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,

then

DSPACE(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