logical agents
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Logical Agents. ECE457 Applied Artificial Intelligence Fall 2007 Lecture #6. Outline. Logical reasoning Propositional Logic Wumpus World Inference Russell & Norvig, chapter 7. Logical Reasoning. Recall: Game-playing with imperfect information Partially-observable environment - PowerPoint PPT PresentationTRANSCRIPT
Logical Agents
ECE457 Applied Artificial IntelligenceFall 2007
Lecture #6
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2
Outline Logical reasoning Propositional Logic Wumpus World Inference
Russell & Norvig, chapter 7
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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)
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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)
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Knowledge Representation Example: Language of arithmetic Syntax describes well-formed
formulas (WFF) X + Y > 7 (WFF) X 7 @ 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
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Knowledge Reasoning Inference
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
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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
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Propositional Logic Atomic 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
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Propositional Logic Complex
sentences evaluate to true or false
Using truth tables Semantics
P Q R P Q (P Q) R
T T T T T
F T T F T
T F T F T
F F T F T
T T F T F
F T F F T
T F F F T
F F F F T
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Propositional Logic Semantics
P Q ¬P
P Q P Q P Q P Q
T T F T T T T
F T T F T T F
T F F F T F F
F F T F F T T
Truth tables for all connectives Given each possible truth value of each
propositional symbol, we can get the possible truth values of the expression
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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
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D E F G ¬E
¬E F D (¬E F) D (¬E F) G
T T T T F T T T
F T T T F T F T
T F T T T T T T
F F T T T T F T
T T F T F F F T
F T F T F F F T
T F F T T T T T
F F F T T T F T
T T T F F T T F
F T T F F T F T
T F T F T T T F
F F T F T T F T
T T F F F F F T
F T F F F F F T
T F F F T T T F
F F F F T T F T
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Wumpus World 2D cave divided
in rooms Gold
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)
4321
1
2
3
4
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Wumpus World Initial state:
(1,1) Goal:
Get the gold and get back to (1,1)
Actions: Turn 90°,
move forward, shoot arrow, pick up gold
Cost: +1000 for getting gold, -1000 for dying,
-1 per action, -10 for shooting the arrow
4321
1
2
3
4
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Exploring the Wumpus World
4321
1
2
3
4
OK
OKPit?
Pit?
Wumpus?
Wumpus?OK
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Wumpus World Logic Propositional symbols
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)
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Wumpus World KB
4321
1
2
3
41. K1,1
2. ¬B1,1
3. ¬S1,1
a. B1,1 (P2,1 P1,2)
b. S1,1 (W2,1 W1,2)
c. K2,1(¬W2,1¬P2,1)
d. K1,2(¬W1,2¬P1,2)
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Wumpus World Inference
B1,
1
P1,2 P2,1 ¬B1,
1
P1,2P2,1 B1,1 (P1,2P2,1)
T T T F T T
T F T F T T
T T F F T T
T F F F F F
F T T T T F
F F T T T F
F T F T T F
F F F T F T
1. K1,1 3. ¬S1,1
2. ¬B1,1
1. K1,1 3. ¬S1,1 5. ¬P2,1
2. ¬B1,1 4. ¬P1,2
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1. K1,1 3. ¬S1,1 5. ¬P2,1 7. ¬W2,1
2. ¬B1,1 4. ¬P1,2 6. ¬W1,2
Wumpus World Inference
S1,
1
W1,
2
W2,
1
¬S1,
1
W1,2W2,1 S1,1 (W1,2W2,1)
T T T F T T
T F T F T T
T T F F T T
T F F F F F
F T T T T F
F F T T T F
F T F T T F
F F F T F T
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1. K1,1 3. ¬S1,1 5. ¬P2,1 7. ¬W2,1
2. ¬B1,1 4. ¬P1,2 6. ¬W1,2
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
Wumpus World Inference
P1,
2
W1,
2
K1,
2
¬P1,
2
¬W1,
2
¬W1,2¬P1,
2
K1,2 (¬W1,2¬P1,2)
T T T F F F F
F T T T F F F
T F T F T F F
F F T T T T T
T T F F F F T
F T F T F F T
T F F F T F T
F F F T T T F
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10.B2,1
11.P3,1
12.¬S2,1
13.¬W2,2
14.¬W3,1
15.¬B1,2
16.¬P1,3
17.¬P2,2
18.S1,2
19.W1,3
20.K2,2
Wumpus World KB1. K1,1
2. ¬B1,1
3. ¬S1,1
4. ¬P1,2
5. ¬P2,1
6. ¬W1,2
7. ¬W2,1
8. K1,2
9. K2,1
4321
1
2
3
4
OK
OKPit?
