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MELJUN CORTES Automata Theory (Automata23)

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MELJUN CORTES Automata Theory (Automata23)

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MELJUN CORTES Automata Theory (Automata23)

1. 1. MELJUN P. CORTES, MBA,MPA,BSCS,ACS Fall 2008 CSC 3130: Automata theory and formal languages NP-complete problems MELJUN CORTES
2. 2. Polynomial-time reductions • Language L polynomial-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 (IS) x (G, k) x∈L (G has IS of size k) L’ (CLIQUE) R y (G’, k’) y ∈ L’ (G’ has clique of size k’)
3. 3. The Cook-Levin Theorem Every L ∈ NP reduces to SAT SAT = {f: f is a satisfiable Boolean formula} f = (x1∨x2 ) ∧ (x2 ∨x3 ∨x4) ∧ (x1) is satisfiable x1 = T x2 = F x3 = T x4 = T f = (x1∨x2 ) ∧ (x1) ∧ (x2) is not satisfiable
4. 4. NP-hardness • Language L is NP-hard if every L’ in NP reduces to L • Intuitively, NP-hard means “harder than all of NP” • The Cook-Levin Theorem says SAT NP SAT is NP-hard P
5. 5. NP-complete • L is NP-complete if L is NP-hard and L ∈ NP • Intuitively, NP-complete means “hardest in NP” • Recall that SAT ∈ NP, so SAT is NP-complete SAT If SAT ∈ P, then P = NP NP P
6. 6. Proof of Cook-Levin Theorem Every L ∈ NP reduces to SAT • To prove it, we have to describe a reduction R: w w∈L R Boolean formula f f is satisfiable
7. 7. Proof of Cook-Levin Theorem • All we know about L: It has a poly-time NTM M – Let’s look at computation tableau of M on input w M S w q0 w1 w2 S-th configuration symbol at time T Since M is nondeterministic, there may be many possible tableaus T qacc …
8. 8. Proof of Cook-Levin Theorem S n = length of input w q0 w1 w2 height of tableau is p(n) for some polynomial p T u width is at most p(n) k possible tableau symbols xT, S, u = qacc true, if the (T, S) cell of tableau contains u false, if not … 1 ≤ S ≤ p(n) 1 ≤ T ≤ p(n) 1≤u≤k
9. 9. Proof of Cook-Levin Theorem w w∈L R Boolean formula f f is satisfiable • We will design a formula f such that: variables of f : xT, S, u assignment to xT, S, u way to fill up the tableau satisfying assignment accepting computation tableau f is satisfiable M accepts w
10. 10. Proof of Cook-Levin Theorem xT, S, u = true, if the (T, S) cell of tableau contains u false, if not 1 ≤ S ≤ p(n) 1 ≤ T ≤ p(n) 1≤u≤k • We want to construct (in time poly(n)) a formula f : f(x1, 1, 1, ..., xp(n), p(n), k) = true, if the xs represent a valid accepting tableau false, if not
11. 11. Proof of Cook-Levin Theorem f = fcell ∧ f0 ∧ fmove ∧ facc fcell: f0: S q0 w1 w2 T “Every cell contains exactly one symbol” “The first row is q0w1w2...wk☐...☐” u qacc fmove: “The moves between rows follow the transitions of M” facc: “qacc appears somewhere in the last row” …
12. 12. Proof of Cook-Levin Theorem • Desired meaning fcell: “Every cell contains exactly one symbol” or: “Exactly one of xS, T, 1 ∨ ... ∨ xS, T, k is true” • Implementation: fcell = fcell 1, 1 ∧ ... ∧ fcell p(n), p(n) fcell T, 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) where at least one symbol no two symbols
13. 13. Proof of Cook-Levin Theorem • Desired meaning f0: “The first row is q0w1w2...wk☐...☐” facc: “qacc appears somewhere in the last row” • Implementation: f0 = x1, 1, q0 ∧ x1, 1, w1 ∧ x1, 1, w2 ∧ ... ∧ x1, p(n),☐ facc = xp(n), 1, qacc ∨ xp(n), 2, qacc ∨ ... ∨ xp(n), p(n), qacc
14. 14. Valid and invalid windows … 6t3t0u0 … … 0t6t0u0 …0 … 6q3t0u0 … … 0k6t0q0 …0 valid window invalid window … 6t3q3u0 … … 0t6a0q7 …0 … 6c3a0t0 … … 0c6a0p0 …0 valid if δ(q3, u) = (q7, a, R) invalid window … 6t3t0u0 … … 0t6t0q3 …0 … 6c3a0t0 … … 0b6a0t0 …0 valid window valid window
15. 15. Proof of Cook-Levin Theorem • Desired meaning fmove: “The moves between rows follow transitions of M” q0 a a b b q3 a b b c q7 b fmove 2, 2 • Implementation: fmove = fmove 1, 1 ∧ ... ∧ fmove p(n)-3, p(n)-3 ∨ qacc (xT, S, a1 ∧ xT, S+1, a2 ∧ xT, S+2, a3 over all valid windows ∧ xT+1, S, a4 ∧ xT+1, S+1, a5 ∧ xT+2, S+1, a6) fmove T, S = a1 a2 a3 a4 a5 a6 …
16. 16. Other NP-complete problems CLIQUE = {(G, k): G is a graph with a clique of k vertices} IS = {(G, k): G is a graph with an independent set of k vertices} VC = {(G, k): G is a graph with a vertex cover of k vertices} CLIQUE, IS and VC are NP-complete SAT NP CLIQUE IS VC
