index.md (1814B)
1 +++ 2 title = "Lecture 11" 3 template = "page-math.html" 4 +++ 5 6 # Time complexity contd. 7 ## Bounded tiling problem 8 Input is natural number n, finite collection of types of tiles, the first row of n tiles 9 is it possible to tile an n × n field with the given first row? 10 when connecting tiles, the touching side must have the same color, tiles must not be rotated. 11 12  13 14 the problem is NP-complete 15 16 ## co-NP 17 Problem L is in co-NP if complement $\bar{L}$ is in NP. 18 i.e. set of instances without solution is in NP. 19 20 we don't know if NP = co-NP in general. 21 but if an NP-complete problem is in co-NP, then NP = co-NP. 22 23 ## EXP and NEXP 24 Nondeterministic TM is exponential time if it runs in time $2^{p(|x|)}$ for some polynomial p. 25 26 NEXP is nondeterministic exponential, EXP is deterministic exponential. 27 28 P ⊆ NP ⊆ PSpace ⊆ EXP ⊆ NEXP. we don't know if this is strict. we know that P ≠ EXP, NP ≠ NEXP. 29 30 # Space complexity 31 Nondeterministic Turing machine M runs in space f if for every input w, every computation of the machine visits at most f(|w|) positions on the tape. 32 i.e. the function f is the upper bound on number of visited cells on tape (length of input word) 33 34 ## PSpace and NPSpace 35 Nondeterministic Turing machine is in polynomial space if it runs in space p for some polynomial p. 36 37 NPSpace is if it runs in nondeterministic polynomial space, PSpace if deterministic. 38 39 Savitch's theorem: if language accepted by nondeterministic TM in f(n) space, then accepted by deterministic TM in f(n)² space. 40 PSpace = NPSpace. 41 42 A language L ∈ PSpace is PSPace-complete if every language L' ∈ PSpace is polynomial-time reducible to L. 43 44 # Complexity hierarchy 45 46  47 48