lec4 - CS151 Complexity Theory Lecture 4 April 7, 2011 A...

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CS151 Complexity Theory Lecture 4 April 7, 2011
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April 7, 2011 2 A puzzle cover up nodes with c colors promise: never color “arrow” same as “blank” determine which kind of tree in poly(n, c) steps? . . .  . . .  depth n A puzzle: two kinds of trees
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April 7, 2011 3 A puzzle . . .  . . .  depth n
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April 7, 2011 4 A puzzle . . .  . . .  depth n
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April 7, 2011 5 Introduction Ideas depth-first-search; stop if see how many times may we see a given “arrow color”? at most n+1 pruning rule? if see a color > n+1 times, it can’t be an arrow node; prune # nodes visited before know answer? at most c(n+2)
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April 7, 2011 6 Sparse languages and NP We often say NP -compete languages are “hard” More accurate: NP -complete languages are “expressive” lots of languages reduce to them
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April 7, 2011 7 Sparse languages and NP Sparse language : one that contains at most poly(n) strings of length ≤ n not very expressive – can we show this cannot be NP -complete (assuming P NP ) ? yes: Mahaney ’82 (homework problem) Unary language : subset of 1* (at most n strings of length ≤ n)
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April 7, 2011 8 Sparse languages and NP Theorem (Berman ’78): if a unary language is NP -complete then P = NP . Proof: let U 1* be a unary language and assume SAT ≤ U via reduction R – φ(x 1 ,x 2 ,…,x n ) instance of SAT
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April 7, 2011 9 Sparse languages and NP self-reduction tree for φ: . . .  φ (x 1 ,x 2 ,…,x n ) φ (1,x 2 ,…,x n ) φ (0,x 2 ,…,x n ) φ (0,0,…,0) φ (1,1,…,1) . . satisfying assignment
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April 7, 2011 10 Sparse languages and NP applying reduction R: . . .  R( φ (x 1 ,x 2 ,…,x n )) R( φ (1,x 2 ,…,x n )) R( φ (0,x 2 ,…,x n )) R( φ (0,0,…,0)) R( φ (1,1,…,1)) . . satisfying assignment
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April 7, 2011 11 Sparse languages and NP on input of length m = |φ(x 1 ,x 2 ,…,x n )|, R produces string of length ≤ p(m) R’s different outputs are “colors” 1 color for strings not in 1 * at most p(m) other colors puzzle solution can solve SAT in poly(p(m)+1, n+1) = poly(m) time!
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April 7, 2011 12 Summary nondeterministic time classes: NP, coNP, NEXP NTIME Hierarchy Theorem: NP NEXP major open questions: P = NP NP = coNP ? ?
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April 7, 2011 13 Summary NP -“intermediate” problems (unless P = NP ) technique: delayed diagonalization unary languages not NP -complete (unless P = NP ) true for sparse languages as well (homework) complete problems: circuit SAT is NP -complete UNSAT is coNP -complete succinct circuit SAT is NEXP -complete
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April 7, 2011 14 Summary EXP PSPACE P L NEXP NP coNP coNEXP
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April 7, 2011 15 Remainder of lecture nondeterminism applied to space reachability two surprises: Savitch’s Theorem Immerman/Szelepcsényi Theorem
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lec4 - CS151 Complexity Theory Lecture 4 April 7, 2011 A...

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