Pit?
Wumpus?
Wumpus?OK
10.B2,1
11.P2,2 P3,1
12.¬S2,1
13.¬W2,2
14.¬W3,115.¬B1,2
16.¬P1,3
17.¬P2,2
18.S1,2
19.W1,3 W2,2
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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)
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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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Rules and Equivalences 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))
Inference rules (α β), α
β (α β)
α α, β
(αβ) (α β), ¬β
α (αβ), (¬βγ)
(α γ))
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Wumpus World & Inference Rules
KB: ¬B1,1
1. B1,1 (P2,1 P1,2) Biconditional elimination
2. (B1,1 (P2,1 P1,2)) ((P2,1 P1,2) B1,1) And elimination
3. (P2,1 P1,2) B1,1 Contraposition
4. ¬B1,1 ¬(P2,1 P1,2) Modus Ponens
5. ¬(P2,1 P1,2) De Morgan’s Rule
¬P2,1 ¬P1,2
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Resolution Inference with rules is sound, but only
complete if we have all the rules Resolution rule is both sound and
complete (αβ), (¬βγ)
(α γ)) But it only works on disjunctions!
Conjunctive normal form (CNF) Eliminate biconditionals:
(αβ) ((αβ)(βα)) Eliminate implications: (α β) (¬α β) Move/Eliminate negations: ¬(¬α) α,
¬(α β) (¬α ¬β), ¬(α β) (¬α ¬β) Distribute over : (α (βγ)) ((αβ) (αγ))
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CNF Example1. B1,1 (P2,1 P1,2)
Eliminate biconditionals
2. (B1,1 (P2,1 P1,2)) ((P2,1 P1,2) B1,1) Eliminate implications
3. (¬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)1. Distribute over
1. (¬B1,1 P2,1 P1,2) (¬P2,1 B1,1) (¬P1,2 B1,1)
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Resolution Algorithm Given a KB Need to answer a query α
KB 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
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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)
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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) ¬P1,2 ¬B1,1 P1,2
(¬B1,1 P1,2 P2,1) (¬P2,1 B1,1) ¬B1,1 Empty clause!
KB ¬P1,2
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Resolution Algorithm Sound Complete Not efficient
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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
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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 goal1. KB: α, β, δ, ε2. (α β) γ3. (δ ε) λ4. (λ γ) q
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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 information1. KB: α, β, δ, ε2. (λ γ) q3. (δ ε) λ4. (α β) γ
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Forward vs. Backward Forward 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
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Assumptions Utility-based agent Environment
Fully observable / Partially observable (approximation)
Deterministic / Strategic / Stochastic Sequential Static / Semi-dynamic Discrete / Continuous Single agent / Multi-agent
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Assumptions Updated Learning agent Environment
Fully observable / Partially observable Deterministic / Strategic / Stochastic Sequential Static / Semi-dynamic Discrete / Continuous Single agent / Multi-agent
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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?
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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 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 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 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
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Exercise: KB, Queries KB
(¬Mythical Immortal) (Mythical ¬Immortal) (Mythical Mammal) (¬Immortal Horned) (¬Mammal Horned) (¬Horned Magical)
Negation of queries ¬Magical ¬Horned ¬Mythical
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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) ¬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
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Exercise: Resolution, ¬Horned (¬Mythical Immortal) (Mythical ¬Immortal)
(Mythical Mammal) (¬Immortal Horned) (¬Mammal Horned) (¬Horned Magical) ¬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
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Exercise: Resolution, ¬Mythical (¬Mythical Immortal) (Mythical ¬Immortal)
(Mythical Mammal) (¬Immortal Horned) (¬Mammal Horned) (¬Horned Magical) ¬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
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Exercise: Resolution, Mythical (¬Mythical Immortal) (Mythical ¬Immortal)
(Mythical Mammal) (¬Immortal Horned) (¬Mammal Horned) (¬Horned Magical) 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 48
Exercise: Note Previous two examples
(KB ¬Mythical) (Horned Magical) (KB Mythical) (Horned Magical)
Therefore KB (Horned Magical)