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Knowledge Representation I
(Propositional Logic)
CSE 473
Some KR Languages
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Propositional Logic
Predicate Calculus
Frame Systems
Rules with Certainty Factors
Bayesian Belief Networks
Influence Diagrams
Semantic Networks
Concept Description Languages
Nonmonotonic Logic
© Daniel S. Weld
2
In Fact…
• All popular knowledge representation
systems are equivalent to (or a subset of)
Logic
• Either Propositional Logic
• Or Predicate Calculus
Probability Theory
© Daniel S. Weld
3
What is Propositional Logic?
• And why have you studied it?
•
And why are we torturing you again?
© Daniel S. Weld
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473 Topics
Perception
Inference
Logic
NLP
Robotics
Supervised
Learning
Knowledge
Representation
Multi-agent
Reinforcement
Learning
Planning
Probability
Search
Problem Spaces
Agency
© Daniel S. Weld
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AI=Knowledge Representation
& Reasoning
• Syntax
• Semantics
• Inference Procedure
Algorithm
Sound?
Complete?
Complexity
© Daniel S. Weld
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Propositional Logic
• Syntax
Atomic sentences: P, Q, …
Connectives:  , , , 
• Semantics
Truth Tables
• Inference
Modus Ponens
Resolution
DPLL
GSAT
• Complexity
© Daniel S. Weld
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Propsitional Logic: Syntax
• Atoms
P, Q, R, …
• Literals
P,  P
• Sentences
Any literal is a sentence
If S is a sentence
• Then (S  S) is a sentence
• Then (S  S) is a sentence
• Conveniences
PQ
© Daniel S. Weld
same as  P  Q
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Semantics
• Syntax: a description of the legal
arrangements of symbols
(Def “sentences”)
• Semantics: what the arrangement of
symbols means in the world
Sentences
Facts
© Daniel S. Weld
Sentences
Semantics
World
Semantics
Representation
Inference
Facts
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Propositional Logic: SEMANTICS
• “Interpretation” (or “possible world”)
Assignment to each variable either T or F
Assignment of T or F to each connective via
defns
P
Q
T F
T T F
F F F
PQ
© Daniel S. Weld
P
Q
T F
T T T
F T F
PQ
Q
P
T T F
F F T
P
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Satisfiability, Validity, & Entailment
• S is satisfiable if it is true in some world
• S is unsatisfiable if it is false all worlds
• S is valid if it is true in all worlds
• S1 entails S2 if wherever S1 is true S2 is
also true
© Daniel S. Weld
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Examples
P => Q
R => R
S  (W  S)
T  T
X => X
© Daniel S. Weld
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Notation




=
}
Implication (syntactic symbol)
Inference
Proves: S1 |-ie S2 if `ie’ algorithm says `S2’ from S1
Entailment
Entails: S1 |= S2 if wherever S1 is true S2 is also true
• Sound
  =
• Complete =  
=

implies
© Daniel S. Weld
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Prop. Logic: Knowledge Engr
1)
2)
3)
4)
5)
One of the women is a biology major
Lisa is not next to Dave in the ranking
Dave is immediately ahead of Jim
Jim is immediately ahead of a bio major
Mary or Lisa is ranked first
1. Choose Vocabulary Universe: Lisa, Dave, Jim, Mary
LD = “Lisa is immediately ahead of Dav
D = “Dave is a Bio Major”
2. Choose initial sentences (wffs)
© Daniel S. Weld
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Reasoning Tasks
• Model finding
KB = background knowledge
S = description of problem
Show (KB  S) is satisfiable
A kind of constraint satisfaction
• Deduction
S = question
Prove that KB |= S
Two approaches:
1. Rules to derive new formulas from old
(inference)
2. Show (KB   S) is unsatisfiable
© Daniel S. Weld
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Propositional Logic: Inference
A mechanical process for computing new sentences
1. Backward & Forward Chaining
Based on rule of modus ponens
If know P1, …, Pn & know (P1 ...
Then can conclude Q
 Pn )=> Q
2. Resolution (Proof by Contradiction)
3. GSAT
4. Davis Putnam
© Daniel S. Weld
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