Resolution Theorem
Proving
G51IAI Introduction to Artificial
Intelligence
Dr Matthew Hyde
Outline
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Propositional Logic Recap
Conjunctive normal form
The Resolution Algorithm
Strategies to Guide the Search
Conclusion
Propositional Logic
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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...
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
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
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)
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
Inference in Propositional Logic
• We can infer new facts from what we already
know
• “Modus Ponens” Rule
We know these two
A => B
A
We can therefore infer
that this is true
B
• Also works for longer sentences
A Λ B Λ C => D
AΛBΛC
D
Example of Inference
CompassNorth => FacingNorth
Snowing => Cold
CompassNorth
Awake
Cold
We can infer that FacingNorth = ????
Example of Inference
CompassNorth => FacingNorth
Snowing => Cold
CompassNorth
Awake
Cold
FacingNorth
We can infer that FacingNorth = True
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
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
Resolution Algorithm
Small example
Is it sunny?
sunny = TRUE?
Knowledge base:
sunny
daytime
sunny V night
Prove sunny
Resolution Algorithm
Small example
Is it sunny?
sunny = TRUE?
Prove sunny
Knowledge base:
Negate it
sunny
Add it to the knowledge base
daytime
CONTRADICTION
sunny V night
¬sunny
¬sunny = FALSE
Therefore: sunny = TRUE
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
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
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
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))
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
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
Resolution Example
¬asleep v fail
asleep
Show that the
knowledge base
entails “fail”
• Negate the theorem
¬asleep v fail
asleep
Very Important:
•NEGATE THE THEOREM
•This means, put a “¬”
symbol in front of it
•If it already has one,
then remove it
Very Important:
Theorem
A
¬B
Fail
Negated Theorem
¬A
B
¬Fail
Then add it to the
knowledge base
and find a
contradiction
Resolution Example
¬asleep v fail
asleep
¬fail
Show that the
knowledge base
entails “fail”
• Negate the theorem
¬asleep v fail
¬asleep
asleep
fail
Empty
Example 2
• Roadrunner and
Coyote
Beep
Beep!
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!
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!
Example 2
Sentence
Expression
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
Coyote is not annoyed
Chase
Smart => ¬Catch
Chase Λ ¬Catch => Annoyed
Smart
¬Annoyed
We are asking: Does the knowledge base entail “Annoyed”
Example 2
Expression
Simplified
Chase
Smart => ¬Catch
Chase Λ ¬Catch => Annoyed
Smart
C
S => ¬B
C Λ ¬B => A
S
¬Annoyed
¬A
• Convert these into Conjunctive Normal Form:
• S => ¬B
• C Λ ¬B => A
Example 2
Expression
CNF
C
C
S => ¬B
C Λ ¬B => A
¬S V ¬B
¬C V B V A
S
S
¬A
¬A
Example 2
Number
Expression
1
C
2
¬S V ¬B
3
¬C V B V A
4
S
5
¬A
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C = Coyote Chases Roadrunner
S = Roadrunner is Smart
B = Coyote Catches Roadrunner
A = Coyote is Annoyed
Example 2
Number
Expression
Number
Expression
1
C
6
BVA
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
Example 2 – Proved in a different way
Number
Expression
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
Example 2 – conclusion
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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”
Resolution Problems
• Can take a very long time
• Depending on the number
of clauses in the knowledge
base
Example 3
BVA
¬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
Example 3
The Knowledge Base
BVA
AVA
¬C V A
¬B V A
C V ¬D
¬A V ¬B V D
¬A
A
Contradiction
Example 3
1) B V A
7) ¬C
2) ¬C V A
8) ¬D
3) ¬B V A
9) ¬A V ¬B
4) C V ¬D
10) B
5) ¬A V ¬B V D
11) ¬A
6) ¬A
12) ¬B
Contradiction
From 2,6
From 4,7
From 5,8
From 1,6
From 9,10
From 3,11
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
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