Lecture 2 Time and Space 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))

1--m 3m Bee dance

Model Independent Classes

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

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, Space M (x) < max{k, s(n)}.

Time and Space For any DTM with k work tapes, Space M (x) < k (Time M (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 For any function s(n) and any constant c > 0, DSPACE(s(n)) = DSPACE(c·s(n))

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 n c 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

PSPACE EXPOLY

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, Space M (x) = s(n).

Space Hierarchy If s 2 (n) is a fully space-constructible function, s 1 (n)/s 2 (n) → 0 as n → infinity, s 1 (n) > log n, then DSPACE(s 2 (n)) DSPACE(s 1 (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, Time M (x) = t(n).

Time Hierarchy If t 1 (n) > n+1, t 2 (n) is fully time-constructible, t 1 (n) log t 1 (n) /t 2 (n) → 0 as n → infinity, then DTIME(t 2 (n)) DTIME(t 1 (n)) ≠ Φ

P EXP

EXP ≠ PSAPACE

PSPACE≠EXP