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Space and time constructible functions.

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.

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  1. Space and time constructible functions. Why do I care? CS 611

  2. announcements • No office hours tommorow. • Qualifying exam policy is out. • you can choose • Schedule changes posted on blog.

  3. 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.

  4. 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)

  5. The Gap Theorem • Borodin, JACM, 1972, 19:1 Example: suppose t(n) = sin(n). Then DTIME(sin(n)) = DTIME(22^sin(n)) MAYBE.

  6. 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).

  7. 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.

  8. The rest of CS 611 • More practice reading and writing proofs • Inclusion results • Separation results • P, NP and other famous classes

  9. Proof Practice • Some scratch work from book, not as much. • Proof project: • scratch work, • the proof, v1.0 • review proofs • the final proof.

  10. 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).

  11. 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.

  12. Famous Complexity Classes • see http://www.mathsci.appstate.edu/~sjg/simpsonsmath/ • Deterministic polynomial time • Nondeterministic polynomial time • Deterministic polynomial space

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