SLIDE 1
Informatics 1
Lecture 8 Searching for Satisfaction
Michael Fourman
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2
¬A ⋁ C ¬B ⋁ D ¬B ⋁ ¬C ⋁ ¬D ¬E ⋁ B E ⋁ B A ⋁ E ¬E ⋁ A
A B C D E
¬A ¬B ¬C ¬D ¬E
Informatics 1 Lecture 8 Searching for Satisfaction Michael Fourman - - PowerPoint PPT Presentation
Informatics 1 Lecture 8 Searching for Satisfaction Michael Fourman - - PowerPoint PPT Presentation
Informatics 1 Lecture 8 Searching for Satisfaction Michael Fourman 1 2 D C A C B D B A E B E A E E A E E B A B B C D C D 3 B C D D C A C B D B D
SLIDE 2
SLIDE 3
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A B C D E
¬A ¬B ¬C ¬D ¬E
¬A ⋁ C ¬B ⋁ D ¬B ⋁ ¬C ⋁ ¬D ¬E ⋁ B E ⋁ B A ⋁ E ¬E ⋁ A ¬E → B B → D ¬B → ¬E ¬D → ¬B ¬E → B B → D ¬E → D E⋁B ¬B⋁D E⋁D
SLIDE 4
4
¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D
A AB ABC ABC̅ ABCD ABCD̅ AB̅ A̅
¬A ⋁ C ¬B ⋁ D ¬B ⋁ ¬C ⋁ ¬D ¬E ⋁ B E ⋁ B A ⋁ E ¬E ⋁ A
SLIDE 5
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D ¬E⋁B E⋁B
A AB ABC ABC̅ ABCD ABCD̅ AB̅ AB̅C AB̅C̅ AB̅CD AB̅CD̅
¬A⋁C
AB̅CDE AB̅CDE̅ AB̅CD̅E AB̅CD̅E̅
¬E⋁B E⋁B
A̅
SLIDE 6
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D ¬E⋁B E⋁B A⋁E A⋁¬E
A AB ABC ABC̅ ABCD ABCD̅ AB̅ AB̅C AB̅C̅ AB̅CD AB̅CD̅
¬A⋁C
AB̅CDE AB̅CDE̅ AB̅CD̅E AB̅CD̅E̅
¬E⋁B E⋁B
A̅ A̅XXX A̅XXXE A̅XXXE̅
SLIDE 7
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D
A AB ABC ABC̅ ABCD ABCD̅ AB̅ A̅
focus on part of this tree.
SLIDE 8
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D
AB ABC ABC̅ ABCD ABCD̅
¬B⋁¬C⋁¬D ¬B⋁D ¬B⋁¬C Premises Conclusion Any assignment of truth values that makes all the premises true will make the conclusion true. The conclusion follows from the premises A valid inference
SLIDE 9
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D
AB ABC ABC̅ ABCD ABCD̅
¬B⋁¬C⋁¬D ¬B⋁D ¬B⋁¬C Premises Conclusion Any assignment of truth values that makes the conclusion false will make at least one of the premises false. For any valid inference
SLIDE 10
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D
AB ABC ABC̅ ABCD ABCD̅
¬B⋁¬C⋁¬D ¬B⋁D ¬B⋁¬C Premises Conclusion If some assignment XYZ of values for ABC makes the conclusion false then the assignments XYZD and XYZD̅ each make one or other of the two premises false. A special property
- f this inference
¬B⋁¬C
SLIDE 11
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¬A⋁C ¬B⋁D ¬B⋁¬C⋁¬D
AB ABC ABC̅ ABCD ABCD̅
¬B⋁¬C ¬B⋁D ¬B⋁¬C⋁¬D ¬B⋁¬C ¬A⋁C ¬B⋁¬A ¬A⋁¬B
Resolution
SLIDE 12
12
U⋁V⋁W⋁X⋁¬C X⋁Y⋁Z⋁C U⋁V⋁W⋁X⋁Y⋁Z
Resolution
SLIDE 13
13
Resolution
¬E⋁B E⋁B A⋁E ¬E⋁A ¬B⋁D ¬B⋁¬C⋁¬D ¬B⋁¬C ¬A⋁C ¬B⋁¬A B ¬A A
⊥
A B E E C D
¬B⋁¬C⋁¬D ¬B⋁D ¬A⋁C ¬E⋁B E⋁B ¬E⋁A A⋁E
Refutation
SLIDE 14
14
¬A ⋁ C ¬B ⋁ D ¬E ⋁ B E ⋁ B A ⋁ E ¬E ⋁ A
A B C D E
¬A ¬B ¬C ¬D ¬E
¬E ⋁ C C ⋁ E E ⋁ D ¬E ⋁ D A D C B
SLIDE 15
Clausal Form
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Resolution uses CNF a conjunction of disjunctions of literals
(¬A⋁C)⋀(¬B⋁D)⋀(¬E⋁B)⋀(¬E⋁A)⋀(A⋁E)⋀(E⋁B)⋀(¬B⋁¬C⋁¬D)
Clausal form is a set of sets of literals
{{¬A,C}, {¬B,D}, {¬E,B}, {¬E,A}, {A,E}, {E,B}, {¬B, ¬C, ¬D}}
Each set of literals represents the disjunction of its literals. An empty set of literals {} represents false ⊥. The clausal form represents the conjunction of these disjunctions
SLIDE 16
Clausal Form
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Clausal form is a set of sets of literals
{ {¬A,C}, {¬B,D}, {¬E,B}, {¬E,A}, {A,E}, {E,B},{¬B, ¬C, ¬D} }
A (partial) truth assignment makes a clause true iff it makes at least one of its literals true (so it can never make the empty clause {} true) A (partial) truth assignment makes a clausal form true iff it makes all of its clauses true ( so the empty clausal form {} is always true ).
SLIDE 17
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Clausal form is a set of sets of literals
{ X0, X1, … , Xn-1 }
Resolution rule for clauses
X Y where ¬A ∈ X, A ∈ Y (X ⋃ Y) \ { ¬A, A }
SLIDE 18
(A?B:C)
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