends, Oracles in Complexity Theory 1 Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode:
More informationLecture 20: PSPACE. November 15, 2016 CS 1010 Theory of Computation
Lecture 20: PSPACE November 15, 2016 CS 1010 Theory of Computation Recall that PSPACE = k=1 SPACE(nk ). We will see that a relationship between time and space complexity is given by: P NP PSPACE = NPSPACE
More informationLecture 23: More PSPACE-Complete, Randomized Complexity
6.045 Lecture 23: More PSPACE-Complete, Randomized Complexity 1 Final Exam Information Who: You On What: Everything through PSPACE (today) With What: One sheet (double-sided) of notes are allowed When:
More information: Computational Complexity Lecture 3 ITCS, Tsinghua Univesity, Fall October 2007
80240233: Computational Complexity Lecture 3 ITCS, Tsinghua Univesity, Fall 2007 16 October 2007 Instructor: Andrej Bogdanov Notes by: Jialin Zhang and Pinyan Lu In this lecture, we introduce the complexity
More informationPOLYNOMIAL SPACE QSAT. Games. Polynomial space cont d
T-79.5103 / Autumn 2008 Polynomial Space 1 T-79.5103 / Autumn 2008 Polynomial Space 3 POLYNOMIAL SPACE Polynomial space cont d Polynomial space-bounded computation has a variety of alternative characterizations
More informationComplexity Theory. Knowledge Representation and Reasoning. November 2, 2005
Complexity Theory Knowledge Representation and Reasoning November 2, 2005 (Knowledge Representation and Reasoning) Complexity Theory November 2, 2005 1 / 22 Outline Motivation Reminder: Basic Notions Algorithms
More information2 Evidence that Graph Isomorphism is not NP-complete
Topics in Theoretical Computer Science April 11, 2016 Lecturer: Ola Svensson Lecture 7 (Notes) Scribes: Ola Svensson Disclaimer: These notes were written for the lecturer only and may contain inconsistent
More informationLecture 22: PSPACE
6.045 Lecture 22: PSPACE 1 VOTE VOTE VOTE For your favorite course on automata and complexity Please complete the online subject evaluation for 6.045 2 Final Exam Information Who: You On What: Everything
More informationCS154, Lecture 17: conp, Oracles again, Space Complexity
CS154, Lecture 17: conp, Oracles again, Space Complexity Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode: Guess string
More informationLecture 10: Boolean Circuits (cont.)
CSE 200 Computability and Complexity Wednesday, May, 203 Lecture 0: Boolean Circuits (cont.) Instructor: Professor Shachar Lovett Scribe: Dongcai Shen Recap Definition (Boolean circuit) A Boolean circuit
More informationCS151 Complexity Theory. Lecture 14 May 17, 2017
CS151 Complexity Theory Lecture 14 May 17, 2017 IP = PSPACE Theorem: (Shamir) IP = PSPACE Note: IP PSPACE enumerate all possible interactions, explicitly calculate acceptance probability interaction extremely
More informationCSE 555 HW 5 SAMPLE SOLUTION. Question 1.
