Space and time constructible functions
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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…

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  • No office hours tommorow.

  • Qualifying exam policy is out.

    • you can choose

  • Schedule changes posted on blog.

Space constructible
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
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
The Gap Theorem

  • Borodin, JACM, 1972, 19:1

    Example: suppose t(n) = sin(n). Then DTIME(sin(n)) = DTIME(22^sin(n))


Hierarchy for space constr fns
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
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
The rest of CS 611

  • More practice reading and writing proofs

  • Inclusion results

  • Separation results

  • P, NP and other famous classes

Proof practice
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
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
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
Famous Complexity Classes

  • see

  • Deterministic polynomial time

  • Nondeterministic polynomial time

  • Deterministic polynomial space