Formal Methods in Computer Science CS1502 FO Equivalence

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Formal Methods in Computer ScienceCS1502FO
Equivalence

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To learn about the notion of FO equivalence
• FO Equivalence
• Equivalent wffs
• Substitution of equivalent wffs
• DeMorgan laws for quantifiers
• Other equivalence laws
• To learn about common FO equivalences and to gain
  skills in using them.

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Equivalence

• Two sentences P and Q are equivalent if they
  have the same truth value in any circumstance.
• P ? Q

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Equivalences

• Tautological equivalence equivalent solely in
  virtue of the meaning of
• the boolean connectives /\, \/,
• FO equivalence equivalent solely in virtue of
  the meaning of
• the boolean connectives /\, \/,
• the quantifiers ?, ?
• the identity symbol

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3 kinds of FO equivalences

• Tautological equivalence
• Equivalence of wffs
• Equivalence involving quantifiers
• Use them with substitution law.
• By classifying them, it should help use remember
  them better.

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Substitution Law

• Suppose S is a complete unit of something.
• Suppose P is a component of S.
• We write S as S(P) to emphasize that P is a
  component of S.
• If Q ? P, we can replace P in S(P) with Q.
• Equivalence depends on context.
• We get S(Q). Every other part of S is the same.
• S(P) ? S(Q).

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• P, Q could be
• Tautologically Equivalent sentences
• Equivalent wffs
• FO equivalent sentences

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Tautological equivalence

• ( ?x Large(x) \/ ?y Tet(y) )
• TTF ( A \/ B )
• ?x Large(x) /\ ?y Tet(y)
• TTF A /\ B

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Taut EQ with Substitution

• Substitution of parts that are Taut EQ.
• Tet(c) \/ (( ?x Large(x) \/ ?y Tet(y) ) /\
  Cube(d))
• C \/ (( A \/ B ) /\ D)
• Tet(c) \/ (?x Large(x) /\ ?y Tet(y) /\ Cube(d))
• C \/ (A /\ B /\ D)

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Equivalence of wffs

• Up until now, when we talk about Equivalence
  between two things, we refer to Equivalence of
  their truth values under any circumstance.
• What kind of things have truth values?
• What about wffs? Do they have truth value?
• How do we adjust the definition of equivalence
  to accommodate this?

satisfaction
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Equivalence of wffs

• P(x) is a wff.
• An object c satisfies wff P(x) if and only if
  sentence P(c) is true.
• Two sentences P and Q are equivalent if they
  have the same truth value in any circumstance.
• Two wffs P(x) and Q(x) are equivalent if they
  are satisfied by exactly the same objects in any
  circumstance.

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Example

• Cube(x) ? Small(x)
• Cube(x) \/ Small(x)

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Wwfs EQ and Substitution

• Substitution of parts that are Equivalent wwfs.
• ?x (Cube(x) ? Small(x))
• ?x (Cube(x) \/ Small(x))
• This is different from substitution with
  tautological equivalence because we are
  substituting wffs rather than sentences.

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Equivalence involving ? ?
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DeMorgan laws for quantifiers

• Tet(a) /\ Tet(b) /\ Tet(c) /\ Tet(d)
• (Tet(a) /\ Tet(b) /\ Tet(c) /\ Tet(d))
• Tet(a)\/Tet(b)\/Tet(c)\/Tet(d)

• Consider a world with 4 blocks, a,b,c,d.
• ?x Tet(x)
• ?x Tet(x)
• ?x Tet(x)

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DeMorgan laws for quantifiers

• ?x P(x) ? ?x P(x)
• ?x P(x) ? ?x P(x)

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Aristotelian forms revisited

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• Homework
• ?x (P(x) ? Q(x))
• Some Os are not Qs
• ?x (P(x) /\ Q(x))

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Aristotelian forms revisited

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• Homework
• ?x (P(x) ? Q(x))
• Some Os are not Qs
• ?x (P(x) ? Q(x))
• ?x (P(x) \/ Q(x))
• ?x (P(x) \/ Q(x))
• ?x (P(x) /\ Q(x))

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Quantifiers and Boolean Connectives

• ?x (P(x) /\ Q(x)) ? ?x P(x) /\ ?x Q(x)
• ?x (P(x) \/ Q(x)) ? ?x P(x) \/ ?x Q(x)
• ?x (P(x) \/ Q(x)) ? ?x P(x) \/ ?x Q(x)
• ?x (P(x) /\ Q(x)) ? ?x P(x) /\ ?x Q(x)
• Find a counterexample.

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Null Quantification

• ?x Cube(b)
• For every object x, b is a cube.
• This is true if and only if b is a cube.
• Cube(b)
• ?x P ? P
• ?x P ? P

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More Null Quantification

• ?x (P \/ Q(x)) ? P \/ ?x Q(x)
• ?x (P /\ Q(x)) ? P /\ ?x Q(x)

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Variable Replacement

• ?x P(x) ? ?y P(y)
• ?x P(x) ? ?y P(y)
• Example
• ?x (?x (?x Cube(x) /\ Tet(x)) /\ Dodec(x))
• ? ?x (?y (?z Cube(z) /\ Tet(y)) /\ Dodec(x))

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Exercise

• ?x (Tet(x) ? ?y (LeftOf(x,y) /\ Larger(b,x)))
• ?x (Tet(x) ? ?y (LeftOf(x,y) /\ Larger(b,x)))
• ?x (Tet(x) \/ ?y (LeftOf(x,y) /\ Larger(b,x)))
• ?x (Tet(x) /\ ?y (LeftOf(x,y) /\ Larger(b,x)))
• ?x (Tet(x) /\ (?y LeftOf(x,y) /\ Larger(b,x)))
• ?x (Tet(x) /\ (?y LeftOf(x,y) \/ Larger(b,x)))
• ?x (Tet(x) /\ (?y LeftOf(x,y) \/ Larger(b,x)))
• ?x (Tet(x) /\ (?y LeftOf(x,y) \/ Larger(b,x)))
• ?x ((Tet(x) /\ ?y LeftOf(x,y)) \/ (Tet(x) /\
  Larger(b,x)))
• ?x (Tet(x) /\ ?y LeftOf(x,y)) \/ ?x(Tet(x) /\
  Larger(b,x))
• ?x (Tet(x) /\ ?y LeftOf(x,y)) \/ ?y(Tet(y) /\
  Larger(b,y))

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• There are a few more equivalences in the textbook
  that you should familiarize yourself with.

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• ?x (P(x) /\ Q(x)) ? ?x P(x) /\ ?x Q(x)
• ?x (P(x) \/ Q(x)) not ? ?x P(x) \/ ?x Q(x)
• ?x (P(x) \/ Q(x)) ? ?x P(x) \/ ?x Q(x)
• ?x (P(x) /\ Q(x)) not ? ?x P(x) \/ ?x Q(x)
• ?x P ? P
• ?x P ? P
• ?x (P \/ Q(x)) ? P \/ ?x Q(x)
• ?x (P /\ Q(x)) ? P /\ ?x Q(x)
• ?x P(x) ? ?y P(y)
• ?x P(x) ? ?y P(y)
• ?x P(x) ? ?y P(y)
• ?x P(x) ? ?y P(y)