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Space and time constructible functions.PowerPoint Presentation

Space and time constructible functions.

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## PowerPoint Slideshow about ' Space and time constructible functions.' - september-dunlap

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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).

- NSPACE(S(n)) subseteq DSPACE (S2(n))

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

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