CSE 555 HW 5 SAMPLE SOLUTION Question 1. Show that if L is PSPACE-complete, then L is NP-hard. Show that the converse is not true. If L is PSPACE-complete, then for all A PSPACE, A P L. We know SAT PSPACE
More informationCSE200: Computability and complexity Space Complexity
CSE200: Computability and complexity Space Complexity Shachar Lovett January 29, 2018 1 Space complexity We would like to discuss languages that may be determined in sub-linear space. Lets first recall
More informationIntroduction to Complexity Theory. Bernhard Häupler. May 2, 2006
Introduction to Complexity Theory Bernhard Häupler May 2, 2006 Abstract This paper is a short repetition of the basic topics in complexity theory. It is not intended to be a complete step by step introduction
More informationLecture 7: The Polynomial-Time Hierarchy. 1 Nondeterministic Space is Closed under Complement
CS 710: Complexity Theory 9/29/2011 Lecture 7: The Polynomial-Time Hierarchy Instructor: Dieter van Melkebeek Scribe: Xi Wu In this lecture we first finish the discussion of space-bounded nondeterminism
More informationLecture 19: Finish NP-Completeness, conp and Friends
6.045 Lecture 19: Finish NP-Completeness, conp and Friends 1 Polynomial Time Reducibility f : Σ* Σ* is a polynomial time computable function if there is a poly-time Turing machine M that on every input
More informationconp, Oracles, Space Complexity
conp, Oracles, Space Complexity 1 What s next? A few possibilities CS161 Design and Analysis of Algorithms CS254 Complexity Theory (next year) CS354 Topics in Circuit Complexity For your favorite course
More informationLecture 24: Randomized Complexity, Course Summary
6.045 Lecture 24: Randomized Complexity, Course Summary 1 1/4 1/16 1/4 1/4 1/32 1/16 1/32 Probabilistic TMs 1/16 A probabilistic TM M is a nondeterministic TM where: Each nondeterministic step is called
More informationPrinciples of Knowledge Representation and Reasoning
Principles of Knowledge Representation and Reasoning Complexity Theory Bernhard Nebel, Malte Helmert and Stefan Wölfl Albert-Ludwigs-Universität Freiburg April 29, 2008 Nebel, Helmert, Wölfl (Uni Freiburg)
More informationCS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 20 February 23, 2018 February 23, 2018 CS21 Lecture 20 1 Outline the complexity class NP NP-complete probelems: Subset Sum NP-complete problems: NAE-3-SAT, max
More informationSOLUTION: SOLUTION: SOLUTION:
Convert R and S into nondeterministic finite automata N1 and N2. Given a string s, if we know the states N1 and N2 may reach when s[1...i] has been read, we are able to derive the states N1 and N2 may
More informationLecture 8: Complete Problems for Other Complexity Classes
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 8: Complete Problems for Other Complexity Classes David Mix Barrington and Alexis Maciel
More informationCS154, Lecture 15: Cook-Levin Theorem SAT, 3SAT
CS154, Lecture 15: Cook-Levin Theorem SAT, 3SAT Definition: A language B is NP-complete if: 1. B NP 2. Every A in NP is poly-time reducible to B That is, A P B When this is true, we say B is NP-hard On
More information-bit integers are all in ThC. Th The following problems are complete for PSPACE NPSPACE ATIME QSAT, GEOGRAPHY, SUCCINCT REACH.
CMPSCI 601: Recall From Last Time Lecture 26 Theorem: All CFL s are in sac. Facts: ITADD, MULT, ITMULT and DIVISION on -bit integers are all in ThC. Th The following problems are complete for PSPACE NPSPACE
More information6.045 Final Exam Solutions
6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch, Nati Srebro 6.045 Final Exam Solutions May 18, 2004 Susan Hohenberger Name: Please write your name on each page. This exam is open
More informationLecture 9: Polynomial-Time Hierarchy, Time-Space Tradeoffs
CSE 531: Computational Complexity I Winter 2016 Lecture 9: Polynomial-Time Hierarchy, Time-Space Tradeoffs Feb 3, 2016 Lecturer: Paul Beame Scribe: Paul Beame 1 The Polynomial-Time Hierarchy Last time
More informationComputational Complexity
p. 1/24 Computational Complexity The most sharp distinction in the theory of computation is between computable and noncomputable functions; that is, between possible and impossible. From the example of
More information9. PSPACE. PSPACE complexity class quantified satisfiability planning problem PSPACE-complete
9. PSPACE PSPACE complexity class quantified satisfiability planning problem PSPACE-complete Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley Copyright 2013 Kevin Wayne http://www.cs.princeton.edu/~wayne/kleinberg-tardos
More informationLecture 4 : Quest for Structure in Counting Problems
CS6840: Advanced Complexity Theory Jan 10, 2012 Lecture 4 : Quest for Structure in Counting Problems Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K. Theme: Between P and PSPACE. Lecture Plan:Counting problems
More informationCS151 Complexity Theory. Lecture 13 May 15, 2017
CS151 Complexity Theory Lecture 13 May 15, 2017 Relationship to other classes To compare to classes of decision problems, usually consider P #P which is a decision class easy: NP, conp P #P easy: P #P
