Satisfy

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Satisfy
An interpretation that makes a sentence true is
said to satisfy that sentence
d
c
onIltd,cgt,ltc,bgt,ltb,agt aboveIltd,cgt,ltc,bgt,ltb,agt,
ltd,bgt,ltd,agt,ltc,agt
b
a
on(X,Y) -gt above(X,Y)
Blocks world
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satisfiability
man mortal kim yes park yes choi yes
?(X) man(X) -gt mortal(X)
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Satisfiability

• An interpretation satisfies a wff if the wff is
  assigned true under that interpretation

X Y Z t t t f t f f f f
X ? Y
not Y ? not Z
Z ? Y
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logically follows

• An expression X logically follows from a set of
  PC expression S if every interpretation that
  satisfies S also satisfies X

logically follows
S
X
satisfy
satisfy
Interpretation
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Logically follows

• An expression X logically follows from a set of
  predicate calculus expressions S if every
  interpretation that satisfies S also satisfies X
• Unfeasible to test every interpretation since
  they are infinite

X
S
I
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Satisfy, Model, Valid, Inconsistent

• For a predicate logic expression X and and
  interpretation I
• If X is true under I and a particular variable
  assignment, then I satisfies X
• If I satisfies X for all variables, I is a model
  of X
• X is satisfiable if and only there is an
  interpretation and variable assignment that
  satisfy it
• A set of expressions is satisfiable if and only
  if there exist an interpretation and variable
  assignment that satisfy every element
• If a set of expressions is not satisfiable then
  it is inconsistent
• If X is true for all possible interpretations
  then it is valid

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Validity

• Truth table can be used to test the validity of
  any expression not containing a variable
• Not always possible to decide the validity of
  expressions that contain variables
• Variables could be numbers that are infinite

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Inference rules
Inference rules
S
X
A sentence
A set of expression
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sound
Inference rules
S
X
A sentence
A set of expression
satisfy
satisfy
Interpretation
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complete

• If an inference rule is able to produce every
  expression that logically follows from S, then it
  is said to be complete.
• Modus Pones is sound and, when used with certain
  appropriate strategy, is to be complete.
• P -gt Q, P Q

Interpretation
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Logically Follows, Sound, Complete

• A predicate logic expression X logically follows
  from a set S of expressions if every
  interpretation and variable assignment that
  satisfies S also satisfies X
• An inference rule is sound if every predicate
  logic expression produced by the rule from the
  set S of expressions also logically follows from
  S
• An inference rule is complete if, given a set S
  of expressions, the rule can infer every
  expression that logically follows from S

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Modus Ponens
is_rainy -gt bring_umbrella is_rainy ?Day,Man
is_rainy(Day)?person(Person) -gt
bring(Person, umbrella) is_rainy(today) person(kim
) Unification
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Inference Rules

• Modus Ponens
• P -gt Q and P are true then Q is true
• Modus Tollens
• P -gt Q is true and Q is false then ?P is true
• Elimination
• If P ? Q is true then both P Q are true
• Induction
• If P Q are true then P ? Q is true

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Logical Inference

• Ability to infer new correct expressions from a
  set of true assertions
• Satisfy
• Interpretation that makes a sentence true

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Modus Ponens

• If it is raining then the ground will be wet
• It is raining
• Can we assume that the ground is wet?
• How about All men are mortal and Socrates is a
  man therefore Socrates is mortal

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Unification

• Need to decide when two expressions match
• Problem with variables
• Eliminate existentially quantified variables
• Skolemization
• Replace each existentially quantified variable
  with a function

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