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Understanding Space and Time Constructible Functions in Computer Science

Explore the concept of space constructible functions and their implications in complexity theory. Learn about the Gap Theorem, hierarchy of space constructible functions, and inclusion and separation results. Dive into proof practice and famous complexity classes.

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Understanding Space and Time Constructible Functions in Computer Science

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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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