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Lecture 23 Space Complexity of DTM

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Space Space M (x) = # of cell that M visits on the work (storage) tapes during the computation on input x. If M is a multitape DTM, then the work tapes do not include the input tape and the write-only output tape.

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Space Bound A DTM is said to have a space bound s(n) if for any input x with |x| < n, SpaceM(x) < max{1, s(n)}.

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Time and Space For any DTM with k work tapes, Space M (x) < K (Time M (x) + 1)

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Complexity Classes A language L has a space complexity s(n) if it is accepted by a multitape with write- only output tape DTM with space bound s(n). DSPACE(s(n)) = {L | L has space complexity s(n)}

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Tape Compression Theorem For any function s(n) and any constant c > 0, DSPACE(s(n)) = DSPACE(c·s(n))

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Model Independent Classes P = U c>0 DTIME(n ) EXP = U c > 0 DTIME(2 ) EXPOLY = U c > 0 DTIME(2 ) PSPACE = U c > 0 DSPACE(n ) c cn n c c

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Extended Church-Turing Thesis A function computable in polynomial time in any reasonable computational model using a reasonable time complexity measure is computable by a DTM in polynomial time.

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P PSPACE

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PSPACE EXPOLY

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A, B ε P imply A U B ε P

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A, B ε P imply AB ε P

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L ε P implies L* ε P

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All regular sets belong to P

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Hierachy Theorem

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Space-constructible function s(n) is fully space-constructible if there exists a DTM M such that for sufficiently large n and any input x with |x|=n, Space M (x) = s(n).

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Space Hierarchy If s 2 (n) is a fully space-constructible function, s 1 (n)/s 2 (n) → 0 as n → infinity, s 1 (n) > log n, then DSPACE(s2(n)) DSPACE(s1(n)) ≠ Φ

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Time-constructible function t(n) is fully time-constructible if there exists a DTM such that for sufficiently large n and any input x with |x|=n, Time M (x) = t(n).

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Time Hierarchy If t 1 (n) > n+1, t 2 (n) is fully time-constructible, t 1 (n) log t 1 (n) /t 2 (n) → 0 as n → infinity, then DTIME(t 2 (n)) DTIME(t 1 (n)) ≠ Φ

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P EXP

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EXP ≠ PSAPACE

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Resource-Bounded Computation Previously: can it be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space, other.

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