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Is DNA Computing Viable for 3-SAT Problems? Dafa Li Theoretical Computer Science, vol. 290, no. 3, pp. 2095-2107, January 2003. Cho, Dong-Yeon

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© 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

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© 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

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© 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.

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© 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 1 1 … 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

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© 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

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© 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)

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© 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

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© 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

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© 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 2- 7 1/3

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© 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- 3 1/2 0.27 The more models a set of clauses has, the more reliable DNA computing is.

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