2. LECTURE 06 Logical Agent Course : T0264 – Artificial Intelligence
10/30/2019
Course : T0264 – Artificial Intelligence
Year : 2013
LECTURE 06 Logical Agent

3. T0264 - Artificial Intelligence
Learning Outcomes
At the end of this session, students will be able to:
LO 2 : Explain various intelligent search algorithms to solve the problems
LO 3 : Explain how to use knowledge representation in reasoning purpose
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4. T0264 - Artificial Intelligence
Outline
Knowledge-Based Agents
Logic in General
Propositional Logic (PL) : A Very Simple Logic
Proportional Theorem Proving (Resolution Algorithm)
Forward and Backward Algorithm
Summary
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5. T0264 - Artificial Intelligence
Knowledge-Based
Agents
Knowledge base = set of sentences in a formal language
Declarative approach to building an agent (or other system):
Tell it what it needs to know
Then it can Ask itself what to do - answers should follow from the KB
Agents can be viewed at the knowledge level
i.e., what they know, regardless of how implemented
Or at the implementation level
i.e., data structures in KB and algorithms that manipulate them
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6. Tugas Presentasi I Logical Agents & Wumpuss World
ANDREAN HARRY RAMADHAN
DIONDY KUSUMA
RIVKY CHANDRA
WIENA MARCELINA
WILSON GUNAWAN
ANDRE IVAN
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7. A Simple Knowledge-Based Agent
Agents
A Simple Knowledge-Based Agent
The agent must be able to:
Represent states, actions, etc.
Incorporate new percepts
Update internal representations of the world
Deduce hidden properties of the world
Deduce appropriate actions
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8. Wumpus World PEAS Description
Knowledge-Based
Agents
Wumpus World PEAS Description
Performance measure
gold +1000, death -1000
-1 per step, -10 for using the arrow
Environment
Squares adjacent to wumpus are smelly
Squares adjacent to pit are breezy
Glitter if gold is in the same square
Shooting kills wumpus if you are facing it
Shooting uses up the only arrow
Grabbing picks up gold if in same square
Releasing drops the gold in same square
Sensors: Stench, Breeze, Glitter, Bump, Scream
Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot
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9. Wumpus World Characterization
Knowledge-Based
Agents
Wumpus World Characterization
Fully Observable No – only local perception
Deterministic Yes – outcomes exactly specified
Episodic No – sequential at the level of actions
Static Yes – Wumpus and Pits do not move
Discrete Yes
Single-agent? Yes – Wumpus is essentially a natural feature
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10. Exploring a Wumpus World
Knowledge-Based
Agents
Exploring a Wumpus World
A = Agent
B = Breeze
G = Glitter, Gold
OK = Safe square
P = Pit
S = Stench
V = Visited
W = Wumpus
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11. Exploring a Wumpus World
Knowledge-Based
Agents
Exploring a Wumpus World
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Logic In General
Logics are formal languages for representing information such that conclusions can be drawn
Syntax defines the sentences in the language
Semantics define the "meaning" of sentences;
i.e., define truth of a sentence in a world
E.g., the language of arithmetic
x+2 ≥ y is a sentence; x2+y > { } is not a sentence
x+2 ≥ y is true if the number x+2 is no less than the number y
x+2 ≥ y is true in a world where x = 7, y = 1
x+2 ≥ y is false in a world where x = 0, y = 6
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Logic In General
Entailment
Entailment means that one thing follows from another:
KB ╞ α
Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true
E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won”
E.g., x+y = 4 entails 4 = x+y
Entailment is a relationship between sentences (i.e., syntax) that is based on semantics
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14. T0264 - Artificial Intelligence
Logic In General
Models
Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated
We say m is a model of a sentence α if α is true in m
M(α) is the set of all models of α
Then
KB ╞ α iff M(KB)  M(α)
E.g.
KB = Giants won and Reds won
α = Giants won
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15. T0264 - Artificial Intelligence
Logic In General
Wumpus Models
KB = wumpus-world rules + observations
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Logic In General
Wumpus Models
KB = wumpus-world rules + observations
α1 = "[1,2] is safe",
KB ╞ α1, proved by model checking
KB = wumpus-world rules + observations
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Logic In General
Wumpus Models
KB = wumpus-world rules + observations
α2 = "[2,2] is safe", KB ╞ α2
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18. T0264 - Artificial Intelligence
Logic In General
Inference
KB ├i α = sentence α can be derived from KB by procedure i
Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α
Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α
Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure.
That is, the procedure will answer any question whose answer follows from what is known by the KB.
