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Problem Solving
Representational Methods
Formal Methods
• Propositional Logic
• Predicate Logic
CS 331 Dr M M Awais
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Problem Solving
Propositional Logic
•
•
•
•
•
Represent FACTS only
Is declarative in nature
Can infer the truth value of fact only
Context free
Compositional: Can combine facts
CS 331 Dr M M Awais
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Problem Solving
Propositional Logic
• Symbols P,Q,R,S ……..
• Truth Symbols True (T), False (F)
• Connectives
^ (and),
v (or),
(Implication),
 (Equality, equivalence)
 (not)
Statements (Propositions) could be true/false
FACTS are also called Atomic Proposition
CS 331 Dr M M Awais
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Problem Solving
Are same
Truth Table
P
P
Q
Q
PvQ
P
T
T
F
F
T
T
T
F
F
T
F
F
F
T
T
F
T
T
F
F
T
T
T
T
CS 331 Dr M M Awais
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Problem Solving
Possible Sentences
(Well Formed Formulas-WFF)
• P^Q
• PvQ
• P Q
Conjuncts
Disjuncts
P=Premise/Antecedent
Q=conclusion/Consequent
• P
• (P v Q) = ( P Q) Inter conversion
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Problem Solving
Propositional logic: Syntax
• Propositional logic is the simplest logic – illustrates basic
ideas
• The proposition symbols P1, P2 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)
CS 331 Dr M M Awais
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Problem Solving
Propositional logic: Semantics
Each model specifies true/false for each proposition symbol
E.g. P1,2
false
P2,2
true
P3,1
false
With these symbols, 8 possible models, can be enumerated automatically.
Rules for evaluating truth with respect to a model m:
S
S1  S2
S1  S2
S1  S2
i.e.,
S1  S2
is true iff
S is false
is true iff
S1 is true
and S2 is true
is true iff
S1is true
or
S2 is true
is true iff
S1 is false or
S2 is true
is false iffS1 is true and
S2 is false
is true iff S1S2 is true and S2S1 is true
CS 331
Dr M M Awais
Simple recursive process evaluates
an arbitrary
sentence, e.g.,
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Problem Solving
Laws of Propositional
Expressions
• Demorgan’s Law (P v Q)=( P ^  Q)
• Distributive Law
P v (Q ^ R) = (P v Q) ^ (P v R)
• Commutative Law P ^ Q= Q ^ P
• Associative Law (P ^ Q) ^ R=P ^ (Q ^ R)
• Contrapositive Law P Q=  Q  P
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Problem Solving
Inferencing
• If simple facts are known to be true one can
find the truth value for the expressions
• Thus INTERPRETATIONS can be done.
• Interpretation is the assignment of truth
values to the sentences
Symbols
T/F
Mapping
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Problem Solving
Expressions for KR
•
•
•
•
•
•
•
Fact 1: Ali likes cars
P
Fact 2: Ali drives cars
Q
P v Q : Ali likes cars or drives cars
P ^ Q : Ali likes cars and drives cars
 Q : Ali does not like cars
P Q: If Ali likes cars then he drives cars
P  Q: ?????
– Above and vice versa
CS 331 Dr M M Awais
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Problem Solving
Implicative Statements
• P
Q: If Ali likes cars then he drives cars
• If P is FALSE and Q is TRUE, it means:
• ALI DOESNOT LIKE CARS BUT STILL HE DRIVES THEM
• Truth Table:
P
Q
T
T
F
F
T
F
F
T
P
T
T
F
T
Q
(of our interest)
(bizarre)
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Problem Solving
Model
• Model:If a sentence “S” is satisfied under a
given interpretation “I” for all possible
variables bindings (values) then I is a model
of S
• Satisfy: If S maps to T (true) under an
interpretation I then I satisfy S
(interpretation that makes the sentence true)
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Problem Solving
Definitions
• Logically Follows: X logically follows
from a set of predicate calculus expressions
S if every interpretation that satisfy S also
satisfy X (X F S)
• Satisfiable: S is satisfiable iff there exists
an interpretation and variable assignments
that satisfy it
CS 331 Dr M M Awais
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Problem Solving
SFX
S: All birds fly
S: Sparrow is bird
X: Sparrow flies
All humans are mortal
Shahid is a human
Shahid is mortal
All students at LUMS are hardworking
Farooq is a student at LUMS
Farooq is a hard working student
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Problem Solving
Definitions
• Inconsistent: If a set of expressions are not satisfiable
• Valid: If any expression has a value T for all
possible interpertations
• Sound: If an expression logically follows from
another expression then the inferential rule is
sound
• Complete: When inferential rule produces every
expression that logically follows a particular
expression
CS 331 Dr M M Awais
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Problem Solving
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
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Problem Solving
Validity and satisfiability
VALID
A sentence is valid if it is true in all models,
e.g., True, A A, A  A, (A  (A  B))  B
SATISFIABLE
A sentence is satisfiable if it is true in some model
e.g., A B
UNSATISFIABLE
A sentence is unsatisfiable if it is true in no models
e.g., A A
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Problem Solving
Example: Wumpus World
• 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 iff gold is in the same square
Shooting kills wumpus if you are facing it
Shooting uses up the only arrow
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Problem Solving
Wumpus world characterization
•
•
•
•
•
•
•
•
•
Fully Observable
Deterministic
Episodic
Static
Discrete
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Problem Solving
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
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
CS 331 Dr M M Awais
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Problem Solving
Exploring a wumpus world
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Problem Solving
Finding wumpus world
4
If Wumpus is in [1,3],
3
W
2
How can we prove it using
Propositional logic
1
1
CS 331 Dr M M Awais
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Problem Solving
Finding wumpus world
4
Knowledge base Contains:
~S1,1
~B1,1
~S2,1
B2,1
S1,2
B1,2
3
W
2
1
1
CS 331 Dr M M Awais
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3
4
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Problem Solving
Finding wumpus world
4
Rules:
R1:
~ S1,1  ~W1,1 ^ ~W1,2 ^ ~W2,1
R2:
R3:
3
W
2
s
~ S2,1  ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1 1
S1,2  W1,3 V W1,2 V W2,2 V ~W1,1
1
2
3
4
Assumption: The square with wumpus is also smelly
CS 331 Dr M M Awais
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Problem Solving
Inference rules in PL
   ,

