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2. Resolution for Propositional Logic CSE2303 Formal Methods I Lecture 22

3. Overview Resolution for Propositional Logic. Arguments

4. Example {P, ¬Q}{P, Q}{¬P, Q}{¬P, ¬Q} {P}{Q} {¬P}

5. DAG (Directed Acyclic Graph) leaves vertices root

6. Resolution A resolution derivation of a formula A is: DAG (directed acyclic graph). The leaves are the clauses in A. The other clauses are resolvents of their parents. A resolution refutation is a resolution derivation whose root is .

7. Arguments Given premises: P 1, …, P n Conclusion: C An argument is valid if P 1  …  P n  C is a tautology. So an argument is valid if and only if ¬(P 1  …  P n )  C is a tautology. So an argument is valid if and only if P 1  …  P n  ¬C is unsatisfiable.

8. Mary Exam Example Today Mary has a Law exam or a Computer Science exam or both. She doesn’t have a Law exam. Therefore she must have a Computer Science exam. L: Mary has a Law exam today. C: Mary has a Computer Science exam today. Premises: L  C, ¬L Conclusion: C

9. Premises: L  C {{L, C}} ¬L {{¬L}} Conclusion: C Negation of Conclusion: {{¬C}} So we want to show {{L, C}, {¬L}, {¬C}} is unsatisfiable

10. {¬L}{L, C}{¬C} {C} Proof

11. Party Example If we have a party then we’ll invite Lana and Bob. If we invite Lana or Bob then we must invite Jake. Therefore if we have a party then we must invite Jake. P: We will have a party. L: We’ll invite Lana. B: We’ll invite Bob. J: We must invite Jake.

12. Premises: P  L  B {{¬P, L}, {¬P, B}} L  B  J {{¬L, J}, {¬B, J}} Conclusion: P  J Negation of Conclusion: {{P}, {¬J}} Need to show {{¬P, L}, {¬P, B}, {¬L, J}, {¬B, J}, {P},{¬J}} is unsatisfiable

13. {¬P, L}{¬L, J} {¬J} {P} Proof {L} {¬L}

14. Advice Example If he had taken my advice or had his wits about him, he would have sold his house and moved to the country. If he had sold his house, Jenny would have bought it. Jenny didn’t buy his house. Therefore he didn’t take my advice. A: He took my advice. W: He had his wits about him. H: He sold his house. C: He moved to the country. J: Jenny bought his house.

15. Premises: (A  W)  (H  C) {{¬A, H}, {¬A, C}, {¬W, H}, {¬W, C}} H  J {{¬H, J}} ¬J {{¬J}} Conclusion: ¬A Negation of Conclusion {{A}} Need to show {{¬A, H}, {¬A, C}, {¬W, H}, {¬W, C}, {¬H, J}, {¬J}, {A}} is unsatisfiable

16. Proof {¬H, J}{¬A, H}{¬J}{A} {¬H} {H}

17. Wumpus World 1 2 3 4 43214321 Start Here Felt a breeze Detect a stench Conclude Wumpus is here

18. Notation W 1,1 –The Wumpus is in square 1,1. S 1,2 –A stench was detected in square 1,2. B 2,1 –A breeze was detected in square 2,1. Etc.

19. Wumpus World P 1 : ¬S 1,1  ¬B 1,1 P 2 : ¬S 2,1  B 2,1 P 3 : S 1,2  ¬B 1,2 P 4 : ¬S 1,1  ¬W 1,1  ¬W 1,2  ¬W 2,1 P 5 : ¬S 2,1  ¬W 1,1  ¬W 2,1  ¬W 2,2  ¬W 3,1 P 6 : ¬S 1,2  ¬W 1,1  ¬W 1,2  ¬W 2,2  ¬W 1,3 P 7 : S 1,2  W 1,3  W 1,2  W 2,2  W 1,1 Conclusion: W 1,3

20. Wumpus World P 1 : {{¬S 1,1 }, {¬B 1,1 }} P 2 : {{¬S 2,1 }, {B 2,1 }} P 3 : {{S 1,2 }, {¬B 1,2 }} P 4 : {{S 1,1,¬W 1,1 }, {S 1,1,¬W 1,2 }, {S 1,1,¬W 2,1 }} P 5 : {{S 2,1,¬W 1,1 },{S 2,1,¬W 2,1 },{S 2,1,¬W 2,2 },{S 2,1,¬W 3,1 }} P 6 : {{S 1,2,¬W 1,1 },{S 1,2,¬W 2,1 },{S 1,2,¬W 2,2 },{S 1,2,¬W 1,3 }} P 7 : {{¬ S 1,2, W 1,3, W 1,2, W 2,2, W 1,1 }} Negation of Conclusion: {{¬W 1,3 }} To show P 1  P 2  P 3  P 4  P 5  P 6  P 7  ¬W 1,3 is unsatisfiable.

21. {¬ S 1,2, W 1,3, W 1,2, W 2,2, W 1,1 }{S 1,2 } {W 1,3, W 1,2, W 2,2, W 1,1 } {¬W 1,3 } {W 1,2, W 2,2, W 1,1 } {¬S 1,1 }{S 1,1,¬W 1,2 } {¬W 1,2 } {¬S 2,1 }{S 2,1,¬W 1,1 } {¬W 1,1 } {S 2,1,¬W 2,2 } {¬W 2,2 } {W 1,1 } {W 2,2, W 1,1 } Proof

22. Revision Know the definition of a resolution derivation. Be able to do resolution for Propositional logic.