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19. T0264 - Artificial Intelligence
Logic In General
Sentences are physical configuration of agent, and reasoning is a process of constructing new physical configurations from old ones.
New configurations represent aspects of the world that actually follow from the aspect that the old configurations represent
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20. T0264 - Artificial Intelligence
Propositional Logic:
A Very Simple Logic
Standard Logic Symbols
 = For all ; [e.g : every one, every body, any time, etc]
 = There exists ; [e.g : some one, some time, etc]
 = Implication ; [ if … then ….]
 = Equivalent ; biconditional [if … and … only … if …]
 = Not ; negation
 = OR ; disjunction
 = AND ; conjunction
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21. T0264 - Artificial Intelligence
Propositional Logic:
A Very Simple Logic
Syntax
Propositional logic is the simplest logic – illustrates basic ideas
The proposition symbols S1, S2 etc are sentences
If S is a sentence, S is a sentence (negation)
If S1 and S2 are sentences, S1  S2 is a sentence (conjunction)
If S1 and S2 are sentences, S1  S2 is a sentence (disjunction)
If S1 and S2 are sentences, S1  S2 is a sentence (implication)
If S1 and S2 are sentences, S1  S2 is a sentence (biconditional)
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22. Propositional Logic: A Very Simple Logic
Semantics
True is true in every model and False is false in every model
The truth value of every other proposition symbol must be specified directly in the model.
For complex sentences, there are five rules for any sub sentences P and Q in any model m :
 P is true iff P is false in m.
 P  Q is true iff both P and Q true in m.
 P  Q is true iff either P or Q is true in m.
 P  Q is true unles P is true and Q is false in m.
i.e., is false iff P is true and Q is false
 P  Q is true iff P and Q are both true or both false in m. i.e., P Q is true and Q  P is true
Simple recursive process evaluates an arbitrary sentence, e.g.,
P1,2  (P2,2  P3,1) = true  (true  false) = true  true = true
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23. Truth Tables for Connectives
Propositional Logic:
A Very Simple Logic
Truth Tables for Connectives
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24. A Truth-Table Enumeration Algorithm
Propositional Logic:
A Very Simple Logic
A Truth-Table Enumeration Algorithm
Depth-first enumeration of all models is sound and complete
For n symbols, time complexity is O(2n), space complexity is O(n)
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25. T0264 - Artificial Intelligence
Propositional Logic:
A Very Simple Logic
Logical Equivalences
Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α
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26. Propositional Theorem Proving (Resolution Algorithm)
Inference procedure based on resolution work by using the principle of proof by contradiction.
That is, to show that KB├ = , we show that (KB   ) is unsatisfied. We do this by contradiction.
Therefore:
First : (KB   ) is convert into CNF (normal clause form)
Second : show clauses obtained by resolving pairs in the row
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27. Propositional Theorem Proving (Resolution Algorithm)
Convert all the proposition logic to normal clause form (CNF).
Negate P and convert the result to clause form. Add it to the set of clause in step
Repeat until either a contradiction is found or no progress can be made, or a predetermined amount of effort has been expended.
Select two clause. Call these the parent clauses.
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28. Propositional Theorem Proving (Resolution Algorithm)
Algorithm: (cont’d)
Resolve them together. The resolvent will be the disjunction of all of the literals of both parent clauses with appropriate substitutions performed and with the following exception:
If there is one pairs of literals T1 and T2 such that one of the parent clauses contains T1 and the other contains  T2 and if T1 and T2 are unifiable, then neither T1 nor  T2 should appear in the resolvent. If there is more than one pair of complementary literals, only one pair should be omitted from the resolvent.
If the resolvent is the empty clause, then a contradiction has been found. If it is not, then add it to the set of clause available to the procedure.
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29. Simple Pseudo Code Resolution algorithm for proportional logic
Propositional Theorem Proving
(Resolution Algorithm)
Simple Pseudo Code Resolution algorithm for proportional logic
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30. Conversion to Normal Clause Form (CNF Algorithm)
Propositional Theorem Proving
(Resolution Algorithm)
Conversion to Normal Clause Form (CNF Algorithm)
Eliminate , using: a  b = a  b.
Reduce the scope of each  to a single term, using de Morgan’s laws: (p) = p
(ab) = a  b
(ab) = a  b
x : P(x) =x : P(x)
x : P(x) = x : P(x)
3. Standardize variables
For sentences like (∀x P(x)) ∨ (∃x Q(x)) which use the same variable name twice, change the name of one of the variables. This avoids confusion later when we drop the quantifiers. For example, from ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃y Loves(y, x)]. we obtain: ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃z Loves(z, x)].
