Space and time constructible functions.

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# Space and time constructible functions. - PowerPoint PPT Presentation

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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### 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…
• 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

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
• 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:
• scratch work,
• the proof, v1.0
• review proofs
• the final proof.
Inclusion Results
• 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
• 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
• see http://www.mathsci.appstate.edu/~sjg/simpsonsmath/
• Deterministic polynomial time
• Nondeterministic polynomial time
• Deterministic polynomial space