Resolution Theorem Proving

1. Resolution Theorem Proving G51IAI Introduction to Artificial Intelligence Dr Matthew Hyde

2. Outline • Propositional Logic Recap • Conjunctive normal form • The Resolution Algorithm • Strategies to Guide the Search • Conclusion

3. Propositional Logic • Proposition symbols (“literals”) • A, B, C, D, Student, North • Each can be either True or False • The name is irrelevant • It is just the name you give to the proposition • In your model, if North = True, it could mean that you are facing north, or everyone is facing north, or that it is possible to go north, etc...

4. Propositional Logic • Logical Connectives form more complex sentences • OR: “V” - True if one of the symbols is true, or both ( Awake V Asleep ) • AND: “Λ” - True if both of the symbols are true ( AwakeΛListening ) • NOT: “¬” ¬Awake, ¬A, ¬B, ¬Student • IMPLIES: “=>” P => Q - True unless P is true and Q is false ( AwakeΛListening ) => ¬Fail

5. Propositional Logic • A “knowledge base” • Simply, a group of logical expressions that we already know to be true. E.g.: North Awake Λ Listening Student Awake V Asleep Literals

6. Propositional Logic • The knowledge base evaluates to true, because it is equivalent to putting an “Λ” between each expression North Awake Λ Listening Student Awake V Asleep North Λ(Awake Λ Listening) ΛStudent Λ(Awake V Asleep)

7. Summary so far • Literals can be either TRUE or FALSE • e.g A, B, ¬C, D, ¬Student, North ... • Knowledge base made up of combinations of literals • Each line of the knowledge base is TRUE • Therefore, ALL of the knowledge base evaluates to TRUE

8. Inference in Propositional Logic • We can infer new facts from what we already know • “Modus Ponens” Rule A => B A B • Also works for longer sentences A Λ B Λ C => D A Λ B Λ C D We know these two We can therefore infer that this is true

9. Example of Inference CompassNorth => FacingNorth Snowing => Cold CompassNorth Awake Cold We can infer that FacingNorth = ????

10. Example of Inference CompassNorth => FacingNorth Snowing => Cold CompassNorth Awake Cold FacingNorth We can infer that FacingNorth = True

11. Resolution in Propositional Logic • Resolution is one method for automated theorem proving • It is important to Artificial Itelligence because it helps logical agents to reason about the world • It helps them to prove new theorems, and therefore helps them to add to their knowledge

12. Resolution Algorithm • Input a knowledge base and an expression • It negates the expression, adds that to the knowledge base, and then finds a contradiction if one exists • If it finds a contradiction, then the negated statement is false • Therefore, the original statement mustbe true

13. Resolution Algorithm Small example Is it sunny? sunny = TRUE? Prove sunny Knowledge base: sunny daytime sunny V night

14. Resolution Algorithm Small example Is it sunny? sunny = TRUE? Prove sunny Knowledge base: sunny daytime sunny V night ¬sunny Negate it Add it to the knowledge base CONTRADICTION ¬sunny = FALSE Therefore: sunny = TRUE

15. Conjunctive Normal Form • Resolution algorithm needs sentences in CNF • A series of “conjunctions” (clauses joined together by “AND”) • (¬ A V B) Λ (B V C) Λ (D V ¬ E V F) Λ (G) Λ ... • Inside the brackets, we only have V (OR) ¬ (NOT) symbols • There are no “implies” (=>) symbols anywhere

16. Conjunctive Normal Form (A V B) Λ (B V C)Λ (D V E V F) Λ (G) Λ ... Clauses The whole thing represents the knowledge base, so it evaluates to TRUE

17. Conjunctive Normal Form • Resolution algorithm ‘resolves’ clauses • In fact, it only applies to clauses • Each pair of clauses that contains complementaryliterals is resolved • Complementary literals have the property that one negates the other • A, ¬A • Student, ¬Student