17. 17. Proving NP-hardness • To show L is NP-hard, it is enough to reduce from some L’ we already know is NP-hard – For now we can take L’ = SAT • To show L is NP-complete, we also need to argue that L is in NP – This is usually the easy part roadmap: CLIQUE VC IS 3SAT SAT
18. 18. 3SAT SAT = {f: f is a satisfiable Boolean formula} 3SAT = {f: f is a satisfiable Boolean formula in conjunctive normal form with at most 3 distinct literals per clause} literal: xi or xi CNF: AND of ORs of literals (conjunctive normal form) 3CNF: CNF with ≤3 lit/clause (x2∨(x1∧x2 ))∧(x1∧(x1∨x2 )) gates (x1∨x2 ) ∧ (x2 ∨x3 ∨x4) ∧ (x1) literals clause
19. 19. NP-hardness of 3SAT • Theorem 3SAT is NP-hard • Proof: We describe a reduction R from SAT Boolean formula f f is satisfiable R 3CNF formula f’ f’ is satisfiable
20. 20. Reducing SAT to 3SAT • Example: f = (x2∨(x1∧x2 ))∧(x1∧(x1∨x2 )) x4 x5 x7 x10 AND x8 x9 OR NOT x6 x7 AND AND x3 NOT x2 x1 x2 x4 x5 NOT T T T T F F F F T T F F T T F F T F T F T F T F x7 = x4 ∧ x5 T F F T F T F T OR x1 x1 x2 We give extra variables to every gate (“wire”) (x4∨x5∨x7) ∧(x4∨x5∨x7) ∧(x4∨x5∨x7) ∧(x4∨x5∨x7) (x4∨x5∨x7) (x4∨x5∨x7) (x4∨x5∨x7) (x4∨x5∨x7)
21. 21. Reducing SAT to 3SAT Boolean formula f R 3CNF formula f’ • Step 1: Add variable xn+j for each gate Gj in f • Step 2: Write 3CNF fj for each gate Gj, j = {1, ..., t} • Step 3: Output f’ = fn+1 ∧ fn+2 ∧ ... ∧ ft ∧ xn+t requires that output of f is true
22. 22. Reducing SAT to 3SAT Boolean formula f f is satisfiable R 3CNF formula f’ f’ is satisfiable • Every satisfying assignment of f extends uniquely to a satisfying assignment of f’ • Conversely, every satisfying assignment of f’ must contain a satisfying assignment of f
23. 23. Independent set IS = {(G, k): G is a graph with an independent set of k vertices} • Theorem IS is NP-hard 1 3 2 4 An independent set is a subset of vertices so that no pair is connected {1, 2}, {1, 3}, {4} are independent sets CLIQUE VC IS 3SAT ✓ SAT
24. 24. Reducing 3SAT to IS • Proof: We describe a reduction from 3SAT to IS 3SAT = {f: f is a satisfiable Boolean formula in 3CNF} IS = {(G, k): G is a graph with an independent set of k vertices} 3CNF formula f f is satisfiable R (G, k) G has an independent set of size k
25. 25. Reducing 3SAT to IS • Example: f = (x1∨x2 ) ∧ (x2 ∨x3 ∨x4) ∧ (x1) x2x3x4 x1x2 TT TF FT FF TTT TTF TFT TFF FTT FTF FFT FFF all interconnected x1 T etc. F Put an edge for every inconsistency
26. 26. Reducing 3SAT to IS • Example: f = (x1∨x2 ) ∧ (x2 ∨x3 ∨x4) ∧ (x1) x2x3x4 x1x2 TT TF FF satisfying assignment of f TTT TTF TFT TFF FTT FFT FFF x1 T G edges = inconsistencies IS of size 3 in G x1 = T x2 = F x3 = T x4 = T any IS of size 3 in G satisfying assignment of f
27. 27. Reducing 3SAT to IS 3CNF formula f R (G, k) • G has a vertex for every clause Ci of f and every satisfying assignment of Ci • G has an edge between any two vertices that represent inconsistent assignments • k is the number of clauses in f
28. 28. Reducing 3SAT to IS 3CNF formula f f is satisfiable R (G, k) G has an IS of size k • Every satisfying assignment of f gives an independent set of size k in G • Conversely, from every IS of size k in G we can “extract” a consistent and satisfying assignment of f
29. 29. Vertex cover VC = {(G, k): G is a graph with a vertex cover of size k} • Theorem VC is NP-hard CLIQUE VC 1 3 2 4 A vertex cover is a set of vertices that touches (covers) all edges {2, 4}, {3, 4}, {1, 2, 3} are vertex covers ✓ IS ✓ 3SAT ✓ SAT
30. 30. Reducing IS to VC • Proof: We describe a reduction from IS to VC (G, k) (G’, k’) R G has an IS of size k G’ has a VC of size k’ • Example 1 2 3 4 independent sets ∅, {1}, {2}, {3}, {4}, {1, 3}, {2, 3} vertex covers {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}
31. 31. Reducing IS to VC • Claim S is an independent set of G if and only if S is a vertex cover of G • Proof S is an independent set of G no edge has both endpoints in S every edge has an endpoint in S S is a vertex cover of G
32. 32. Reducing IS to VC (G, k) R (G’, k’) • Set G’ = G, k’ = n – k (n = number of vertices) G has an IS of size k by previous Claim. G’ has a VC of size k’
33. 33. The ubiquity of NP-complete problems • We saw a few examples of NP-complete problems, but there are many more • A surprising fact of life is that most CS problems are either in P or NP-complete • A 1979 book by Garey and Johnson lists 100+ NP-complete problems