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R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ??????

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Presentation on theme: "R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ??????"— Presentation transcript:

1 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? SATISFIABILITY Eric L. Frederich

2 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? Overview Definitions K-Satisfiability 3-Satisfiability –Three Colorable Problem Transformation –Davis – Putnam Algorithm 2-Satisfiability –Algorithm Summary References

3 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? Definitions Satisfiability –An instance of the problem is defined by a Boolean expression written using only AND, OR, NOT, variables, and parentheses. The question is: given the expression, is there some assignment of TRUE and FALSE values to the variables that will make the entire expression true? * Product of Sums –A way of arranging a boolean expression such that the final output is a result of ‘AND’ing some ‘OR’ clauses where each term within an ‘OR’ clause may or may not be ‘NOT’ed. n –The number of boolean variables found in the expression (with or without a NOT) m –The number of ‘OR’ clauses ‘AND’ed together in the boolean expression K –The maximum number of variables found within each OR clause * Definition from

4 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? K-Satisfiability K is the maximum number of terms allowed per ‘OR’ clause. For any K > 2, the problem lies in NP-Complete –Solution of order 2 n Special K’s –3SAT 3 Colorable Graph problem Still NP-Complete –2SAT Polynomial time solution K = 5 –(A + B’ + C)(B + C’ + E’)(A’ + B + C’+ D’ + E)(B + D) K = 3 (3SAT) –(X’ + Y)(X + Y’ + Z)(X’ + Z’)(Y + Z’) K = 2 (2SAT) –(X1 + X2)(X2’ + X3)(X1’ + X2’)(X3 + X4)(X3’ + X5)(X4’ + X5’)(X3’ + X4)

5 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? 3 – Colorable Transformation Procedure Transform(G) for each vertex v i in G output(r i + y i + b i ) for each edge (v i,v j ) in G output(r i ’ + r j ’)(y i ’ + y j ’)(b i ’ + b j ’) output : boolean expression with K = 3

6 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? (R2 + Y2 + B2) (R3 + Y3 + B3) (R4 + Y4 + B4) (R5 + Y5 + B5) (R7 + Y7 + B7) (R1’ + R2’)(Y1’ + Y2’)(B1’ + B2’) (R1’ + R3’)(Y1’ + Y3’)(B1’ + B3’) (R1’ + R4’)(Y1’ + Y4’)(B1’ + B4’) (R1’ + R5’)(Y1’ + Y5’)(B1’ + B5’) (R2’ + R5’)(Y2’ + Y5’)(B2’ + B5’) (R2’ + R6’)(Y2’ + Y6’)(B2’ + B6’) (R3’ + R4’)(Y3’ + Y4’)(B3’ + B4’) (R4’ + R5’)(Y4’ + Y5’)(B4’ + B5’) (R5’ + R6’)(Y5’ + Y6’)(B5’ + B6’) (R7’ + R4’)(Y7’ + Y4’)(B7’ + B4’) (R7’ + R5’)(Y7’ + Y5’)(B7’ + B5’) (R6 + Y6 + B6) (R1 + Y1 + B1)( Y1 )( B2)(R3 )( B4)(R5 ) ( Y6 )( Y7 ) (R1' + R2’)( + Y2’)(B1’ + )(R1’ + )( + Y3’)(B1’ + B3’) (R1' + R4’)( + Y4’)(B1’ + )(R1’ + )( + Y5’)(B1’ + B5’) (R2' + )(Y2’ + Y5’)( + B5’)(R2’ + R6’)(Y2’ + )( + B6’) ( + R4’)(Y3’ + Y4’)(B3’ + )(R4’ + )(Y4’ + Y5’)( + B5’) ( + R6’)(Y5’ + )(B5’ + B6’)(R7’ + R4’)( + Y4’)(B7’ + ) (R7' + )( + Y5’)(B7’ + B5’) (R1 + Y1 + B1)(R2 + Y2 + B2)(R3 + Y3 + B3)(R4 + Y4 + B4)(R5 + Y5 + B5) (R6 + Y6 + B6)(R7 + Y7 + B7) (R1' + R2’)(Y1’ + Y2’)(B1’ + B2’)(R1’ + R3’)(Y1’ + Y3’)(B1’ + B3’) (R1' + R4’)(Y1’ + Y4’)(B1’ + B4’)(R1’ + R5’)(Y1’ + Y5’)(B1’ + B5’) (R2' + R5’)(Y2’ + Y5’)(B2’ + B5’)(R2’ + R6’)(Y2’ + Y6’)(B2’ + B6’) (R3’ + R4’)(Y3’ + Y4’)(B3’ + B4’)(R4’ + R5’)(Y4’ + Y5’)(B4’ + B5’) (R5’ + R6’)(Y5’ + Y6’)(B5’ + B6’)(R7’ + R4’)(Y7’ + Y4’)(B7’ + B4’) (R7' + R5’)(Y7’ + Y5’)(B7’ + B5’) 2 (21) = 2,097,152