More informationCircuits. Lecture 11 Uniform Circuit Complexity
Circuits Lecture 11 Uniform Circuit Complexity 1 Recall 2 Recall Non-uniform complexity 2 Recall Non-uniform complexity P/1 Decidable 2 Recall Non-uniform complexity P/1 Decidable NP P/log NP = P 2 Recall
More information1 From previous lectures
CS 810: Introduction to Complexity Theory 9/18/2003 Lecture 11: P/poly, Sparse Sets, and Mahaney s Theorem Instructor: Jin-Yi Cai Scribe: Aparna Das, Scott Diehl, Giordano Fusco 1 From previous lectures
More informationChapter 9. PSPACE: A Class of Problems Beyond NP. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 9 PSPACE: A Class of Problems Beyond NP Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Geography Game Geography. Alice names capital city c of country she
More informationCOMPLEXITY THEORY. Lecture 17: The Polynomial Hierarchy. TU Dresden, 19th Dec Markus Krötzsch Knowledge-Based Systems
COMPLEXITY THEORY Lecture 17: The Polynomial Hierarchy Markus Krötzsch Knowledge-Based Systems TU Dresden, 19th Dec 2017 Review: ATM vs. DTM Markus Krötzsch, 19th Dec 2017 Complexity Theory slide 2 of
More informationUmans Complexity Theory Lectures
Complexity Theory Umans Complexity Theory Lectures Lecture 1a: Problems and Languages Classify problems according to the computational resources required running time storage space parallelism randomness
More informationP = k T IME(n k ) Now, do all decidable languages belong to P? Let s consider a couple of languages:
CS 6505: Computability & Algorithms Lecture Notes for Week 5, Feb 8-12 P, NP, PSPACE, and PH A deterministic TM is said to be in SP ACE (s (n)) if it uses space O (s (n)) on inputs of length n. Additionally,
More information: On the P vs. BPP problem. 30/12/2016 Lecture 11
03684155: On the P vs. BPP problem. 30/12/2016 Lecture 11 Promise problems Amnon Ta-Shma and Dean Doron 1 Definitions and examples In a promise problem, we are interested in solving a problem only on a
More informationINAPPROX APPROX PTAS. FPTAS Knapsack P
CMPSCI 61: Recall From Last Time Lecture 22 Clique TSP INAPPROX exists P approx alg for no ε < 1 VertexCover MAX SAT APPROX TSP some but not all ε< 1 PTAS all ε < 1 ETSP FPTAS Knapsack P poly in n, 1/ε
More information9. PSPACE 9. PSPACE. PSPACE complexity class quantified satisfiability planning problem PSPACE-complete
Geography game Geography. Alice names capital city c of country she is in. Bob names a capital city c' that starts with the letter on which c ends. Alice and Bob repeat this game until one player is unable
More information15-855: Intensive Intro to Complexity Theory Spring Lecture 7: The Permanent, Toda s Theorem, XXX
15-855: Intensive Intro to Complexity Theory Spring 2009 Lecture 7: The Permanent, Toda s Theorem, XXX 1 #P and Permanent Recall the class of counting problems, #P, introduced last lecture. It was introduced
More information1 Circuit Complexity. CS 6743 Lecture 15 1 Fall Definitions
CS 6743 Lecture 15 1 Fall 2007 1 Circuit Complexity 1.1 Definitions A Boolean circuit C on n inputs x 1,..., x n is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x 1,..., x n ),
More informationCS151 Complexity Theory. Lecture 1 April 3, 2017
CS151 Complexity Theory Lecture 1 April 3, 2017 Complexity Theory Classify problems according to the computational resources required running time storage space parallelism randomness rounds of interaction,
More informationLecture 8: Alternatation. 1 Alternate Characterizations of the Polynomial Hierarchy
CS 710: Complexity Theory 10/4/2011 Lecture 8: Alternatation Instructor: Dieter van Melkebeek Scribe: Sachin Ravi In this lecture, we continue with our discussion of the polynomial hierarchy complexity
More information6.045: Automata, Computability, and Complexity (GITCS) Class 17 Nancy Lynch
6.045: Automata, Computability, and Complexity (GITCS) Class 17 Nancy Lynch Today Probabilistic Turing Machines and Probabilistic Time Complexity Classes Now add a new capability to standard TMs: random
More informationAlgorithms & Complexity II Avarikioti Zeta
Algorithms & Complexity II Avarikioti Zeta March 17, 2014 Alternating Computation Alternation: generalizes non-determinism, where each state is either existential or universal : Old: existential states
More informationIntroduction to Computational Complexity
Introduction to Computational Complexity George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 400 George Voutsadakis (LSSU) Computational Complexity September
More information15.1 Proof of the Cook-Levin Theorem: SAT is NP-complete
CS125 Lecture 15 Fall 2016 15.1 Proof of the Cook-Levin Theorem: SAT is NP-complete Already know SAT NP, so only need to show SAT is NP-hard. Let L be any language in NP. Let M be a NTM that decides L
More informationLecture Notes Each circuit agrees with M on inputs of length equal to its index, i.e. n, x {0, 1} n, C n (x) = M(x).