• Modus Ponens
• And-elimination: from a conjuction any
conjunction can be inferred:
 


• Resolution:
Unit resolution when
l3 is empty

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Problem Solving
4
Proof of W presence
Knowledge base Contains:
~ S1,1
~B1,1
~S2,1
B2,1
S1,2
B1,2
Rules:
R1:
~ S1,1  ~W1,1 ^ ~W1,2 ^ ~W2,1
R2:
~ S2,1  ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1
R3:
S1,2  W1,3 V W1,2 V W2,2 V ~W1,1
Modus Ponen on
W
2
1
1
2
3
4
Results:
~W1,1
~W1,2
~W2,1
R1: ~ S1,1  ~W1,1 ^ ~W1,2 ^ ~W2,1
~W1,1 ^ ~W1,2 ^ ~W2,1
And Elimination:
~W1,1
~W1,2
~W2,1
3
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Problem Solving
4
Proof of W presence
Knowledge base Contains:
~ S1,1
~B1,1
~S2,1
B2,1
S1,2
B1,2
Rules:
R1:
~ S1,1  ~W1,1 ^ ~W1,2 ^ ~W2,1
R2:
~ S2,1  ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1
R3:
S1,2  W1,3 V W1,2 V W2,2 V ~W1,1
Modus Ponen on
R2:
~ S2,1  ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1
~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1
And Elimination:
~W1,1
~W2,1
~W2,2
~W3,1
CS 331 Dr M M Awais
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W
2
1
1
2
3
4
Results:
~W1,1
~W1,2
~W2,1
~W1,1
~W2,1
~W2,2
~W3,1
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Problem Solving
Proof of W presence
Knowledge base Contains:
~ S1,1
~B1,1
~S2,1
B2,1
S1,2
B1,2
Rules:
R1:
~ S1,1  ~W1,1 ^ ~W1,2 ^ ~W2,1
R2:
~ S2,1  ~W1,1 ^ ~W2,1 ^ ~W2,2 ^ ~W3,1
R3:
S1,2  W1,3 V W1,2 V W2,2 V W1,1
Modus Ponen on
R3: S1,2  W1,3 V W1,2 V W2,2 V W1,1
W1,3 V W1,2 V W2,2 V W1,1
Unit Resolution:
Alpha:W1,3 V W1,2 V W2,2 and Beta: W1,1
W1,3 V W1,2 V W2,2
Unit Resolution again with : W1,2 and W2,2
CS 331 Dr M M Awais
W1,3
4
3
W
2
1
1
2
3
~W1,1
~W2,1
~W2,2
~W3,1
W1,3
4
Results:
~W1,1
~W1,2
~W2,1
Simply drop all facts
that can create conflict
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Problem Solving
Pros and cons of propositional
logic
 Propositional logic is declarative
 Propositional logic allows partial/disjunctive/negated
information
– (unlike most data structures and databases)
 Propositional logic is compositional:
– meaning of (B  P ) is derived from meaning of B and of P
 Meaning in propositional logic is context-independent
– (unlike natural language, where meaning depends on context)
 Propositional logic has very limited expressive power
– (unlike natural language)
– E.g., cannot say “Music cause disturbance to all neighbors“
• except by writing one sentence for each neighbor
•
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