Announcements. Office Hours Swap: OH schedule has been updated to reflect this.

Transcription

1 SA Solving

2 Announements Offie Hours Swap: Zavain has offie hours from 4-6PM toay in builing 460, room 040A. Rose has offie hours tonight from 7-9PM in Gates B26B. Keith has offie hours hursay from 2-4PM in Gates 178. OH sheule has been upate to reflet this.

3 A Note on the Chekpoints...

4 his Doesn't Work! heorem: If R is transitive, then R -1 is transitive. Proof: Consier any a, b, an suh that arb an br. Sine R is transitive, we have ar. Sine arb an br, we have br -1 a an R -1 b. Sine we have ar, we have R -1 a. hus R -1 b, br -1 a, an R -1 a. his proves a. b.. (arb br R -1 b br -1 a R -1 a) You nee to show a. b.. (ar -1 b br -1 ar -1 )

5 his Doesn't Work! heorem: If R is transitive, then R -1 is transitive. Proof: Consier any a, b, an suh that arb an br. Sine R is transitive, we have ar. Sine arb an br, we have br -1 a an R -1 b. Sine we have ar, we have R -1 a. hus R -1 b, br -1 a, an R -1 a. his proves a. b.. (arb br R -1 b br -1 a R -1 a) You nee to show a. b.. (ar -1 b br -1 ar -1 )

6 his Doesn't Work! heorem: If R is transitive, then R -1 is transitive. Proof: Consier any a, b, an suh that arb an br. Sine R is transitive, we have ar. Sine arb an br, we have br -1 a an R -1 b. Sine we have ar, we have R -1 a. hus R -1 b, br -1 a, an R -1 a. his proves a. b.. (arb br R -1 b br -1 a R -1 a) You nee to show a. b.. (ar -1 b br -1 ar -1 )

7 A Corret Proof heorem: If R is transitive, then R -1 is transitive. Proof: Consier any a, b, an suh that ar -1 b an br -1. We will prove ar -1. Sine ar -1 b an br -1, we have that bra an Rb. Sine Rb an bra, by transitivity we know Ra. Sine Ra, we have ar -1, as require.

8 he akeaway Point Don't get trippe up by efinitions! If you want to iretly prove that P Q, assume that P is true an prove that Q is true as well.

9 An now... SA!

10 Is his Formula Ever rue? p p

11 Is his Formula Ever rue? p p

12 Is his Formula Ever rue? (r s t) (s t r) (t r s) t s

13 Is his Formula Ever rue? (x 0 (x 1 x 0 )) (x 2 x 1 x 0 ) (x 1 x 1 )

14 Satisfiability A propositional logi formula φ is alle satisfiable if there is some assignment to its variables that makes it evaluate to true. An assignment of true an false to the variables of φ that makes it evaluate to true is alle a satisfying assignment. Similar terms: φ is tautologial if it is always true. φ is satisfiable if it an be mae true. φ is unsatisfiable if it is always false.

15 SA he boolean satisfiability problem (SA) is the following: Given a propositional logi formula φ, is φ satisfiable? Note: Goal is just to get a yes/no answer, not to atually fin a satisfying assignment. hough usually we an obtain an assignment easily when given a SA solver. Extremely important problem both in theory an in pratie.

16 Appliations of SA

17

18

19

20

21 Solving SA: ake One

22 A Simple Algorithm Given a formula φ, we an just buil a truth table for φ an hek all of the rows. If any of them evaluate to true, then φ is satisfiable. If none of them evaluate to true, then φ is unsatisfiable. So what might this look like?

23 he ruth able Algorithm p p p F

24 he ruth able Algorithm p p p F F

25 he ruth able Algorithm p p p F F F

26 he ruth able Algorithm p p p F F F

27 he ruth able Algorithm p p p F F F

28 he ruth able Algorithm p q (p q) q F F F F F F F F F

29 he ruth able Algorithm p q (p q) q F F F F F F F F F

30 q (r q) r p q F p F F F F F F F F F F F r F F F F F F F F F F F F F F F F F F F F F F F F

31 q (r q) r p q F p F F F F F F F F F F F r F F F F F F F F F F F F F F F F F F F F F F F F

32 A Large Problem ruth tables an get very big very quikly! With n variables, there will be 2 n rows. Many real-wor SA instanes have hunres of thousans of variables; this is ompletely infeasible!

