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- 1. By: Mohammad Khosravi , Reza Ramezani To: Dr Moosavi 1 Electronic & Computer Department Isfahan University Of Technology
- 2. Input: given a list of clauses C1, . . . ,Cm Output: the maximum number of clauses which satisfied. Unit Propagation (UP) algorithm use for compute LB when we have at least one unit clause, otherwise we will use Failed literal detection. 2
- 3. Let 1 be the Max-SAT instance {x1, x2, x3,¬x1 x4,¬x1 x5,¬x4 ¬x5,¬x1 ¬x2 ¬x3, x1 ¬x2} Q = [x1, x2, x3] x1 propagate { x2, x3, x4, x5,¬x4 ¬x5, ¬x2 ¬x3} Q = [x2,x3,x4,x5] x2 propagate {x3, x4, x5,¬x4 ¬x5,¬x3} Q =[x3, x4, x5,¬x3] 3
- 4. x3 propagate {#,x4, x5,¬x4 ¬x5} Inconsistent subformula detected by UP is {x1, x2, x3,¬x1 ¬x2 ¬x3} The remaining clauses {¬x1 x4,¬x1 x5,¬x4 ¬x5, x1 ¬x2} 4
- 5. {x1, x2, x3,¬x1 x4,¬x1 x5,¬x4 ¬x5,¬x1 ¬x2 ¬x3, x1 ¬x2} Q1 = [x1, x2, x3] x1 propagate Q1 = [ x2, x3] { x2, x3, x4, x5,¬x4 ¬x5, ¬x2 ¬x3} Q2 = [x4, x5] x4 propagate Q2 = [x5,¬x5] x5 propagate 5
- 6. Inconsistent subformula : {x1,¬x1 x4,¬x1 x5,¬x4 ¬x5} remaining clauses: {x2, x3,¬x1 ¬x2 ¬x3, x1 ¬x2} Q1 = [x2, x3] x2 propagate {x3,¬x1 ¬x3, x1 } Q1 = [x3] Q2 = [x1] x1 propagate 6
- 7. {x3, ¬x3 } Q2 = [¬x3 ] inconsistent subformula : {x2, x3,¬x1 ¬x2 ¬x3, x1 ¬x2}. 7
- 8. P={#,#, y ¬x,¬y z,¬y ¬z, y x} P U {x},with UP inconsistent subformula 1 = {x, y ¬x,¬y z,¬y ¬z}, P U {¬ x}, with UP inconsistent Subformula 2= {¬x,¬y z,¬y ¬z, y x} Final incosistant subformula=(1 U 2) {x,¬x} = {y ¬x,¬y z,¬y ¬z, y x} We can use FL whit UP & UP* 8
- 9. Solutions for solving maxsat efficiency: Rules Unit Propagation Failed Literal Detection Selecting variables for propagation Data Structure 9
- 10. Each variable order computed from: Binary-Length with two value. Unit-Length with one value. Total-Length that contain the number of clauses that variable occurs. W(i) is sum of all above weights. This weight will calculate for Var(i) and Var(-i). Final Variable’s weight is W(i) + W(-i) Each Variable with higher weight with select earlier for propagation 10
- 11. Each variable order computed from: Binary-Length with three value. Unit-Length with two value. Third-Length with one value. Total-Length that contain the number of clauses that variable occurs. AKMax-Sat solver ReCompute this weights at each level, such as: propagate variable, remove variable and restore variable. 11

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