1 / 19

CSCI 3130: Formal languages and automata theory

Fall 2010. The Chinese University of Hong Kong. CSCI 3130: Formal languages and automata theory. The Cook-Levin Theorem. Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130. The class NP. A verifier for L is a TM V such that s is a potential solution for z

ivana
Download Presentation

CSCI 3130: Formal languages and automata theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Fall 2010 The Chinese University of Hong Kong CSCI 3130: Formal languages and automata theory The Cook-Levin Theorem Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130

  2. The class NP • A verifier for L is a TM V such that • s is a potential solution for z • We say V runs in polynomial time if on every input z, it runs in time polynomial in |z| (for every s) z ∈ L V accepts 〈z, s〉 for some s NP is the class of all languages that have polynomial-time verifiers

  3. NP-complete problems NP-complete DOM SET MIN COVER VERTEX COVER SUBSET SUM IND SET HAM CYCLE HAM PATH PARTITION CLIQUE 3SAT NP SAT P

  4. Polynomial-time reductions • Language Lpolynomial-time reduces to L’ if there exists a polynomial-time computable map R that takes an instance x of L into instance y of L’ s.t. x ∈ L if and only if y ∈ L’ L (CLIQUE) L’ (IS) R (G, k) x y (G’, k’) x ∈ L y ∈ L’ (G has clique of size k) (G’ has IS of size k’)

  5. NP-completeness • A language C is NP-complete if: • Cook-Levin Theorem: 1. C is in NP, and 2.For every L in NP, L reduces to C. C NP SAT is NP-complete P

  6. SAT and 3SAT SAT = {〈f〉: fis a satisfiable Boolean formula} ((x1∨x2 ) ∧ (x1∨x2)) ∨ ((x1∨(x2 ∧ x3)) ∧x3) 3SAT = {〈f〉: fis a satisfiable 3CNF formula} (x1∨x2∨x2 ) ∧ (x2∨x3∨x4) ∧ (x2∨x3∨x5) ∧ (x1∨x5∨x5)

  7. The Cook-Levin Theorem • To show this, we need a reduction R such that: Every L∈NP reduces to SAT R z Boolean formula f f is satisfiable z ∈ L

  8. NP-completeness of SAT • All we know: L has a poly-time verifierV • Look at the computation tableau of V on input 〈z, s〉 S z | s V … q0 z1 # # S-th configuration symbol at time T # T … qacc # #

  9. NP-completeness of SAT S … q0 n = length of z x1 # # # height of tableau is nc for some constantc T u width is at most nc k possible tableau symbols … qacc # # true, if the (T, S) cell of tableau contains u 1 ≤ S ≤ nc xT, S, u = 1 ≤ T ≤ nc false, if not 1 ≤ u ≤ k

  10. NP-completeness of SAT • We will design a formula f such that: R z Boolean formula f f is satisfiable z ∈ L variables of f : xT, S, u assignment to xT, S, u way to fill up the tableau accepting computation tableau satisfying assignment f is satisfiable V accepts 〈z, s〉for some s

  11. NP-completeness of SAT • We want to construct (in time O(n2c)) a formula f : true, if the (T, S) cell of tableau contains u 1 ≤ S ≤ nc xT, S, u = 1 ≤ T ≤ nc false, if not 1 ≤ u ≤ k true, if the xs represent a valid and accepting tableau f(x1, 1, 1, ..., xnc, nc, k) = false, if not Symbols in computation tableau come from the alphabet G∪Q∪{#} ={a1,...,ak}

  12. NP-completeness of SAT S f = fcell ∧ f0 ∧ fmove ∧ facc … q0 z1 # # # T fcell: “Every cell contains exactly one symbol” u f0: “The first row is #q0z|s☐...☐#” for some s … qacc # # fmove: “The moves between rows follow the transitions of V” facc: “qacc appears somewhere in the last row”

  13. NP-completeness of SAT • Desired meaning • Implementation: fcell: “Every cell contains exactly one symbol” or: “Exactly one of xS, T, 1 ∨ ... ∨ xS, T, k is true” fcell = fcell1, 1 ∧ ... ∧ fcellnc,nc where fcellT, S = (xT, S, 1 ∨ ... ∨ xT, S, k) ∧ (xT, S, 1 ∧ xT, S, 2) ∧ (xT, S, 1 ∧ xT, S, 3) ∧ ... ∧ (xT, S, k-1 ∧ xT, S, k) at least one symbol no cell containstwo or more symbols

  14. NP-completeness of SAT • Desired meaning • Implementation: f0: “The first row is #q0z1z2...zn|s1s2...sm☐...☐# for some (possible solution) s1s2...sm” f0 = x1, 1, # ∧ x1, 2, q0 ∧ x1, 3, z1 ∧ ... ∧ x1, n+2,zn ∧ x1, n+3, | ∧ x1, nc, #

  15. NP-completeness of SAT • Desired meaning • Implementation: facc: “qacc appears somewhere in the last row” facc = xnc, 1, qacc ∨ xnc, 2, qacc ∨ ... ∨ xnc, nc, qacc

  16. Valid and invalid windows valid windows invalid windows … 6a3b0x0 … … 0a6b0x0…0 … 6q2a0b0 … … 0a6b0q2 …0 … 6a3q2a0 … … 0q5a6x0…0 … 6q2q2a0 … … 0q2q2x3…0 q2 a/xL … 6#3b0a0 … … 0#6b0q5 …0 … 6a3q2a0 … … 0q5a6b0…0 q5 … 6a3a0☐0 … … 0x6a0☐0…0 … 6#3q2a0 … … 0q5#6x0…0

  17. NP-completeness of SAT • Desired meaning • Implementation: q0 b a # # a2 a3 a1 # fmove: “The moves between rows follow transitions of V” a4 a5 a6 # fmove2, 2 … qacc # # fmove = fmove1, 1 ∧ ... ∧ fmovenc-3, nc-3 ∨ fmoveT, S = (xT, S, a1 ∧ xT, S+1, a2 ∧ xT, S+2, a3 ∧ xT+1, S, a4 ∧ xT+1, S+1, a5 ∧ xT+2, S+1, a6) over all valid windows a1 a2 a3 a4 a5 a6

  18. NP-completeness of SAT R z Boolean formula f ✔ f is satisfiable z ∈ L Let V be a poly-time verifier TM for L. R := On input z, Construct the formulas fcell,f0,fmove, andfacc Output the formula f = fcell ∧ f0 ∧ fmove ∧ facc. Then R is computable in time O(n2c) and V accepts z if and only if f is satisfiable.

  19. NP-complete problems NP-complete DOM SET MIN COVER VERTEX COVER SUBSET SUM IND SET HAM CYCLE HAM PATH PARTITION CLIQUE 3SAT NP SAT P

More Related