33 Clause-Base Algorithms

34 Simplifying Our Formulas Arbitrary formulas in propositional logi an be omplex. Lots of ifferent onnetives. Arbitrary nesting of formulas. Can be iffiult to see how they all interrelate. Goal: Convert formulas into a simpler format.

35 Literals an Clauses A literal in propositional logi is a variable or its negation: x y But not x y. A lause is a many-way OR (isjuntion) of literals. x y z x But not x ( y z)

36 Conjuntive Normal Form A propositional logi formula φ is in onjuntive normal form (CNF) if it is the many-way AND (onjuntion) of lauses. (x y z) ( x y) (x y z w) x z But not (x (y z)) (x y) Only legal operators are,,. No nesting allowe.

37 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

38 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

39 he Struture of CNF ( x y z ) ( x y z ) ( x y z ) Eah lause must have at least one true literal in it.

40 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

41 he Struture of CNF ( x y z ) ( x y z ) ( x y z ) We shoul pik at least one true literal from eah lause

42 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

43 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

44 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

45 he Struture of CNF ( x y z ) ( x y z ) ( x y z )

46 he Struture of CNF ( x y z ) ( x y z ) ( x y z ) subjet to the onstraint that we never hoose a literal an its negation

47 Getting to CNF here are exellent algorithms for solving SA formulas in CNF. How o we onvert an arbitrary propositional logi formula into a formula in CNF?

48 Negation Normal Form A formula φ in propositional logi is in negation normal form (NNF) iff he only onnetives are,, an. is only applie iretly to variables. Examples: (a ( b )) ( a) a b Non-Examples: (a b) a ( b ) a b

49 Getting to NNF NNF is a stepping stone towar CNF: Only have,, an. All negations pushe onto variables. Buil an algorithm to get from arbitrary propositional logi own to NNF. Our onversion proess will work as follows: Eliminate omplex onnetives. Simplify negations.

50 Eliminating Complex Connetives NNF only allows,,. First step: Replae other onnetives with these three. Replae φ ψ with ( φ ψ). Replae with (p p) for any variable p. Replae with (p p) for any variable p. For now, let's ignore ; etails are triky. Result: A new formula that is logially equivalent to the original an not muh bigger.

51 Eliminating Complex Connetives (p q r) (s t q) (p q r) (s t q) (p q r) ( s t q)

52 Simplifying Negations NNF only allows negations in front of variables. Now that we have just,, an, repeately apply these rules to ahieve this result: Replae φ with φ Replae (φ ψ) with φ ψ Replae (φ ψ) with φ ψ his proess eventually terminates; the height of the negations keeps ereasing.

53 Simplifying Negations (p q r) ( s t q) (p q r) ( s t q) (p q r) ( s t q) (p q r) (s t q)

54 From NNF to CNF Now that we an get to NNF, let's get own to CNF. Reall: CNF is the onjuntion of lauses: (x 1 x 2 x 3 ) (x 1 x 4 ) We'll use an inutive approah to onvert NNF into CNF.

55 From NNF to CNF Every NNF formula is either A literal, he onjuntion of two NNF formulas: φ ψ he isjuntion of two NNF formulas: φ ψ Let's work through some examples an see if we an fin a pattern.

