reasoning in propositional logic
TRANSCRIPT
Reasoning inPropositional Logic
Logic reasoning (inference)
Logical entailment (implication)
Logical deduction (inference algorithms)
BC a
BC a
Sentences Sentences
Facts Facts
Representation
Modeled world
Se
man
tics
Se
man
tics
logicallyentails
emerge
INFERENCE
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Logical implication
One thing follows another
KB |= a
A KB logically entails the sentence a if and only if a is true in the world where the KB is true.
Models: Formally structured worlds with respect to which
truth can be evaluated.
m is a model of a sentence a if a is true in m.
M(a) is the set of all the models of a .
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Example of logical entailment
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Possible models
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Models for the KB
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Logically entailed sentence
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Sentence no logically entailed
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Inferential processes
Characteristics
• Sound or truth preserving: an inference algorithm that derives only entailed sentences.
• Complete: an inference algorithm that derive any sentence that is entailed.
Example: Model Checking
1. List the models (combinations according to the propositional symbols).
2.Mark those in which KB is true.
3.Verify if in these models the sentence to be inferred is also true.
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Logical concepts
Logical equivalence
Two sentences α and β are logically
equivalent if they are true in the same set of
models.
Validity
A sentence is valid if it is true
in all models.
Satisfiability
A sentence is satisfiable if it is
true in, or satisfied by, some model.
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Standard logical equivalences
Associativity Double-negation elimination
ContrapositionCommutativity
PQQP
PQQP
PP )( PQQP
)(())((
))(())((
RQPRQP
RQPRQP
Distributivity De MorganImplication elimination
Biconditional elimination
)()()(
)()()(
RPQPRQP
RPQPRQP
QPQP
QPQP
)(
)(QPQP )()( PQQPQP
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Validity
A sentence is valid (tautology) if it is always true.
qp qqp )(p q
T
F
T
F
T
F
T
T
T
T
F
F
F
F
F
T
T
T
T
T
pqqp ))((
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