Lecture 22. Time of DTM

Time of DTM

• TimeM (x) = # of moves that DTM M takes on input x.

• TimeM(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,

TimeM(x) < max {n+1, t(n)}

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

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)}

Model

• Multitape TM with write-only output.

Linear Speed Up

Suppose t(n)/n → infinity as n → infinity. Then for any constant c > 0,

DTIME(t(n)) = DTIME(ct(n))