# lec4 - A puzzle A puzzle two kinds of trees CS151...

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1 CS151 Complexity Theory Lecture 4 April 7, 2011 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 April 7, 2011 3 A puzzle . . . . . . depth n April 7, 2011 4 A puzzle . . . . . . depth n 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) 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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2 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) 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 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 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 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! April 7, 2011 12 Summary • nondeterministic time classes: NP, coNP, NEXP • NTIME Hierarchy Theorem: NP NEXP • major open questions: P = NP NP = coNP ? ?
3 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 April 7, 2011

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lec4 - A puzzle A puzzle two kinds of trees CS151...

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