CS 221: Computational Complexity Prof. Salil Vadhan Lecture Notes 4 February 3, 2010 Scribe: Jonathan Pines 1 Agenda P-/NP- Completeness NP-intermediate problems NP vs. co-np L, NL 2 Recap Last time, we
More informationLecture 17: Cook-Levin Theorem, NP-Complete Problems
6.045 Lecture 17: Cook-Levin Theorem, NP-Complete Problems 1 Is SAT solvable in O(n) time on a multitape TM? Logic circuits of 6n gates for SAT? If yes, then not only is P=NP, but there would be a dream
More informationLimits to Approximability: When Algorithms Won't Help You. Note: Contents of today s lecture won t be on the exam
Limits to Approximability: When Algorithms Won't Help You Note: Contents of today s lecture won t be on the exam Outline Limits to Approximability: basic results Detour: Provers, verifiers, and NP Graph
More informationNP Completeness. CS 374: Algorithms & Models of Computation, Spring Lecture 23. November 19, 2015
CS 374: Algorithms & Models of Computation, Spring 2015 NP Completeness Lecture 23 November 19, 2015 Chandra & Lenny (UIUC) CS374 1 Spring 2015 1 / 37 Part I NP-Completeness Chandra & Lenny (UIUC) CS374
More informationAdvanced topic: Space complexity
Advanced topic: Space complexity CSCI 3130 Formal Languages and Automata Theory Siu On CHAN Chinese University of Hong Kong Fall 2016 1/28 Review: time complexity We have looked at how long it takes to
More informationCS151 Complexity Theory. Lecture 4 April 12, 2017
CS151 Complexity Theory Lecture 4 A puzzle A puzzle: two kinds of trees depth n...... cover up nodes with c colors promise: never color arrow same as blank determine which kind of tree in poly(n, c) steps?
More informationDefinition: Alternating time and space Game Semantics: State of machine determines who
CMPSCI 601: Recall From Last Time Lecture 3 Definition: Alternating time and space Game Semantics: State of machine determines who controls, White wants it to accept, Black wants it to reject. White wins
More informationMTAT Complexity Theory October 13th-14th, Lecture 6
MTAT.07.004 Complexity Theory October 13th-14th, 2011 Lecturer: Peeter Laud Lecture 6 Scribe(s): Riivo Talviste 1 Logarithmic memory Turing machines working in logarithmic space become interesting when
More informationIdentifying an Honest EXP NP Oracle Among Many
Identifying an Honest EXP NP Oracle Among Many Shuichi Hirahara The University of Tokyo CCC 18/6/2015 Overview Dishonest oracle Honest oracle Queries Which is honest? Our Contributions Selector 1. We formulate
More informationDatabase Theory VU , SS Complexity of Query Evaluation. Reinhard Pichler
Database Theory Database Theory VU 181.140, SS 2018 5. Complexity of Query Evaluation Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 17 April, 2018 Pichler
More informationDefinition: conp = { L L NP } What does a conp computation look like?
