Conversion to Conjunctive Normal Form

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Conversion to Conjunctive Normal Form
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Assertion

• All Romans who know Marcus either hate Caesar or
  think than anyone who hates anyone is crazy. All
  Romans wish they were Greeks.

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Expressed in FOPL
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For the transformation, lets simplify

• Let
• P roman(X)
• Q know(X, marcus)
• R hate(X, caesar)
• S hate(Y,Z)
• T thinkcrazy(X,Y)
• V wish_greek(X)

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Giving
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Step 1 Eliminate implication using the
identity
So
becomes
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Applied to the Original Expression
Now eliminate the second gt
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Applied to the original expression
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Eliminating the Third ImplicationGives
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Step 2 Invoke deMorgan
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Step 3 Standardize the quantifiers so that each
binds a unique variable

• For Example
• Given

We write
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Step 4 Move all quantifiers to the left without
changing their order

• Step 3 makes this legal

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Step 5 Eliminate Existential Quantifiers
Skolemization

• Type 1
• Given

Tells us that there is an individual assignment
to X drawn from its domain under which school(X)
is satisfied. .
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• So, invent a function that goes into the domain
  of X and picks out just that item that satisfies
  school.
• Call it pick
• The original expression is transformed to
  school(pick())
• Where
• Pick is a function with no arguments
• That returns the value from the domain of X that
  satisfies School
• We might not know how to get the value.
• But we give a name to the method that we know
  exists

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Type 2

• Suppose we have

Where P,Q are elements of the set of integers In
English, given an integer P, there is another
integer Q, such that Q gt P We cant invent a
single function, because Q depends on P Instead,
invent a function whose single argument is the
universally quantified variable.
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• get(P) returns an integer gt P

Becomes
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Our example has one instance of an existentially
quantified variable within the scope of a
universally quantified variable
Becomes
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Giving
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Step 6 Drop the remaining quantifiers

• Legal since everything is universally quantified

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Step 7 Rewrite the expression as a conjunct of
disjuncts using dist. And assoc. laws

• This Gives

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Step 8 Rewrite each conjunct as a separate
clause that are implicitly anded
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Step 9 Rename variables in clauses so that no
two clauses use the same variable name

• This is already the case
• The Expression is now in conjunctive normal form.

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