18. Procedure for converting to CNF • (a) To eliminate ↔, • (a ↔ b) ≡ (a → b) Λ(b→ a) • (b) To eliminate →, • (a → b) ≡ ¬ a ν b • (c) Double negation ¬ (¬a) ≡ a • (d) De Morgan • ¬ (a Λ b) ≡(¬a ν¬b) ¬(a ν b) ≡(¬a Λ¬b) • (e) Distributivity of Λover ν • (a Λ(b ν c )) ≡((aΛ b) ν (a Λ c)) • (f) Distributivity of νover Λ • (a ν(b Λ c )) ≡((a ν b) Λ(a ν c))

19. Resolution Rule • Given two clauses: ( A V B ) ( ¬B V C ) • Produce one clause containing all of the literals except the two complementary literals: A V C

20. Resolution Rule • Given two clauses: ( A V B V C V D ) ( E V F V ¬B V G V H ) • Produce one clause containing all of the literals except the two complementary literals: A V C V D V E V F V G V H

21. Resolution Example ¬asleep v fail asleep Show that the knowledge base entails “fail” • Negate the theorem ¬asleep v fail asleep

22. Very Important: • NEGATE THE THEOREM • This means, put a “¬” symbol in front of it • If it already has one, then remove it

23. Very Important: Theorem Negated Theorem A ¬A ¬B B Fail ¬Fail Then add it to the knowledge base and find a contradiction

24. Resolution Example ¬asleep v fail asleep Show that the knowledge base entails “fail” • Negate the theorem ¬fail ¬asleep v fail asleep ¬asleep fail Empty

25. Example 2 Beep Beep! • Roadrunner and Coyote

26. Example 2 • Coyote chases Roadrunner • If Roadrunner is smart, Coyote does not catch it • If coyote chases Roadrunner and does not catch it, then Coyote is annoyed. • Roadrunner is smart • Theorem: Coyote is annoyed ???? Beep Beep!

27. Example 2 • Theorem: Coyote is annoyed • We try to prove that “Coyote is NOT annoyed” is false • We add “Coyote is NOT annoyed” to the knowledge base, and prove false • So, the original theorem must be true Beep Beep!

28. Example 2 We are asking: Does the knowledge base entail “Annoyed”

29. Example 2 • Convert these into Conjunctive Normal Form: • S => ¬B • C Λ ¬B => A

30. Example 2

31. Example 2 • C = Coyote Chases Roadrunner • S = Roadrunner is Smart • B = Coyote Catches Roadrunner • A = Coyote is Annoyed

32. Example 2 ¬Annoyed Λ Annoyed

33. Example 2 – Proved in a different way Coyote catches Roadrunner AND Coyote does not catch Roadrunner

34. Example 2 – conclusion Annoyed Λ ¬Annoyed • This cannot be true, therefore adding “¬Annoyed” causes a contradiction in the knowledge base • Theorem was: “Coyote is annoyed” • We added the opposite and proved FALSE • We proved that Annoyed = TRUE, by proving that ¬Annoyed = FALSE • Therefore Coyote is annoyed • The knowledge base entails “annoyed”

35. Resolution Problems • Can take a very long time • Depending on the number of clauses in the knowledge base

36. Example 3 B V A ¬C V A ¬B V A C V ¬D ¬A V ¬B V D • Does this entail A? • In other words: Does all of this mean that A is TRUE? • If we set A to FALSE, and find a contradiction, then A must be TRUE

37. Example 3 The Knowledge Base B V A ¬C V A ¬B V A C V ¬D ¬A V ¬B V D ¬A A V A A Contradiction

38. Example 3 1) B V A 2) ¬C V A 3) ¬B V A 4) C V ¬D 5) ¬A V ¬B V D 6) ¬A 7) ¬C From 2,6 8) ¬D From 4,7 9) ¬A V ¬B From 5,8 10) B From 1,6 11) ¬A From 9,10 12) ¬B From 3,11 Contradiction

39. Example 3 - conclusion • So you can find the answer in 1 step or 6 steps, depending on the order in which you resolve the clauses • The speed of the resolution algorithm depends on the order • But, resolution will always find a proof if one exists. You just have to keep going until there are no more clauses to resolve

40. What you need to know • The resolution algorithm • Uses the principle of proof by contradiction • Why the knowledge base must be in Conjunctive Normal Form • The speed of the algorithm depends on the order in which you resolve the clauses