Space and time constructible
functions.
Why do I care?
CS 611
announcements
• No office hours tommorow.
• Qualifying exam policy is out.
– you can choose
• Schedule changes posted on blog.
Space Constructible
• A function S(n) is space constructible if…
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there is an S(n) space bound TM , that…
for each n there is
an input of size n for which
M uses exactly S(n) cells
• Example: Mlog computes log(m).
– Mlog uses log(digits-in(x)) cells to compute
log(x) for some x with n digits.
Why you care.
• The following statement is false:
“For every space bound t(n), all TMs with
space bound g(n) such that g(n) > t(n) can
solve more problems than TMs with space
bound t(n)”
(i.e., more time always gives more power)
The Gap Theorem
• Borodin, JACM, 1972, 19:1
 t ( n ). DTIME ( t ( n ))  DTIME ( 2
2
t(n)
Example: suppose t(n) = sin(n). Then
DTIME(sin(n)) = DTIME(22^sin(n))
MAYBE.
)
Hierarchy for Space Constr. Fns.
• For fully space constructible functions s1 and s2
If s1(n) in o(s2(n)) then
DTIME(s1) subset DTIME (s2).
(theorem 5.15 in our book).
Which functions are
space constructible?
• log(n), nk, 2n and n!
• If f,g are space constructible, then
f(n)*g(n), 2f(n) and f(n)g(n)
are space constructible too.
The rest of CS 611
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More practice reading and writing proofs
Inclusion results
Separation results
P, NP and other famous classes
Proof Practice
• Some scratch work from book, not as much.
• Proof project:
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scratch work,
the proof, v1.0
review proofs
the final proof.
Inclusion Results
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Of the form: X is a subset of or equal to Y.
Y is at least as powerful as X, or,
X is no more powerful than Y.
Example:
– NSPACE(S(n)) subseteq DSPACE (S2(n))
(for fully space constructible S(n), of course).
Separation Results
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Of the form X subset Y or X != Y.
Y is more powerful than X, or,
X and Y have different power.
Example:
– Space hierarchy theorem
• Rare results in complexity theory.
– lower bounds are hard to prove.
Famous Complexity Classes
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see http://www.mathsci.appstate.edu/~sjg/simpsonsmath/
Deterministic polynomial time
Nondeterministic polynomial time
Deterministic polynomial space