Lecture 23 Space Complexity of DTM Space Space

Lecture 23 Space Complexity of DTM

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.

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

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

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

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

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

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.

P PSPACE

PSPACE EXPOLY

A, B ε P imply A U B ε P

A, B ε P imply AB ε P

L ε P implies L* ε P

All regular sets belong to P

Hierachy Theorem

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).

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(s 2(n)) DSPACE(s 1(n)) ≠ Φ

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).

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

P EXP

EXP ≠ PSAPACE