56 Examples: NNF to CNF (x y z w) ( x y)

57 Examples: NNF to CNF x ( y z) (x y) ( x z)

58 y x (y z ) F x F F F F F F F F F F F z F F F F F F F F F (x y) ( x z) F F F F F F F F F F F F F F F F F

59 Examples: NNF to CNF (w x) ( y z) ((w x) y) ((w x) z) (w y) ( x y) ( w z) ( x z)

60 Examples: NNF to CNF (v w x) ( y z) ((v w x) y) (( v w x) z) ((v w) y) ( x y) (( v w) z) (x z) (v y) ( w y) ( x y) ( v z) ( w z) ( y z)

61 Converting NNF to CNF Apply the following reasoning inutively: If the formula is a literal, o nothing. If the formula is φ ψ: Convert φ an ψ to CNF, all it φ' an ψ'. Yiel φ' ψ' If the formula is φ ψ: Convert φ an ψ to CNF, all it φ' an ψ'. Repeately apply the istributive law x (y z) (x y) (x z) to φ' ψ' until simplifie.

62 A Problem (a b) ( ) ( e f) (g h) ((a ) (a ) (b ) (b )) (e f) (g h) ((a e) (a f) (a e) (a f) ) ((b e) (b f) (b e) (b f)) (g h) (a e g) (a f g) (a e g) (a f g) (b e g) (b f g) (b e g) (b f g) (a e h) (a f h) (a e h) (a f h) (b e h) (b f h) (b e h) (b f h)

63 A Problem (a b) ( ) ( e f) (g h) ((a ) (a ) (b ) (b )) (e f) (g h) ((a e) (a f) (a e) (a f) ) ((b e) (b f) (b e) (b f)) (g h) (a e g) (a f g) (a e g) (a f g) (b e g) (b f g) (b e g) (b f g) (a e h) (a f h) (a e h) (a f h) (b e h) (b f h) (b e h) (b f h)

64 A Problem (a b) ( ) ( e f) (g h) ((a ) (a ) (b ) (b )) (e f) (g h) ((a e) (a f) (a e) (a f) ) ((b e) (b f) (b e) (b f)) (g h) (a e g) (a f g) (a e g) (a f g) (b e g) (b f g) (b e g) (b f g) (a e h) (a f h) (a e h) (a f h) (b e h) (b f h) (b e h) (b f h)

65 A Problem (a b) ( ) ( e f) (g h) ((a ) (a ) (b ) (b )) (e f) (g h) ((a e) (a f) (a e) (a f) ) ((b e) (b f) (b e) (b f)) (g h) (a e g) (a f g) (a e g) (a f g) (b e g) (b f g) (b e g) (b f g) (a e h) (a f h) (a e h) (a f h) (b e h) (b f h) (b e h) (b f h)

66 A Problem (a b) ( ) ( e f) (g h) ((a ) (a ) (b ) (b )) (e f) (g h) ((a e) (a f) (a e) (a f) ) ((b e) (b f) (b e) (b f)) (g h) (a e g) (a f g) (a e g) (a f g) (b e g) (b f g) (b e g) (b f g) (a e h) (a f h) (a e h) (a f h) (b e h) (b f h) (b e h) (b f h)

67 Exponential Blowup Our logi for eliminating an lea to exponential size inreases. Not a problem with the algorithm; some formulas proue exponentially large CNF formulas. We will nee to fin another approah.

68 Equivalene an Equisatisfiability Reall: wo logial formulas φ an ψ are are equivalent (enote φ ψ) if they always take on the same truth values. wo logial formulas φ an ψ are equisatisfiable (enote φ ψ) if φ is satisfiable iff ψ is satisfiable. o solve SA for a formula φ, we an instea solve SA for an equisatisfiable ψ.

