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Lecture 22. Time of DTM

Lecture 22. Time of DTM. Time of DTM. Time M (x) = # of moves that DTM M takes on input x. Time M (x) < infinity iff x ε L(M). Time Bound. M is said to have a time bound t(n) if for every x with |x| < n, Time M (x) < max {n+1, t(n)}. Theorem.

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Lecture 22. Time of DTM

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  1. Lecture 22. Time of DTM

  2. Time of DTM • TimeM (x) = # of moves that DTM M takes on input x. • TimeM(x) < infinity iff x ε L(M).

  3. Time Bound M is said to have a time bound t(n) if for every x with |x| < n, TimeM(x) < max {n+1, t(n)}

  4. Theorem • For any multitape DTM M, there exists a one-tape DTM M’ to simulate M within time TimeM’(x) < c + (TimeM(x)) c is a constant. 2

  5. Complexity Class • A language L has a (deterministic) time-complexity t(n) if there is a multitape DTM M accepting L, with time bound t(n). • DTIME(t(n)) = {L | L has a time bound t(n)}

  6. Model • Multitape TM with write-only output.

  7. Linear Speed Up Suppose t(n)/n → infinity as n → infinity. Then for any constant c > 0, DTIME(t(n)) = DTIME(ct(n))

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