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 For any multitape DTM M, there exists a one-tape DTM M’ to simulate M within time Time M’ (x) < c + (Time M (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))