69 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b )

70 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b )

71 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a b )

72 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a ) ( b ) ( )

73 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q )

74 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a )

75 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a ) ( )

76 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

77 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

78 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

79 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

80 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

81 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

82 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

83 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

84 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

85 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

86 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

87 Equisatisfiably from NNF to CNF ( a b ) ( a ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

88 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

89 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b )

90 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b )

91 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) )

92 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q ) ( b q ) ( q ) ( a q ) ( q )

93 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r)

94 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) a b

95 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a ) ( b )

96 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

97 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

98 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

99 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

100 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

101 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

102 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

103 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

104 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

105 Equisatisfiably from NNF to CNF ( ( a q ) ( b q ) ( q ) ( a q ) ( q ) ) ( a b ) ( a q r) ( b q r ) ( q r) ( a q r) ( q r) ( a r ) ( b r )

106 Equisatisfiably from NNF to CNF If the formula is a literal, o nothing. If the formula is φ ψ: Convert φ an ψ to CNF, all it φ' an ψ'. Yiel φ' ψ' If the formula is φ ψ: Convert φ an ψ to CNF, all it φ' an ψ'. Create a new variable q. A q to eah lause of φ'. A q to eah lause of ψ'. Yiel φ' ψ'. As at most n new variables to eah lause, where n is the number of lauses. Size inrease at worst quarati.

107 Where We Are Now We've foun a way to onvert arbitrary propositional logi formulas into equisatisfiable CNF formulas. he size may inrease, but at most quaratially. Now, let's atually try to solve SA!

108 A Simple Baktraking Algorithm

109 ( a b ) ( a ) ( a ) ( a ) ( a ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

110 ( a b ) ( a ) ( a ) ( a ) ( a ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

111 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

112 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

113 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

114 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

115 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

116 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

117 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

118 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

119 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

120 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

121 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

122 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

123 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

124 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

125 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

126 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

127 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

128 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

129 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

130 ( a b ) ( ) ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) b Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

131 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

132 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

133 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

134 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

135 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

136 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

137 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

138 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

139 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

140 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

141 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

142 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

143 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

144 ( a b ) ( ) b ( ) ( ) ( ) a ( ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

145 ( a b ) ( ) b ( ) ( ) ( ) a ( ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

146 ( a b ) ( ) b ( ) ( ) ( ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

147 ( a b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

148 ( a b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( a b ) ( a b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

149 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( b ) ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

150 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( b ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

151 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( b ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

152 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

153 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

154 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

155 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

156 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

157 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

158 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

159 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

160 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

161 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

162 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

163 ( ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

164 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( b ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

165 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( b ) ( b ) b ( b ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

166 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

167 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

168 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

169 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

170 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

171 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

172 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

173 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

174 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

175 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

176 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

177 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

178 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

179 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

180 ( b ) ( a ) b ( a ) ( a ) ( a ) a ( ) ( b ) b ( ) Formula from he Quest for Effiient Boolean Satisfiability Solvers by Shara Malik of Prineton University

181 Baktraking SA Solving If φ is an empty set of lauses, return true. All lauses are satisfie. If φ ontains ( ), return false, Some lause is unsatisfiable. Otherwise: Choose some variable x. Return whether φ x is satisfiable or φ x is satisfiable, where φ x is φ with x set to false an φ x is φ with x set to true.

182 Analyzing Baktraking he baktraking solver works reasonably well on most inputs. Low memory usage just nee to remember one potential path along the tree. For formulas with many satisfying assignments, typially fins one very quikly. But it has its weaknesses. Completely blin searhing might miss obvious hoies. In the worst-ase, must explore the entire tree, whih has 2 n leaves.

183 Aing Heuristis A heuristi is an approah to solving a problem that may or may not work orretly. Contrast with an algorithm, whih has efinitive guarantees on its behavior. he simpliity of CNF makes it possible to a heuristis to our baktraking solver. What sorts of heuristis might we a?

184 Pure Literal Elimination ( a b ) ( a ) ( a b ) ( a b ) ( b )

185 Pure Literal Elimination ( a b ) ( a ) ( a b ) ( a b ) ( b )

186 Pure Literal Elimination ( a b ) ( a ) ( a b ) ( a b ) ( b ) he he variable variable is is never never negate negate here. here. here here is is no no reason reason not not to to set set it it to to true. true.

187 Pure Literal Elimination ( a b ) ( a b ) ( b ) he he variable variable is is never never negate negate here. here. here here is is no no reason reason not not to to set set it it to to true. true.

