1. ECE457 Applied Artificial Intelligence Fall 2007 Lecture #6
Logical Agents
ECE457 Applied Artificial Intelligence
Fall 2007
Lecture #6

2. Outline Logical reasoning Propositional Logic Wumpus World Inference
Russell & Norvig, chapter 7
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2

3. Logical Reasoning Recall: Game-playing with imperfect information
Partially-observable environment
Need to infer about hidden information
Two new challenges
How to represent the information we have (knowledge representation)
How to use the information we have to infer new information and make decisions (knowledge reasoning)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 3

4. Knowledge Representation
Represent facts about the environment
Many ways: ontologies, mathematical functions, …
Statements that are either true or false
Language
To write the statements
Syntax: symbols (words) and rules to combine them (grammar)
Semantics: meaning of the statements
Expressiveness vs. efficiency
Knowledge base (KB)
Contains all the statements
Agent can TELL it new statements (update)
Agent can ASK it for information (query)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 4

5. Knowledge Representation
Example: Language of arithmetic
Syntax describes well-formed formulas (WFF)
X + Y > 7 (WFF)
X Y + (not a WFF)
Semantics describes meanings of formulas
“X + Y > 7” is true if and only if the value of X and the value of Y summed together is greater than 7
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 5

6. Knowledge Reasoning Inference Entailment
Discovering new facts and drawing conclusions based on existing information
During ASK or TELL
“All humans are mortal” “Socrates is human”
Entailment
A sentence  is inferred from sentences 
 is true given that the  are true
 entails 
  
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 6

7. Propositional Logic Sometimes called “Boolean Logic”
Sentences are true (T) or false (F)
Words of the syntax include propositional symbols…
P, Q, R, …
P = “I’m hungry”, Q = “I have money”, R = “I’m going to a restaurant”
… and logical connectives
¬ negation NOT
 conjunction AND
 disjunction OR
 implication IF-THEN
 biconditional IF AND ONLY IF
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 7

8. Propositional Logic Atomic sentences Complex sentences
Propositional symbols
True or false
Complex sentences
Groups of propositional symbols joined with connectives, and parenthesis if needed
(P  Q)  R
Well-formed formulas following grammar rules of the syntax
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 8

9. Propositional Logic Complex sentences evaluate to true or false
Using truth tables
Semantics P Q R
P  Q
(P  Q)  R T F
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 9

10. Propositional Logic Semantics
Truth tables for all connectives
Given each possible truth value of each propositional symbol, we can get the possible truth values of the expression P Q ¬P
P  Q
P  Q
P  Q
P  Q T F
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 10

11. Propositional Logic Example
Propositional symbols:
A = “The car has gas”
B = “I can go to the store”
C = “I have money”
D = “I can buy food”
E = “The sun is shining”
F = “I have an umbrella”
G = “I can go on a picnic”
If the car has gas, then I can go to the store
A  B
I can buy food if I can go to the store and I have money
(B  C)  D
If I can buy food and either the sun is not shining or I have an umbrella, I can go on a picnic
(D  (¬E  F))  G
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 11

12. D E F G ¬E ¬E  F D  (¬E  F) D  (¬E  F)  G T
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 12

13. Wumpus World 4 3 2 1 2D cave divided in rooms Gold Pits Wumpus
Glitters
Agent has to pick it up
Pits
Agent falls in and dies
Agent feels breeze near pit
Wumpus
Agent gets eaten and dies if Wumpus alive
Agent can kill Wumpus with arrow
Agent smells stench near Wumpus (alive or dead)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 13

14. Wumpus World 4 3 2 1 Initial state: Goal: Actions: Cost: (1,1)
Get the gold and get back to (1,1)
Actions:
Turn 90°, move forward, shoot arrow, pick up gold
Cost:
+1000 for getting gold, for dying, -1 per action, -10 for shooting the arrow
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15. Exploring the Wumpus World4 3 2 1
Wumpus?
Pit? OK OK
Wumpus? OK
Pit?
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16. Wumpus World Logic Propositional symbols Rules
Pi,j = “there is a pit at (i,j)”
Bi,j = “there is a breeze at (i,j)”
Si,j = “there is a stench at (i,j)”
Wi,j = “there is a Wumpus at (i,j)”
Ki,j = “(i,j) is ok”
Rules
Bi,j  (Pi+1,j  Pi-1,j  Pi,j+1  Pi,j-1)
Si,j  (Wi+1,j  Wi-1,j  Wi,j+1  Wi,j-1)
Ki,j  (¬Wi,j  ¬Pi,j)
Have to be written out for every (i,j)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 16

