lecture 23 space complexity of dtm
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Lecture 23 Space Complexity of DTM
Space
• SpaceM(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,
SpaceM(x) < max{1, s(n)}.
Time and Space
• For any DTM with k work tapes, SpaceM(x) < K (TimeM(x) + 1)
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)}
Tape Compression Theorem
• For any function s(n) and any constant c > 0,
DSPACE(s(n)) = DSPACE(c·s(n))
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
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,
SpaceM(x) = s(n).
Space Hierarchy
If • s2(n) is a fully space-constructible function,• s1(n)/s2(n) → 0 as n → infinity,• s1(n) > log n,thenDSPACE(s2(n)) DSPACE(s1(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,
TimeM(x) = t(n).
Time Hierarchy
If• t1(n) > n+1,• t2(n) is fully time-constructible,• t1(n) log t1(n) /t2(n) → 0 as n → infinity,then DTIME(t2(n)) DTIME(t1(n)) ≠ Φ
P EXP
EXP ≠ PSAPACE