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 ¬D → ¬B B A ¬B → ¬E ¬E ⋁ B ¬E ⋁ A E ¬E A ⋁ E ¬E → B E ⋁ B ¬A ¬B ¬E → B B → D E ⋁ B ¬B ⋁ D ¬E → D E ⋁ D ¬C ¬D

4 A ̅ A A B ̅ AB AB C ̅ ABC ¬A ⋁ C ABC D ̅ ABCD ¬A ⋁ C ¬B ⋁ D ¬E ⋁ B ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D ¬E ⋁ A A ⋁ E E ⋁ B ¬B ⋁ ¬C ⋁ ¬D

5 A ̅ A A B ̅ AB A B ̅ C A B ̅ C ̅ AB C ̅ ABC ¬A ⋁ C ABC D ̅ ABCD ¬A ⋁ C A B ̅ C D ̅ A B ̅ CD ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D A B ̅ C D ̅ E ̅ A B ̅ CDE A B ̅ C D ̅ E A B ̅ CD E ̅ E ⋁ B ¬E ⋁ B E ⋁ B ¬E ⋁ B

6 A ̅ A A B ̅ AB A ̅ XXX A B ̅ C A B ̅ C ̅ AB C ̅ ABC A ̅ XXX E ̅ ¬A ⋁ C ABC D ̅ A ̅ XXXE ABCD ¬A ⋁ C A B ̅ C D ̅ A B ̅ CD A ⋁ ¬E A ⋁ E ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D A B ̅ C D ̅ E ̅ A B ̅ CDE A B ̅ C D ̅ E A B ̅ CD E ̅ E ⋁ B ¬E ⋁ B ¬E ⋁ B E ⋁ B

7 focus on part of this tree. A ̅ A A B ̅ AB AB C ̅ ABC ABC D ̅ ABCD ¬A ⋁ C ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D

AB 8 Premises AB C ̅ ABC ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D ABC D ̅ ABCD ¬A ⋁ C ¬B ⋁ ¬C ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D Conclusion Any assignment of truth values that A valid makes all the premises true inference will make the conclusion true. The conclusion follows from the premises

AB 9 Premises AB C ̅ ABC ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D ABC D ̅ ABCD ¬A ⋁ C ¬B ⋁ ¬C ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D Conclusion Any assignment of truth values that For any valid makes the conclusion false will make inference at least one of the premises false.

AB 10 Premises AB C ̅ ABC ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D ¬B ⋁ ¬C ABC D ̅ ABCD ¬A ⋁ C ¬B ⋁ ¬C Conclusion ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D A special property If some assignment of this inference XYZ of values for ABC makes the conclusion false then the assignments XYZ D and XYZ D ̅ each make one or other of the two premises false.

11 ¬A ⋁ ¬B AB Resolution ¬B ⋁ ¬C AB C ̅ ABC ABC D ̅ ABCD ¬A ⋁ C ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D ¬A ⋁ C ¬B ⋁ ¬C ¬B ⋁ ¬A

12 Resolution U ⋁ V ⋁ W ⋁ X ⋁ ¬C X ⋁ Y ⋁ Z ⋁ C U ⋁ V ⋁ W ⋁ X ⋁ Y ⋁ Z

13 Resolution ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D E ⋁ B ¬B ⋁ ¬C ¬A ⋁ C ¬E ⋁ B ¬B ⋁ ¬A B ¬E ⋁ A A ⋁ E ¬A A ⊥ A Refutation B E ¬E ⋁ A A ⋁ E C E ¬A ⋁ C ¬E ⋁ B E ⋁ B D ¬B ⋁ ¬C ⋁ ¬D ¬B ⋁ D

14 E ⋁ B ¬E ⋁ B ¬B ⋁ D D C E ⋁ D ¬E ⋁ D B A D B E ¬E A ⋁ E ¬A ⋁ C ¬E ⋁ A C ⋁ E ¬E ⋁ C ¬A ¬B C ¬C ¬D A

Clausal Form 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 15

Clausal Form 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 ). 16

Clausal form is a set of sets of literals { X 0 , X 1 , … , X n-1 } Resolution rule for clauses X Y where ¬A ∈ X , A ∈ Y (X ⋃ Y) \ { ¬A, A } 17

18 (A?B:C)