Is DNA Computing Viable for 3-SAT Problems? Dafa Li Theoretical Computer Science, vol. 290, no. 3, pp , January Cho, Dong-Yeon

© 2003 SNU CSE Biointelligence Lab 2 Abstract 3-SAT problems  NP-hard problem  2 n Lipton’s model of DNA computing  The separate (or extract) operation  Imperfect operation DNA computing is not viable for 3-SAT

© 2003 SNU CSE Biointelligence Lab 3 Lipton ’ s Model How to use Lipton’s model of DNA computing to solve 3-SAT  Synthesize large number of copies of any single DNA strand  Create a double DNA strand by annealing  Extract (or separate) those DNA strands that contain some pattern  Detect if a test tube is empty by PCR

© 2003 SNU CSE Biointelligence Lab 4 The Key Issue The error for DNA computing  There are 10 l (l = 13 proposed by Adleman) copies of s for each distinct strand s in a test tube.  The separate (or extract) operation  p: success rate, 0 < p < 1, (q = 1 - p)  Lipton thinks that a typical percentage might be 90.  The paper will report  No matter how large l is and no matter how close to 1 p is, there always exists a class of 3-SAT problems such that DNA computing error must occur with Lipton’s model.

© 2003 SNU CSE Biointelligence Lab 5 There is a class of 3-SAT problems such that for Lipton ’ s model DNA computing error arises. Example 1  For the set of clauses C 1, C 2, …, C n, where C i is the variable x i, i=1,2,…,n.  Only one solution: … 1  There always exists a natural number [l/lg(1/p)] such that p n 10 l [l/lg(1/p)].  This means that the last test tube t n contains nothing.  There for error arises. l = 13

© 2003 SNU CSE Biointelligence Lab 6 Example 2  Let the set of clauses C 1, C 2, …, C m, where each C i is the same variable x it is intuitive to show that DNA computing error arises.  There always exists a natural number [l/lg(1/p)] such that p m 10 l [l/lg(1/p)].  This means that the last test tube t m contains nothing.  Then an error arises because the set of the clauses is satisfiable. l = 13

© 2003 SNU CSE Biointelligence Lab 7 Example 3  Let’s consider the set of the clauses x 1  x 2, x 2  x 3, x 3  x 4,…,x n-1  x n.  Step 1: x 1  x 2  t 1 contains p2 n-1 + p(1+q)2 n-2 DNA strands. p2 n-1 (x 1 =1) q2 n-1 (x 1 =1) 2 n-1 (x 1 =0) 2 n (all) pq2 n-2 (x 1 =1, x 2 =0) p2 n-2 (x 1 =0, x 2 =0) (x 1 =1, x 2 =0)

© 2003 SNU CSE Biointelligence Lab 8 Example 3 (continued)  Step 2: x 2  x 3  t 2 contains p 2 2 n-2 + 2p 2 (1+q)2 n-3 DNA strands. p2 n-1 (x 1 =1) pq2 n-2 (x 1 =1, x 2 =0) p2 n-2 (x 1 =0, x 2 =0) p 2 2 n-2 (x 1 =1, x 2 =1) pq2 n-2 (x 1 = 1, x 2 =1) p2 n-2 (x 1 =1, x 2 =0) pq2 n-2 (x 1 =1, x 2 =0) p2 n-2 (x 1 =0, x 2 =0) p 2 q2 n-3 (x 1 =1, x 2 =1, x 3 =0) p2 n-3 (x 1 =1, x 2 =0, x 3 =0) p 2 q2 n-3 (x 1 =1, x 2 =0, x 3 =0) p 2 2 n-3 (x 1 =0, x 2 =0, x 3 =0) Remainder

© 2003 SNU CSE Biointelligence Lab 9 Example 3 (continued)  Further operations  t 3 contains p 3 2 n-3 + 3p 3 (1+q)2 n-4 DNA strands.  t 4 contains p 4 2 n-4 + 4p 4 (1+q)2 n-5 DNA strands.  …  t n-1 contains 2p n-1 + (n-1)p n-1 (1+q)  p n-1 (n(1+q)+p)10 l DNA strands.  When n , the limit of the latter is 0.  For any p  (0,1), no matter how close to 1, and for any l > 0, no matter how large, there always exist a natural number N such that when n > N, p n-1 (n(1+q)+p)10 l < 1. l = 13

© 2003 SNU CSE Biointelligence Lab 10 There is a class of 3-SAT problems for which Lipton ’ s model is reliable. Example 4  For the set of clauses C 1, C 2, …, C m with n variables, where each C i is x 3i-2  x 3i-1  x 3i, Lipton’s model is reliable.  For each clause  T = (1-(1-p/2) 3 )2 n  t m contains DNA strands. (n = 3m)  2 3 -(2-p) 3  1 whenever p  /3

© 2003 SNU CSE Biointelligence Lab 11 Example 5  For the set of clauses C 1, C 2, …, C m with n variables, where each C i is x 2i-1  x 2i, Lipton’s model is reliable.  ((2 2 -(2-p) 2 )/2 2 ) m 2 n 10 l = (4p-p 2 ) n/2 10l  4p-p 2  1 when p  /2  0.27 The more models a set of clauses has, the more reliable DNA computing is.