188 Pure Literal Elimination ( a b ) ( a b ) ( b )

189 Pure Literal Elimination ( a b ) ( a b ) ( b )

190 Pure Literal Elimination ( a b ) ( a b ) ( b ) he he variable variable a is is always always negate negate here. here. here here is is no no reason reason not not to to set set it it to to false. false.

191 Pure Literal Elimination ( b ) he he variable variable a is is always always negate negate here. here. here here is is no no reason reason not not to to set set it it to to false. false.

192 Pure Literal Elimination ( b )

193 Pure Literal Elimination ( b )

194 Pure Literal Elimination ( b ) he he variable variable b is is always always negate negate here. here. here here is is no no reason reason not not to to set set it it to to false. false.

195 Pure Literal Elimination he he variable variable b is is always always negate negate here. here. here here is is no no reason reason not not to to set set it it to to false. false.

196 Pure Literal Elimination

197 Pure Literal Elimination All All lauses lauses have have been been satisfie, so so the the formula formula is is satisfiable.

198 Pure Literal Elimination A literal is alle pure if its negation appears nowhere in the formula. Setting that literal to true will satisfy some number of lauses automatially an simplify the formula. Many formulas an be satisfie by iteratively applying pure literal elimination.

199 Unit Propagation ( a b ) ( a ) ( a b ) ( a b )

200 Unit Propagation ( a b ) ( a ) ( a b ) ( a b )

201 Unit Propagation ( a b ) ( a ) ( a b ) ( a b ) is is all all by by itself. itself. has has to to be be false false for for this this formula formula to to be be true. true.

202 Unit Propagation ( a b ) ( a ) ( a b ) ( a b ) is is all all by by itself. itself. has has to to be be false false for for this this formula formula to to be be true. true.

203 Unit Propagation ( a b ) ( a ) ( a b ) ( a b )

204 Unit Propagation ( a b ) ( a ) ( a b ) ( a b )

205 Unit Propagation ( a b ) ( a ) ( a b ) ( a b ) a is is all all by by itself. itself. a has has to to be be true true for for this this formula formula to to be be true. true.

206 Unit Propagation ( b ) ( b ) ( b ) a is is all all by by itself. itself. a has has to to be be true true for for this this formula formula to to be be true. true.

207 Unit Propagation ( b ) ( b ) ( b )

208 Unit Propagation ( b ) ( b ) ( b )

209 Unit Propagation ( b ) ( b ) ( b ) b is is all all by by itself. itself. b has has to to be be true true for for this this formula formula to to be be true. true.

210 Unit Propagation ( ) b is is all all by by itself. itself. b has has to to be be true true for for this this formula formula to to be be true. true.

211 Unit Propagation ( )

212 Unit Propagation ( )

213 Unit Propagation ( ) We We are are left left with with an an empty empty lause. lause. he he formula formula is is unsatisfiable.

214 Unit Propagation A unit lause is a lause ontaining just one literal. For the formula to be true, that literal must be set to true. his might expose other unit lauses.

215 DPLL he DPLL algorithm is a moifiation of the simple baktraking searh algorithm. Name for Davis, Putnam, Logemann, an Lovelan, its inventors. Inorporates the two heuristis we just saw.

216 DPLL Simplify φ with unit propagation. Simplify φ with pure literal elimination. If φ is empty, return true. If φ ontains ( ), return false. Otherwise: Choose some variable x. Return whether φ x is satisfiable or φ x is satisfiable, where φ x is φ with x set to false an φ x is φ with x set to true.