Space Complexity 28 Definition: conp = { L L NP } What does a conp computation look like? In NP algorithms, we can use a guess instruction in pseudocode: Guess string y of x k length and the machine accepts
More informationSAT, NP, NP-Completeness
CS 473: Algorithms, Spring 2018 SAT, NP, NP-Completeness Lecture 22 April 13, 2018 Most slides are courtesy Prof. Chekuri Ruta (UIUC) CS473 1 Spring 2018 1 / 57 Part I Reductions Continued Ruta (UIUC)
More informationCOMPLEXITY THEORY. PSPACE = SPACE(n k ) k N. NPSPACE = NSPACE(n k ) 10/30/2012. Space Complexity: Savitch's Theorem and PSPACE- Completeness
15-455 COMPLEXITY THEORY Space Complexity: Savitch's Theorem and PSPACE- Completeness October 30,2012 MEASURING SPACE COMPLEXITY FINITE STATE CONTROL I N P U T 1 2 3 4 5 6 7 8 9 10 We measure space complexity
More informationsatisfiability (sat) Satisfiability unsatisfiability (unsat or sat complement) and validity Any Expression φ Can Be Converted into CNFs and DNFs
Any Expression φ Can Be Converted into CNFs and DNFs φ = x j : This is trivially true. φ = φ 1 and a CNF is sought: Turn φ 1 into a DNF and apply de Morgan s laws to make a CNF for φ. φ = φ 1 and a DNF
More informationIntroduction to Advanced Results
Introduction to Advanced Results Master Informatique Université Paris 5 René Descartes 2016 Master Info. Complexity Advanced Results 1/26 Outline Boolean Hierarchy Probabilistic Complexity Parameterized
More informationNP-Complete Problems. More reductions
NP-Complete Problems More reductions Definitions P: problems that can be solved in polynomial time (typically in n, size of input) on a deterministic Turing machine Any normal computer simulates a DTM
More information6.840 Language Membership
6.840 Language Membership Michael Bernstein 1 Undecidable INP Practice final Use for A T M. Build a machine that asks EQ REX and then runs M on w. Query INP. If it s in P, accept. Note that the language
More information6.045J/18.400J: Automata, Computability and Complexity Final Exam. There are two sheets of scratch paper at the end of this exam.
6.045J/18.400J: Automata, Computability and Complexity May 20, 2005 6.045 Final Exam Prof. Nancy Lynch Name: Please write your name on each page. This exam is open book, open notes. There are two sheets
More informationDe Morgan s a Laws. De Morgan s laws say that. (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2.
De Morgan s a Laws De Morgan s laws say that (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2. Here is a proof for the first law: φ 1 φ 2 (φ 1 φ 2 ) φ 1 φ 2 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 a Augustus DeMorgan
More informationLecture Notes 4. Issued 8 March 2018
CM30073 Advanced Algorithms and Complexity 1. Structure of the class NP Lecture Notes 4 Issued 8 March 2018 Recall that it is not known whether or not P = NP, the widely accepted hypothesis being that
More informationITCS:CCT09 : Computational Complexity Theory Apr 8, Lecture 7
ITCS:CCT09 : Computational Complexity Theory Apr 8, 2009 Lecturer: Jayalal Sarma M.N. Lecture 7 Scribe: Shiteng Chen In this lecture, we will discuss one of the basic concepts in complexity theory; namely
More informationLecture 12: Interactive Proofs
princeton university cos 522: computational complexity Lecture 12: Interactive Proofs Lecturer: Sanjeev Arora Scribe:Carl Kingsford Recall the certificate definition of NP. We can think of this characterization
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 25 Last time Class NP Today Polynomial-time reductions Adam Smith; Sofya Raskhodnikova 4/18/2016 L25.1 The classes P and NP P is the class of languages decidable
More informationNon-Approximability Results (2 nd part) 1/19
Non-Approximability Results (2 nd part) 1/19 Summary - The PCP theorem - Application: Non-approximability of MAXIMUM 3-SAT 2/19 Non deterministic TM - A TM where it is possible to associate more than one