17. Wumpus World KB 4 3 2 1 K1,1 ¬B1,1 ¬S1,1 B1,1  (P2,1  P1,2)
S1,1  (W2,1  W1,2)
K2,1(¬W2,1¬P2,1)
K1,2(¬W1,2¬P1,2) 4 3 2 1
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 17

18. Wumpus World Inference
1. K1,1 3. ¬S1,1 5. ¬P2,1
2. ¬B1,1 4. ¬P1,2
1. K1,1 3. ¬S1,1
2. ¬B1,1
B1,1
P1,2
P2,1
¬B1,1
P1,2P2,1
B1,1  (P1,2P2,1) T F
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 18

19. Wumpus World Inference
1. K1,1 3. ¬S1,1 5. ¬P2,1 7. ¬W2,1
2. ¬B1,1 4. ¬P1,2 6. ¬W1,2
S1,1
W1,2
W2,1
¬S1,1
W1,2W2,1
S1,1  (W1,2W2,1) T F
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 19

20. Wumpus World Inference
1. K1,1 3. ¬S1,1 5. ¬P2,1 7. ¬W2,1 9. K2,1
2. ¬B1,1 4. ¬P1,2 6. ¬W1,2 8. K1,2
1. K1,1 3. ¬S1,1 5. ¬P2,1 7. ¬W2,1
2. ¬B1,1 4. ¬P1,2 6. ¬W1,2
P1,2
W1,2
K1,2
¬P1,2
¬W1,2
¬W1,2¬P1,2
K1,2  (¬W1,2¬P1,2) T F
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 20

21. Wumpus World KB 4 3 2 1 K1,1 ¬B1,1 ¬S1,1 ¬P1,2 ¬P2,1 ¬W1,2 ¬W2,1 K1,2
Pit? OK OK
Wumpus?
¬B1,2
¬P1,3
¬P2,2
S1,2
W1,3  W2,2 OK
Pit?
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 21

22. Inference with Truth Tables
Sound
Only infers true conclusions from true premises
Complete
Finds all facts entailed by KB
Time complexity = O(2n)
Checks all truth values of all symbols
Space complexity = O(n)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 22

23. Inference with Rules Speed up inference by using inference rules
Use along with logical equivalences
No need to enumerate and evaluate every truth value
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24. Rules and Equivalences
Inference rules
(α  β), α β
(α  β) α
α, β (αβ)
(α  β), ¬β α
(αβ), (¬βγ) (α  γ)
Logical equivalences
(α  β)  (β  α)
(α  β)  (β  α)
((α  β)  γ)  (α  (β  γ))
((α  β)  γ)  (α  (β  γ))
¬(¬α)  α
(α  β)  (¬β  ¬α)
(α  β)  (¬α  β)
(α  β)  ((α  β)  (β  α))
¬(α  β)  (¬α  ¬β)
¬(α  β)  (¬α  ¬β)
(α  (β  γ))  ((α  β)  (α  γ))
(α  (β  γ))  ((α  β)  (α  γ))
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 24

25. Wumpus World & Inference Rules
KB: ¬B1,1
B1,1  (P2,1  P1,2)
Biconditional elimination
(B1,1  (P2,1  P1,2))  ((P2,1  P1,2)  B1,1)
And elimination
(P2,1  P1,2)  B1,1
Contraposition
¬B1,1  ¬(P2,1  P1,2)
Modus Ponens
¬(P2,1  P1,2)
De Morgan’s Rule
¬P2,1  ¬P1,2
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 25