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31. Convert to Normal Clause Form (cont’d)
Propositional Theorem Proving
(Resolution Algorithm)
Convert to Normal Clause Form (cont’d)
Move all quantifiers to the left of the formula without changing their relative order.
Eliminate existential quantifiers by inserting Skolem functions.
∃x P(x) into P(A), where A is a new constant
Drop universal quantifiers
Convert the matrix into a conjunction of disjoints, using associativity and distributivity (distribute ORs over ANDs)
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32. Example : Conversion to Normal Clause Form
Propositional Theorem Proving
(Resolution Algorithm)
Example : Conversion to Normal Clause Form
Sentence:
“Every body who know Hitler, either like Hitler or think that anyone who killed some one is crazy”
Proportional Logic is :
x : [body(x)  know(x, Hitler)] [like(x, Hitler) (y:z: killed(y, z)  crazy(x, y)]
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33. Applied of Clause Form Algorithm
Propositional Theorem Proving
(Resolution Algorithm)
Applied of Clause Form Algorithm
x : [body(x)  know(x, Hitler)]
[like(x, Hitler) (y:z: killed(y, z)  crazy(x, y)]
x : [body(x)  know(x, Hitler)]  [like(x, Hitler) (y: (z: killed(y, z))  crazy(x, y))]
Ok !
x : y : z : [body(x)  know(x, Hitler)]  [like(x, Hitler) (killed(y, z))  crazy(x, y)]
[body(x)  know(x, Hitler)]  [like(x, Hitler) (killed(y, z))  crazy(x, y)]
body(x)  know(x, Hitler)  like(x, Hitler) killed(y, z)  crazy(x, y)
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34. Conversion to Normal Form
Propositional Theorem Proving
(Resolution Algorithm)
Conversion to Normal Form
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35. Propositional Theorem Proving (Resolution Algorithm)
Conversion to Normal Form
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36. Forward and Backward Chaining
Algorithm
Forward and Backward Chaining
Idea:
fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found
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37. Forward Chaining Algorithm
Forward and Backward
Algorithm
Forward Chaining Algorithm
Forward chaining is sound and complete for Horn KB
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38. Forward Chaining Example
Forward and Backward
Algorithm
Forward Chaining Example
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39. Forward Chaining Example
Forward and Backward
Algorithm
Forward Chaining Example
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40. Forward Chaining Example
Forward and Backward
Algorithm
Forward Chaining Example
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41. T0264 - Artificial Intelligence
Forward and Backward
Algorithm
Proof of Completeness
FC derives every atomic sentence that is entailed by KB
FC reaches a fixed point where no new atomic sentences are derived
Consider the final state as a model m, assigning true/false to symbols
Every clause in the original KB is true in m
a1  …  ak  b
Hence m is a model of KB
If KB╞ q, q is true in every model of KB, including m
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Forward and Backward
Algorithm
Backward Chaining
Idea: work backwards from the query q:
to prove q by BC,
check if q is known already, or
prove by BC all premises of some rule concluding q
Avoid loops: check if new sub goal is already on the goal stack
Avoid repeated work: check if new sub goal
has already been proved true, or
has already failed
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43. Backward Chaining Example
Forward and Backward
Algorithm
Backward Chaining Example
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44. Backward Chaining Example
Forward and Backward
Algorithm
Backward Chaining Example
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Forward and Backward
Algorithm
Backward Chaining Example
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46. T0264 - Artificial Intelligence
Forward and Backward
Algorithm
Backward Chaining Example
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47. T0264 - Artificial Intelligence
Summary
Logical agents apply inference to a knowledge base to derive new information and make decisions
Basic concepts of logic:
syntax: formal structure of sentences
semantics: truth of sentences in the models
entailment: necessary truth of one sentence given another
inference: deriving sentences from other sentences
soundness: derivations produce only entailed sentences
completeness: derivations can produce all entailed sentences
Resolution is complete for propositional logic Forward, backward chaining are linear-time, complete for Horn clauses
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48. T0264 - Artificial Intelligence
References
Stuart Russell, Peter Norvig, Artificial intelligence : a modern approach. PE. New Jersey. ISBN: , Chapter 7
Elaine Rich, Kevin Knight, Shivashankar B. Nair Artificial Intelligence. MHE. New York. , Chapter 5
Propositional Logic:
The Resolution Method:
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49. << CLOSING >>
End of Session 06
Good Luck
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