Lecture 1-2 Time and Space of DTM

Model Multitape TM with write-only output.

Time of DTM

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)}

Linear Speed Up

1--m Bee dance 3m q

1--m a b c d e f initial 3m q

1--m a b c d e f 1st bee dance 3m c d q

1--m a b c d e f 1st bee dance 3m c d e f q

1--m a b c d e f 1st bee dance 3m c d e f q

1--m a b c d e f 1st bee dance 3m a b c d e f q

1--m a b c d e f 3m a’ b’ c’ d’ e’ f’ p

1--m a b c’ d’ e f 2nd bee dance 3m a’ b’ e’ f’ p

1--m a b c’ d’ e’ f’ 2nd bee dance 3m a’ b’ p

1--m a b c’ d’ e’ f’ 2nd bee dance 3m a’ b’ p

1--m a’ b’ c’ d’ e’ f’ 2nd bee dance 3m p

1--m a’ b’ c’ d’ e’ f’ initial 3m p

Model Independent Classes

Space SpaceM(x) = total # of cells that M visits on all working (storage) tapes during the computation on input x. If M is a multitape DTM, then the working tapes do not include the input tape and the write-only output tape.

Multi-tape DTM Input tape (read only) working tapes Output tape (possibly, write only)

Space Bound A DTM with k work tapes is said to have a space bound s(n) if for any input x with |x| < n, SpaceM(x) < max{k, s(n)}.

Time and Space For any DTM with k work tapes, SpaceM(x) < k (TimeM(x) + 1)

Complexity Classes A language L has a space complexity s(n) if it is accepted by a multitape with write-only output tape DTM with space bound s(n). DSPACE(s(n)) = {L | L has space complexity s(n)}

Tape Compression Theorem

1--m 3m

Model Independent Classes P = U c>0 DTIME(n ) EXP = U c > 0 DTIME(2 ) EXPOLY = U c > 0 DTIME(2 ) PSPACE = U c > 0 DSPACE(n ) c cn c n c

Extended Church-Turing Thesis A function computable in polynomial time in any reasonable computational model using a reasonable time complexity measure is computable by a DTM in polynomial time.

P PSPACE SpaceM(x) < k (TimeM(x) + 1)

PSPACE EXPOLY

Input tape (read only) working tapes Output tape (possibly, write only)

“Sufficiently large”

A, B P imply A U B P

A, B P imply AB P

L P implies L* P

All regular sets belong to P

Space Hierarchy Theorem

Space-constructible function s(n) is fully space-constructible if there exists a DTM M such that for sufficiently large n and any input x with |x|=n, SpaceM(x) = s(n).

Space Hierarchy If s2(n) is a fully space-constructible function, s1(n)/s2(n) → 0 as n → infinity, s1(n) > log n, then DSPACE(s2(n)) DSPACE(s1(n)) ≠ Φ

Time Hierarchy

Time-constructible function t(n) is fully time-constructible if there exists a DTM such that for sufficiently large n and any input x with |x|=n, TimeM(x) = t(n).

Time Hierarchy If t1(n) > n+1, t2(n) is fully time-constructible, t1(n) log t1(n) /t2(n) → 0 as n → infinity, then DTIME(t2(n)) DTIME(t1(n)) ≠ Φ

P EXP Could you prove

PSPACE≠EXP