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.

)2(DTIME))((DTIME).()(2 nt

ntnt

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