7 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 (R1 + Y1 + B1)(R2 + Y2 + B2)(R3 + Y3 + B3)(R4 + Y4 + B4)(R5 + Y5 + B5) (R6 + Y6 + B6)(R7 + Y7 + B7) (R1' + R2’)(Y1’ + Y2’)(B1’ + B2’)(R1’ + R3’)(Y1’ + Y3’)(B1’ + B3’) (R1' + R4’)(Y1’ + Y4’)(B1’ + B4’)(R1’ + R5’)(Y1’ + Y5’)(B1’ + B5’) (R2' + R5’)(Y2’ + Y5’)(B2’ + B5’)(R2’ + R6’)(Y2’ + Y6’)(B2’ + B6’) (R3’ + R4’)(Y3’ + Y4’)(B3’ + B4’)(R4’ + R5’)(Y4’ + Y5’)(B4’ + B5’) (R5’ + R6’)(Y5’ + Y6’)(B5’ + B6’)(R7’ + R4’)(Y7’ + Y4’)(B7’ + B4’) (R7' + R5’)(Y7’ + Y5’)(B7’ + B5’) 3-Colorable R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7'

8 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? Davis - Putnam Recursive but still 2 n Prunes off branches which will not yield a result. Does not examine the entire expression at each guess Procedure split (E) 1. if E has an empty clause, then return 2. if E has no clauses, then exit with current partial assignment 3. select next unassigned variable, x i, in E 4. split ( E ( x i = 0 ) ) 5. split ( E ( x i = 1 ) )

9 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? Davis - Putnam (X1 + X3’ + X4)(X1’ + X2’ + X4’)(X2’ + X3)(X1’ + X2 + X4) (X3’ + X4)(X2’ + X3) (X2’ + X4’)(X2’ + X3)(X2 + X4) X1 = 0 X1 = 1 (X3’ + X4) X2 = 0 (X3’ + X4)(X3) X2 = 1 (X4) (X4’)(X3) X2 = 0 X2 = 1 X3 = 0 X3 = 1 X3 = 0 X3 = 1 X3 = 0 X3 = 1 X3 = 0 X3 = 1 X4 = 0 X4 = 1 (X4) X4 = 0 (X4) X4 = 0 (X4) X4 = 0 (X4) X4 = 0 (X4’) X4 = 0

10 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? 2- Satisfiability Pick an assignment Drop terms that become true Make assignments based on initial assignment. Repeat…. (X1 + X2)(X2’ + X3)(X1’ + X2’)(X3 + X4)(X3’ + X5)(X4’ + X5’)(X3’ + X4)

11 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? Summary For K-Satisfiability where K > 2 it is an NP- Complete Problem 2-Satisfiability is solvable in polynomial time Davis-Putnam improves efficiency but does not decrease order of complexity 3-Colorable problem is synonymous to 3- Satisfiability

12 R1 R2 R3 R4 R5 R6 R7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 B1 B2 B3 B4 B5 B6 B7 R1' R2' R3' R4' R5' R6' R7' Y1' Y2' Y3' Y4' Y5' Y6' Y7' B1' B2' B3' B4' B5' B6' B7' ?????? References "The New Turing Omnibus", by A.K. Dewdney A.W.H. Freeman/Owl Book, 2001 New York


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