217 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a )

218 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a ) SAR

219 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a ) SAR Guess a

220 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a ) SAR Guess a

221 ( a b ) ( a ) ( b ) ( a ) ( b ) ( b ) ( b ) ( a b ) ( a ) SAR Guess a

222 ( b ) ( ) ( b ) ( ) ( b ) ( b ) ( b ) ( b ) ( ) SAR Guess a

223 ( b ) ( ) ( b ) ( ) ( b ) ( b ) ( b ) ( b ) ( ) SAR Guess a

224 ( b ) ( ) ( b ) ( ) ( b ) ( b ) ( b ) ( b ) ( ) SAR Guess a Propagate

225 ( b ) ( ) ( b ) ( ) ( b ) ( b ) ( b ) ( b ) ( ) SAR Guess a Propagate

226 ( b ) ( b ) ( ) ( b ) ( b ) ( ) SAR Guess a Propagate

227 ( b ) ( b ) ( ) ( b ) ( b ) ( ) SAR Guess a Propagate

228 ( b ) ( b ) ( ) ( b ) ( b ) ( ) SAR Guess a Propagate

229 ( b ) ( b ) ( ) ( b ) ( b ) ( ) SAR Guess a Propagate Propagate

230 ( b ) ( b ) ( ) ( b ) ( b ) ( ) SAR Guess a Propagate Propagate

231 ( b ) ( b ) ( ) SAR Guess a Propagate Propagate

232 ( b ) ( b ) ( ) SAR Guess a Propagate Propagate

233 ( b ) ( b ) ( ) SAR Guess a Propagate Propagate

234 ( b ) ( b ) ( ) SAR Guess a Propagate Propagate FAILURE

235 ( b ) ( b ) ( ) ( b ) ( b ) ( ) SAR Guess a Propagate Propagate FAILURE

236 ( b ) ( ) ( b ) ( ) ( b ) ( b ) ( b ) ( b ) ( ) SAR Guess a Propagate Propagate FAILURE

237 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a ) SAR Guess a Propagate Propagate FAILURE

238 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a ) SAR Guess a Propagate Propagate FAILURE Fore: a

239 ( a b ) ( a b ) ( b ) ( b ) ( a ) ( b ) ( a b ) ( b ) ( a b ) ( a ) ( a b ) ( a b ) ( a ) SAR Guess a Propagate Propagate FAILURE Fore: a

240 ( a b ) ( b ) ( b ) ( b ) ( a b ) ( b ) ( a b ) ( a b ) SAR Guess a Propagate Propagate FAILURE Fore: a

241 ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) SAR Guess a Propagate Propagate FAILURE Fore: a

242 ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) SAR Guess a Propagate Propagate FAILURE Fore: a

243 ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b

244 ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b

245 ( b ) ( b ) ( b ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b

246 ( ) ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b

247 ( ) ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b

248 ( ) ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate

249 ( ) ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate

250 ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate

251 ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate

252 ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate

253 ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate

254 ( ) ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate

255 ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate

256 ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate

257 ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate

258 ( ) SAR Guess a Propagate Propagate FAILURE Fore: a Propagate b Propagate Propagate FAILURE

259 DPLL is Powerful DPLL was invente 50 years ago this July, but is still the basis for most SA solvers. he two heuristis aggressively simplify many ommon ases. However, still has an exponential worst-ase runtime.

260 A Ranomize Approah

261 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b

262 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b

263 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false false false false

264 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false false false false

265 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false false false false

266 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false false true false

267 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false false true false

268 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false false true false

269 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true false

270 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true false

271 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true false

272 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true false true

273 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true false true

274 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true false true

275 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true true false true

276 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true true false true

277 Ranomize SA Solving ( a b ) ( a ) ( a ) ( a ) ( a ) ( b ) ( a b ) ( a b ) a b false true true true false true

278 Ranomize SA Solving We an attempt to solve SA ranomly: Make a ompletely ranom guess at a satisfying assignment. If the formula is satisfiable, output true. For eah one-variable hange possible: If that hange improves the assignment, make the hange an repeat. If stuk, make another ompletely ranom guess at a satisfying assignment. After some number of tries, output probably not. Many inustrial SA solving algorithms (suh as GSA an WalkSA) are base on this approah.

279 How har is SA? We'll see more on this later on...

280 How har is SA? We'll see more on this later on...

281 Next ime Introution to Computability heory Strings an eision problems. DFAs.