Lecture 2 Time and Space of DTM

1
Lecture 2 Time and Space of DTM
2
Time of DTM

• TimeM (x) of moves that DTM M takes on input
  x.
• TimeM(x) lt infinity iff x e L(M).

3
Time Bound

• M is said to have a time bound t(n) if for
  every x with x lt n,
• TimeM(x) lt max n1, t(n)

4
Theorem

• For any multitape DTM M, there exists a one-tape
  DTM M to simulate M within time
• TimeM(x) lt c (TimeM(x))
• c is a constant.

2
5
Complexity Class

• A language L has a (deterministic)
  time-complexity t(n) if there is a multitape DTM
  M accepting L, with time bound t(n).
• DTIME(t(n)) L L has a time bound t(n)

6
Model

• Multitape TM with write-only output.

7
Linear Speed Up

• Suppose t(n)/n ? infinity as n ? infinity. Then
  for any constant c gt 0,
• DTIME(t(n)) DTIME(ct(n))

8
1--m
Bee dance
3m
9
Model Independent Classes
10
Space

• SpaceM(x) total of cells that M visits on all
  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.

11
Space Bound

• A DTM with k work tapes is said to have a space
  bound s(n) if for any input x with x lt n,
• SpaceM(x) lt maxk, s(n).

12
Time and Space

• For any DTM with k work tapes,
• SpaceM(x) lt k (TimeM(x) 1)

13
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)

14
Tape Compression Theorem

• For any function s(n) and any constant c gt 0,
• DSPACE(s(n)) DSPACE(cs(n))

15
Model Independent Classes
c

• P U cgt0 DTIME(n )
• EXP U c gt 0 DTIME(2 )
• EXPOLY U c gt 0 DTIME(2 )
• PSPACE U c gt 0 DSPACE(n )

cn
c
n
c
16
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.

17
P c PSPACE
18
PSPACE c EXPOLY
19
A, B e P imply A U B e P
20
A, B e P imply AB e P
21
L e P implies L e P
22
All regular sets belong to P
23
Space Hierarchy Theorem
24
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 xn,
• SpaceM(x) s(n).

25
Space Hierarchy

• If
• s2(n) is a fully space-constructible function,
• s1(n)/s2(n) ? 0 as n ? infinity,
• s1(n) gt log n,
• then
• DSPACE(s2(n)) DSPACE(s1(n)) ? F

26
Time Hierarchy
27
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 xn,
• TimeM(x) t(n).

28
Time Hierarchy

• If
• t1(n) gt n1,
• t2(n) is fully time-constructible,
• t1(n) log t1(n) /t2(n) ? 0 as n ? infinity,
• then
• DTIME(t2(n)) DTIME(t1(n)) ? F

29
P c EXP
30
EXP ? PSAPACE
31
PSPACE?EXP