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Prove: SAT is NPcomplete.
To prove SAT is NPcomplete, we must show that every NP problem instance can be
polynomially transformed into a SAT instance.
So, suppose we have a NP problem P, and its instance w = w
1
...w
n.
Since P is NP, there exist a NonDeterministic Turing Machine PTM, that will take in w
and produce correctly accept or reject in polynomial time.
Since PTM exist, we will use PTM in addition to w to construct our SAT instance.
Our SAT instance is defined as follows:
phi = phi
cell
AND phi
start
AND phi
move
AND phi
accept
where each of the phi has a function:
phi
cell
: guarantee that each cell is valid. (see below for what “cells” are)
phi
start
: guarantee start configuration is accurate.
phi
move
: guarantee each transition is valid.
phi
accept
: guarantee there is an accepting configuration.
In essence, since PTM exist, we can run w on PTM, and it will consist of a set of TM
configurations, starts from #q
0
w
1
w
2
…w
n
B…B#, and if PTM(w) is accept, than one of the
configuration will contain a q
accept
.
We put all configurations in a table L, where each row is a configuration:
Start configuration
1 of the possible 2
nd
configuration
1 of the possible 3
rd
configuration, derived
from the 2
nd
configuration on the previous
row.
.
.
.
Since P runs in NP time, there is at max O(n
k
) rows, where n is the length of input to P, or
n = w. As a result, there can be at max O(n
k
) columns, because you can only add 1
symbol to the tape at each step (or each row). For simplicity, we will address the number
of rows and columns as n
k
in the subsequence of the prove.
A cell is simply 1 cell of the above table. In each cell, there is a limited amount of
character that can present. The set of cell alphabets C is the following:
C = Q U T U {#}, where Q is the set of states, T is the tape alphabet of PTM (which
includes input alphabet), and {#} is the front and end mark of each string configuration.
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View Full DocumentEach variable in our SAT instance will correspond to a particular character in the table L.
Thus X
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 Spring '08
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