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By: Mohammad Khosravi , Reza RamezaniTo: Dr Moosavi

An improved to Ak-MaxSat

Electronic & Computer Department

Isfahan University Of Technology

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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.

Max-Sat Problem

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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]

UP, using one queue

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x3 propagate{#,x4, x5,¬x4 ¬x5}Inconsistent subformula detectedby UP is {x1, x2, x3,¬x1 ¬x2 ¬x3}The remaining clauses {¬x1 x4,¬x1 x5,¬x4 ¬x5, x1 ¬x2}

Continue UP processing

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{x1, x2, x3,¬x1 x4,¬x1 x5,¬x4 ¬x5,¬x1 ¬x2 ¬x3, x1 ¬x2}Q1 = [x1, x2, x3] x1 propagateQ1 = [ x2, x3]{ x2, x3, x4, x5,¬x4 ¬x5, ¬x2 ¬x3}Q2 = [x4, x5] x4 propagateQ2 = [x5,¬x5] x5 propagate

UP*, using two queues

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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

Continue UP* processing

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{x3, ¬x3 }Q2 = [¬x3 ] inconsistent subformula : {x2, x3,¬x1 ¬x2 ¬x3, x1 ¬x2}.

Continue UP* processing

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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*

Failed Literal Detection

Solutions for solving maxsat efficiency:RulesUnit PropagationFailed Literal DetectionSelecting variables for propagationData Structure

Max-Sat Problem effective ways:

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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

AK Max-Sat idea

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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.

Our Max-Sat idea