More information6.841/18.405J: Advanced Complexity Wednesday, February 12, Lecture Lecture 3
6.841/18.405J: Advanced Complexity Wednesday, February 12, 2003 Lecture Lecture 3 Instructor: Madhu Sudan Scribe: Bobby Kleinberg 1 The language MinDNF At the end of the last lecture, we introduced the
More informationSpace and Nondeterminism
CS 221 Computational Complexity, Lecture 5 Feb 6, 2018 Space and Nondeterminism Instructor: Madhu Sudan 1 Scribe: Yong Wook Kwon Topic Overview Today we ll talk about space and non-determinism. For some
More informationU.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan August 30, Notes for Lecture 1
U.C. Berkeley CS278: Computational Complexity Handout N1 Professor Luca Trevisan August 30, 2004 Notes for Lecture 1 This course assumes CS170, or equivalent, as a prerequisite. We will assume that the
More informationMTAT Complexity Theory October 20th-21st, Lecture 7
MTAT.07.004 Complexity Theory October 20th-21st, 2011 Lecturer: Peeter Laud Lecture 7 Scribe(s): Riivo Talviste Polynomial hierarchy 1 Turing reducibility From the algorithmics course, we know the notion
More informationCS151 Complexity Theory
Introduction CS151 Complexity Theory Lecture 5 April 13, 2004 Power from an unexpected source? we know PEXP, which implies no polytime algorithm for Succinct CVAL poly-size Boolean circuits for Succinct
More informationNP Complete Problems. COMP 215 Lecture 20
NP Complete Problems COMP 215 Lecture 20 Complexity Theory Complexity theory is a research area unto itself. The central project is classifying problems as either tractable or intractable. Tractable Worst
More informationQuantum Complexity Theory. Wim van Dam HP Labs MSRI UC Berkeley SQUINT 3 June 16, 2003
Quantum Complexity Theory Wim van Dam HP Labs MSRI UC Berkeley SQUINT 3 June 16, 2003 Complexity Theory Complexity theory investigates what resources (time, space, randomness, etc.) are required to solve
More informationNon-Deterministic Time
Non-Deterministic Time Master Informatique 2016 1 Non-Deterministic Time Complexity Classes Reminder on DTM vs NDTM [Turing 1936] (q 0, x 0 ) (q 1, x 1 ) Deterministic (q n, x n ) Non-Deterministic (q
More informationCSCI 1590 Intro to Computational Complexity
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
More informationSpace Complexity. Huan Long. Shanghai Jiao Tong University
Space Complexity Huan Long Shanghai Jiao Tong University Acknowledgements Part of the slides comes from a similar course in Fudan University given by Prof. Yijia Chen. http://basics.sjtu.edu.cn/ chen/
More informationTheory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death
Theory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory
More informationSpace is a computation resource. Unlike time it can be reused. Computational Complexity, by Fu Yuxi Space Complexity 1 / 44
Space Complexity Space is a computation resource. Unlike time it can be reused. Computational Complexity, by Fu Yuxi Space Complexity 1 / 44 Synopsis 1. Space Bounded Computation 2. Logspace Reduction
More informationComputational Complexity of Bayesian Networks
Computational Complexity of Bayesian Networks UAI, 2015 Complexity theory Many computations on Bayesian networks are NP-hard Meaning (no more, no less) that we cannot hope for poly time algorithms that
More informationLecture 26: Arthur-Merlin Games
CS 710: Complexity Theory 12/09/2011 Lecture 26: Arthur-Merlin Games Instructor: Dieter van Melkebeek Scribe: Chetan Rao and Aaron Gorenstein Last time we compared counting versus alternation and showed
More informationEssential facts about NP-completeness:
CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions
More informationKernel-Size Lower Bounds. The Evidence from Complexity Theory
: The Evidence from Complexity Theory IAS Worker 2013, Warsaw Part 2/3 Note These slides are a slightly revised version of a 3-part tutorial given at the 2013 Workshop on Kernelization ( Worker ) at the
More information