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Non-deterministic time

Non-deterministic time. [AB 2]. Non-deterministic Turing Machines. Excluding “ _ ”.  and _. q reject q accept. 3. The accepted language. Def: A Non-deterministic Turing machine (NDTM) accepts L iff there exists a path from an initial configuration to an accepting configuration

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Non-deterministic time

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  1. Non-deterministic time [AB 2]

  2. Non-deterministic Turing Machines Excluding “_”  and _ qrejectqaccept 3

  3. The accepted language Def: A Non-deterministic Turing machine (NDTM) accepts L iff there exists a path from an initial configuration to an accepting configuration In particular if it rejects x, there is no path from an initial configuration to an accepting configuration in the configuration graph of x

  4. Time complexity Def: The running time of a NDTM T on input x is the length of the longest path from an initial configuration to a terminating configuration. Def: A NDTM T runs in time f(n) if for every input x, the running time of T on x is at most f(|x|).

  5. Time-Complexity

  6. Deterministic vs. Nondeterministic

  7. Examples • Perfect Matching – • Input G=(V,E). • Yes instances: G has a perfect matching. • No instances: G does not have a perfect matching. • MaxClique-atleast-k • Input G=(V,E),k. • Yes instances: G has a k-clique. • No instances: G does not have a k-clique.

  8. What about • No Perfect Matching – • Input G=(V,E). • No instances: G has a perfect matching. • Yes instances: G does not have a perfect matching. • MaxClique-atmost-k • Input G=(V,E),k. • No instances: G has a k-clique. • Yes instances: G does not have a k-clique.

  9. P, NP and co-NP coL= *-L coNP== {coL | LNP}

  10. NP – second definition A language L belongs to NP if there exists • A polynomial p(n) • A polynomial time TM M Such that xLiff u  {0,1}p(n) s.t. M(x,u)=1 Any u s.t. M(x,u)=1 is called a witness for xL

  11. Witness Verification Program

  12. Nodeterministic

  13. NP  PSPACE Proof: • Suppose L belongs to NP, solvable by M(x,u). • Algorithm in PSPACE: Given x  {0,1}n , try sequentially all u  {0,1}p(n) and accept iff for some u, M(x,u)=1.

  14. Name the Class

  15. SAT

  16. SIP 254-259, AB 2.3 Cook/Levin

  17. Tableaux cne

  18. Example

  19. M(Q{#})4

  20. M,xvariables cne

  21. M,w

  22. Q.E.D.

  23. SAT is NPC

  24. P, NP, co-NP and NPC

  25. NPC

  26. NPC

  27. Summary so far

  28. 3SAT

  29. SIP 259-260 CNF is NPC

  30. CNF3CNF (xy)(x1x2... xt)... clauses with 1 or 2 literals clauses with more than 3 literals replication split (xyx) (x1  x2  c11)(c11 x3 c12)... (c1t-3 xt-1xt)

  31. CLIQUE is NPC

  32. 3SATL CLIQUE 001 010 111 101

  33. SIP 251-253 SATp CLIQUE: proof

  34. INDEPENDENT-SET is NPC

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