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Unformatted text preview: ac c epts .
3. Otherwis e, rejec t.”
Sinc e s tage1 and s tage2 both run in poly nomial time nondeterminis tic ally , NTM N c an dec ide
the union of L1 and L2 in poly nomial time. Henc e, N P is c los ed under union.
Without los s of generality , let N 1 and N 2 be the NTMs that dec ide two languages L1 and L2
in NP. Then we c ons truc t an NTM N that dec ides the c onc atenation of L1 and L2 in poly nomial
N = ”on input w
1. For eac h pos s ible s plit of w = w1 w2
a. Run N 1 on w1 nondeterminis tic ally and run N 2 on w2 nondeterminis tic ally .
Ac c ept if both ac c ept.
2. Otherwis e, rejec t.”
As s ume the jwj = n . There are O(n) pos s ible s plits of w at mos t. Bes ides , the s um of the time
c omplex ities of N 1 and N 2 is als o poly nomial. Thus , s tep1 run in poly nomial time and NTM N
c an dec ide the c onc atenation of L1 and L2 in poly nomial time. Henc e, N P is c los ed under
c onc atenation.
Sinc e ab is ex ponential in the length of b , we need to break down ab and apply the modulo
operation s ev eral times s o that the res ulting s iz e nev er goes ex ponential. Firs t, we treat b in its
binary repres entation. There are at mos t log2 b bits . So
Q b 2i
Q bi2 i
ab mod p = (
a i ) mod p = (
(a mod p )) mod p
i i In the wors t c as e, we as s ume bi = 1 for all i . Howev er, abi+12 mod p = (abi2 )2 mod p = (abi2 mod p )2 mod p : That is , eac h term c an be c omputed by
s quaring the remainder of the prev ious term mod p . Henc e, for eac h modulo operation, the
c omputation is O(p 2 ) whic h is in poly nomial time. Additionally , for eac h multiplic ation of
neighboring terms , we als o apply the modulo operation in O(p 2 ): Sinc e there are at mos t log2 b
terms , ab Ñ c (mod p ) is O(2 log2 b p 2 ): Therefore, M ODEX P 2 P :
i+1 i i 7.16
1.) In the proof of theorem 7.56, the SUBSETSUM problem ins tanc e c ons truc ted c ontains
numbers of large magnitude pres ented in dec imal notation. Howev er, c ompared with
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This note was uploaded on 10/26/2013 for the course CS 22C:135 taught by Professor Zhang during the Spring '13 term at University of Iowa.
- Spring '13