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
( Awake Λ Listening )
NOT: “¬”
¬Awake, ¬A, ¬B, ¬Student
IMPLIES: “=>”
P => Q - True unless P is true and Q is false
( Awake Λ Listening ) => ¬Fail

5. Propositional Logic Literals 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) Λ Student 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 must be true

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

14. Resolution Algorithm CONTRADICTION Small example ¬sunny = FALSE
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 complementary literals 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 Λ over ν
(a Λ (b ν c )) ≡ ((a Λ b) ν (a Λ c))
(f) Distributivity of ν over 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. This means, put a “¬” symbol in front of it
Very Important:
NEGATE THE THEOREM
This means, put a “¬” symbol in front of it
If it already has one, then remove it

23. A ¬A ¬B B Fail ¬Fail Very Important: Theorem Negated Theorem
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 Sentence Expression Coyote chases Roadrunner Chase
If Roadrunner is smart, Coyote does not catch it
Smart => ¬Catch
If coyote chases Roadrunner and does not catch it, then Coyote is Annoyed
Chase Λ ¬Catch => Annoyed
Roadrunner is smart
Smart
Coyote is not annoyed
¬Annoyed
We are asking: Does the knowledge base entail “Annoyed”

29. Example 2 Convert these into Conjunctive Normal Form: S => ¬B
Expression
Simplified
Chase C
Smart => ¬Catch
S => ¬B
Chase Λ ¬Catch => Annoyed
C Λ ¬B => A
Smart S
¬Annoyed ¬A
Convert these into Conjunctive Normal Form:
S => ¬B
C Λ ¬B => A

30. S => ¬B ¬S V ¬B C Λ ¬B => A ¬C V B V A Example 2 Expression CNF

31. Example 2 C = Coyote Chases Roadrunner S = Roadrunner is Smart
Number
Expression 1 C 2
¬S V ¬B 3
¬C V B V A 4 S 5 ¬A
C = Coyote Chases Roadrunner
S = Roadrunner is Smart
B = Coyote Catches Roadrunner
A = Coyote is Annoyed

32. Example 2 ¬Annoyed Λ Annoyed Number Expression 1 C 6
B V A from 1 and 3 2
¬S V ¬B 7
¬B from 2 and 4 3
¬C V B V A 8
A from 6 and 7 4 S 9
False from 5 and 8 5 ¬A
¬Annoyed Λ Annoyed

33. Example 2 – Proved in a different way
Number
Expression 1 C 6
¬S V ¬C V A from 2 and 3 2
¬S V ¬B 7
¬C V A from 4 and 6 3
¬C V B V A 8
¬C from 5 and 7 4 S 9
False from 1 and 8 5 ¬A
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. Can take a very long time
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 7) ¬C From 2,6 3) ¬B V A 8) ¬D From 4,7
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