26. Resolution
Inference with rules is sound, but only complete if we have all the rules
Resolution rule is both sound and complete
(αβ), (¬βγ) (α  γ)
But it only works on disjunctions!
Conjunctive normal form (CNF)
Eliminate biconditionals: (αβ)  ((αβ)(βα))
Eliminate implications: (α  β)  (¬α  β)
Move/Eliminate negations: ¬(¬α)  α, ¬(α  β)  (¬α  ¬β), ¬(α  β)  (¬α  ¬β)
Distribute  over : (α  (βγ))  ((αβ)  (αγ))
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 26

27. CNF Example
B1,1  (P2,1  P1,2)
Eliminate biconditionals
(B1,1  (P2,1  P1,2))  ((P2,1  P1,2)  B1,1)
Eliminate implications
(¬B1,1  P2,1  P1,2)  (¬(P2,1  P1,2)  B1,1)
Move/Eliminate negations
(¬B1,1  P2,1  P1,2)  ((¬P2,1  ¬P1,2)  B1,1)
Distribute  over 
(¬B1,1  P2,1  P1,2)  (¬P2,1  B1,1)  (¬P1,2  B1,1)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 27

28. Resolution Algorithm Given a KB Need to answer a query α
Proof by contradiction
Show that (KB  ¬α) is unsatisfiable
i.e. leads to a contradiction
If (KB  ¬α), then (KB  α) must be true
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 28

29. Resolution Algorithm Convert (KB  ¬α) into CNF
For every pair of clauses that contain complementary symbols
Apply resolution to generate a new clause
Add new clause to KB
End when
Resolution gives the empty clause (KB  α)
No new clauses can be added (fail)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 29

30. Wumpus World & Resolution
(¬B1,1  P1,2  P2,1)  (¬P1,2  B1,1)  (¬P2,1  B1,1)
CNF form of B1,1  (P2,1  P1,2)
¬B1,1
Query: ¬P1,2
(¬B1,1  P1,2  P2,1)  (¬P2,1  B1,1)  (¬P1,2  B1,1)  ¬B1,1  P1,2
(¬B1,1  P1,2  P2,1)  (¬P2,1  B1,1)  ¬P1,  ¬B1,1  P1,2
(¬B1,1  P1,2  P2,1)  (¬P2,1  B1,1)  ¬B1,1  Empty clause!
KB  ¬P1,2
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 30

31. Resolution Algorithm Sound Complete Not efficient
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 31

32. Horn Clauses
Resolution algorithm can be further improved by using Horn clauses
Disjunction clause with at most one positive symbol
¬α  ¬β  γ
Can be rewritten as implication
(α  β)  γ
Inference in linear time!
Using Modus Ponens
Forward or backward chaining
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 32

33. Forward Chaining Data-driven reasoning
Start with known symbols
Infer new symbols and add to KB
Use new symbols to infer more new symbols
Repeat until query proven or no new symbols can be inferred
Work forward from known data, towards proving goal
KB: α, β, δ, ε
(α  β)  γ
(δ  ε)  λ
(λ  γ)  q
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 33

34. Backward Chaining Goal-driven reasoning
Start with query, try to infer it
If there are unknown symbols in the premise of the query, infer them first
If there are unknown symbols in the premise of these symbols, infer those first
Repeat until query proven or its premise cannot be inferred
Work backwards from goal, to prove needed information
KB: α, β, δ, ε
(λ  γ)  q
(δ  ε)  λ
(α  β)  γ
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 34

35. Forward vs. Backward Forward chaining Backward chaining
Proves everything
Goes to work as soon as new information is available
Expands the KB a lot
Improves understanding of the world
Typically used for proving a world model
Backward chaining
Proves only what is needed for the goal
Does nothing until a query is asked
Expands the KB as little as needed
More efficient
Typically used for proofs by contradiction
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 35

36. Assumptions Utility-based agent Environment
Fully observable / Partially observable (approximation)
Deterministic / Strategic / Stochastic
Sequential
Static / Semi-dynamic
Discrete / Continuous
Single agent / Multi-agent
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 36

