Lecture 23 Space Complexity of DTMSpaceSpace BoundTime and SpaceComplexity ClassesTape Compression TheoremModel Independent ClassesExtended Church-Turing ThesisP PSPACEPSPACE EXPOLYA, B ε P imply A U B ε PA, B ε P imply AB ε PL ε P implies L* ε PAll regular sets belong to PHierachy TheoremSpace-constructible functionSpace HierarchyTime-constructible functionTime HierarchyP EXPEXP ≠ PSAPACELecture 23 Space Complexity of DTMSpace•SpaceM(x) = # of cell that M visits on the 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 is said to have a space bound s(n) if for any input x with |x| < n, SpaceM(x) < max{1, 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•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 )ccnnccExtended 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 PSPACEPSPACE EXPOLYA, B ε P imply A U B ε PA, B ε P imply AB ε PL ε P implies L* ε PAll regular sets belong to PHierachy TheoremSpace-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 HierarchyIf •s2(n) is a fully space-constructible function,•s1(n)/s2(n) → 0 as n → infinity,•s1(n) > log n,thenDSPACE(s2(n)) DSPACE(s1(n)) ≠ ΦTime-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 HierarchyIf•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)) ≠ ΦP EXPEXP ≠
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