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The Padding Argument

The Padding Argument. Motivation: Scaling-Up Complexity Claims. We have:. can be simulated by…. space. space. + non-determinism. + determinism. We want:. can be simulated by…. space. space. + non-determinism. + determinism. Formally. s i (n) can be computed with space s i (n).

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The Padding Argument

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  1. The Padding Argument Complexity

  2. Motivation: Scaling-Up Complexity Claims We have: can be simulated by… space space + non-determinism + determinism We want: can be simulated by… space space + non-determinism + determinism Complexity

  3. Formally si(n) can be computed with space si(n) Claim: For any two space constructible functions s1(n),s2(n)logn, f(n)n: NSPACE(s1(n))  SPACE(s2(n))  NSPACE(s1(f(n)))  SPACE(s2(f(n))) simulation overhead E.g NSPACE(n)SPACE(n2)  NSPACE(n2)SPACE(n4) Complexity

  4. space: s1(.) in the size of its input Idea DTM space: O(s2(f(n))) NTM n n . . . . . . . . . space: O(s1(f(n))) f(n) 0 . . . . . . . 0 Complexity

  5. Padding argument • Let LNPSPACE(s1(f(n))) • There is a 3-Tape-NTM ML: |x| Input babba Work  O(s1(f(|x|))) Complexity

  6. Padding argument • Let L’ = { x0f(|x|)-|x|| xL } • We’ll show a NTM ML’which decides L’in the same number of cells as ML. f(|x|) babba00000000000000000000000000000000 Input Work  O(s1(f(|x|)) Complexity

  7. Padding argument – ML’ In O(log(f(|x|)) space • Count backwards the number of 0’s and check there are f(|x|)-|x| such. 2. RunML on x. in O(s1(f(|x|))) space f(|x|) Input babba00000000000000000000000000000000 Work  O(s1(f(|x|))) Complexity

  8. Padding argument Total space: O(s1(f(|x|))) f(|x|) Input babba00000000000000000000000000000000 Work  O(s1(f(|x|))) Complexity

  9. Padding Argument • We started with LNSPACE(s1(f(n))) • We showed: L’NSPACE(s1(n)) • Thus, L’SPACE(s2(n)) • Using the DTM for L’ we’ll construct a DTM for L, which will work in O(s2(f(n))) space. Complexity

  10. Padding Argument • The DTM for L’ will simulate the DTM for L when working on its input concatenated with zeros Input babba 00000000000000000000000 Complexity

  11. Padding Argument • When the input head leaves the input part, just pretend it encounters 0s. • keeping track after the simulated position takes O(log(f(|x|))) space. • Thus our machine uses O(s2(f(|x|))) space. •  NSPACE(s1(f(n)))SPACE(s2(f(n))) Complexity

  12. Savitch: Generalized Version Theorem (Savitch): S(n) ≥ log(n) NSPACE(S(n))  SPACE(S(n)2) Proof: We proved NLSPACE(log2n). The theorem follows from the padding argument.  Complexity

  13. Corollary Corollary:PSPACE = NPSPACE Proof: Clearly, PSPACENPSPACE. By Savitch’s theorem, NPSPACEPSPACE.  Complexity

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