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Lecture 24 Time and Space of NTM

This lecture covers the time and space complexity of non-deterministic Turing machines (NTMs) and their relationship with complexity classes such as NTIME, NP, and NSPACE. It also discusses the concepts of time and space bounds, the Speed Up Theorem, the Hierarchy Theorem, and Savich's Theorem.

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Lecture 24 Time and Space of NTM

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  1. Lecture 24 Time and Space of NTM

  2. Time • For a NDM M and an input x, TimeM(x) = the minimum # of moves leading to accepting x if x ε L(M) = infinity if x not in L(M)

  3. Time Bound A NTM M is said to have a time bound t(n) if for sufficiently large n and every x ε L(M) With |x|=n, TimeM(x) < max {n+1, t(n)} .

  4. Complexity Classes NTIME(t(n)) = {L(M) | M is a NTM with time bound t(n)} NP = U c > 0 NTIME(n ) c

  5. Relationship • P NP • NP ≠ EXP • NP EXPOLY

  6. Theorem • Speed Up Theorem still holds. • Hierarchy Theorem may not.

  7. Space For a NTM M and an input x, SpaceM(x) = the minimum, over all computation paths, of maximum space taken in each work tape on input x if x ε L(M) = infinity otherwise

  8. Space bound • A NTM M is said to have a space bound s(n) if sufficiently large n and every input x with |x|=n, SpaceM(x) ≤ max{1, s(n)}

  9. Complexity Classes • NSPACE(s(n)) = {L(M) | M is a NTM with space bound s(n)} • NSPACE = Uc>0 NSPACE(n ) c

  10. Relationship • NP NSPACE • PSAPACE = NPSPACE (why?)

  11. Savich’s Theorem • If s(n) ≥ log n, then NSPACE (s(n)) DSPACE(s(n) ) The proof will be given in next lecture! 2

  12. Theorems • Compresion Theorem holds. • Hierarchy Theorem may not.

  13. Translation Lemma • Let s1(n), s2(n) and f(n) be fully space-constructible functions with s2(n) > n and f(n)> n. Then NSPACE(s1(n)) NSPACE(s2(n)) implies NSPACE(s1(f(n))) NSPACE(s2(f(n)))

  14. Hierarchy 4 8 • NSPACE(n ) DSPACE(n ) DSPACE(n ) NSPACE(n ) • For r > 1 and a > 0, NSPACE(n ) ≠ NSPACE (n ) 9 ≠ 9 r+a r

  15. Proof of Savitch’s Theorem

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