37. Assumptions Updated Learning agent Environment
Fully observable / Partially observable
Deterministic / Strategic / Stochastic
Sequential
Static / Semi-dynamic
Discrete / Continuous
Single agent / Multi-agent
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 37

38. Exercise
If the unicorn is mythical, then it is immortal, but if it is not mythical then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned.
Is the unicorn
Magical?
Horned?
Mythical?
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 38

39. Exercise: CNF Propositional symbols
Mythical = “The unicorn is mythical”
Immortal = “The unicorn is immortal”
Mammal = “The unicorn is a mammal”
Horned = “The unicorn is horned”
Magical = “The unicorn is magical”
If the unicorn is mythical, then it is immortal
Mythical  Immortal
¬Mythical  Immortal
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 39

40. Exercise: CNF Propositional symbols
Mythical = “The unicorn is mythical”
Immortal = “The unicorn is immortal”
Mammal = “The unicorn is a mammal”
Horned = “The unicorn is horned”
Magical = “The unicorn is magical”
If it is not mythical then it is a mortal mammal
¬Mythical  (¬Immortal  Mammal)
Mythical  (¬Immortal  Mammal)
(Mythical  ¬Immortal)  (Mythical  Mammal)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 40

41. Exercise: CNF Propositional symbols
Mythical = “The unicorn is mythical”
Immortal = “The unicorn is immortal”
Mammal = “The unicorn is a mammal”
Horned = “The unicorn is horned”
Magical = “The unicorn is magical”
If the unicorn is either immortal or a mammal, then it is horned
(Immortal  Mammal)  Horned
¬(Immortal  Mammal)  Horned
(¬Immortal  ¬Mammal)  Horned
(¬Immortal  Horned)  (¬Mammal  Horned)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 41

42. Exercise: CNF Propositional symbols
Mythical = “The unicorn is mythical”
Immortal = “The unicorn is immortal”
Mammal = “The unicorn is a mammal”
Horned = “The unicorn is horned”
Magical = “The unicorn is magical”
The unicorn is magical if it is horned
Horned  Magical
¬Horned  Magical
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 42

43. Exercise: KB, Queries KB Negation of queries ¬Magical ¬Horned
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)
Negation of queries
¬Magical
¬Horned
¬Mythical
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 43

44. Exercise: Resolution, ¬Magical
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  ¬Magical
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  ¬Horned  ¬Magical
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  ¬Immortal  ¬Mammal  ¬Horned  ¬Magical
¬Mythical  (Mythical  ¬Immortal)  Mythical  ¬Immortal  ¬Mammal  ¬Horned  ¬Magical
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 44

45. Exercise: Resolution, ¬Horned
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  ¬Horned
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  ¬Immortal  ¬Mammal  (¬Horned  Magical)  ¬Horned
¬Mythical  (Mythical  ¬Immortal)  Mythical  ¬Immortal  ¬Mammal  (¬Horned  Magical)  ¬Horned
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 45

46. Exercise: Resolution, ¬Mythical
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  ¬Mythical
(¬Mythical  Immortal)  ¬Immortal  Mammal  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  ¬Mythical
¬Mythical  ¬Immortal  Mammal  (¬Immortal  Horned)  Horned  (¬Horned  Magical)  ¬Mythical
¬Mythical  ¬Immortal  Mammal  (¬Immortal  Horned)  Horned  Magical  ¬Mythical
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 46

47. Exercise: Resolution, Mythical
(¬Mythical  Immortal)  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  Mythical
Immortal  (Mythical  ¬Immortal)  (Mythical  Mammal)  (¬Immortal  Horned)  (¬Mammal  Horned)  (¬Horned  Magical)  Mythical
Immortal  Mythical  (Mythical  Mammal)  Horned  (¬Mammal  Horned)  (¬Horned  Magical)  Mythical
Immortal  Mythical  (Mythical  Mammal)  Horned  (¬Mammal  Horned)  Magical  Mythical
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 47

48. Exercise: Note Previous two examples Therefore
(KB  ¬Mythical)  (Horned  Magical)
(KB  Mythical)  (Horned  Magical)
Therefore
KB  (Horned  Magical)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 48