CS 1502 Formal Methods in Computer Science

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CS 1502 Formal Methods in Computer Science

• Lecture Notes 12
• Variables and Quantifiers

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Variables

• A variable is a placeholder for a name in a
  sentence
• FOL gives us an infinite number of variables
• Tarskis World/Fitch/Boole gives us only u, v, w,
  x, y, and z as variables

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Atomic wffs

• Cube(b)
• Cube(x)
• LeftOf(c,y)
• Between(z,b,d)
• Larger(x,y)

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Variables

• Cube(b) ? Cube(d) The truth of this sentence
  is determined by the truth- values of
  Cube(b) and Cube(d) .
• Cube(x) ? Cube(d) Cube(x) is neither true
  nor false, since x is a placeholder for
  the name of an object. x is said to be
  free or unbound.

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Forming wff

• If P is wff then so is ?P.
• If P1, P2, , Pn are wffs then so is (P1 ? P2 ?
  ? Pn).
• If P1, P2, , Pn are wffs then so is (P1 ? P2 ?
  ? Pn).
• If P and Q are wffs, so is (P ? Q).
• If P and Q are wffs, so is (P ? Q).
• If P is a wff and v is a variable, then ?v P is a
  wff. Every occurrence of v is said to be bound.
• If P is a wff and v is a variable, then ?v P is a
  wff. Every occurrence of v is said to be bound.

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Sentences

• A sentence is a well-formed formula with no
  unbound (free) variables.
• The scope of a quantifier is defined as those
  variables that fall under the quantifiers
  influence as indicated by the enclosing
  parentheses.

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Which are Sentences?

• ?x LeftOf(x, y)
• ?x (Cube(x) ? Large(x))
• ?x (Small(x) ? Tet(y))
• ?x (Cube(c) ? SameRow(x, c))

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Satisfaction (diff meaning!) of a wff

• Let S(x) be a wff with free variable x. Object b
  is said to satisfy S(x) if and only if S(b) is a
  true sentence. EG, b satisfies Cube(x).

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Semantics of Quantifiers

• ?x S(x) is true if and only if there is at least
  one object that satisfies S(x).
• ?x S(x) is true if and only if every object
  satisfies S(x).

The truth-value is determined with respect to a
domain of discourse (or world, or universe)
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Example

• Universe 2, 5, 17, 93, 67

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Some Tetrahedron is to the left of c.
Some Cube is to the left of c.
a is to the left of every Cube.
Every Cube is in back of c.
C is supposed to be back one square as it is on
the next slide
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Aristotelian Forms

• All Ps are Qs

?x P(x) ? Q(x)
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Aristotelian Forms

• Some Ps are Qs

?x P(x) ? Q(x)
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Aristotelian Forms

• No Ps are Qs

?x P(x) ? ?Q(x)
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Aristotelian Forms

• Some Ps are not Qs

?x P(x) ? ?Q(x)
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Translation

• ?x (Tet(x) ? LeftOf(x, c))
• ?x (Tet(x) ? Large(x) ? LeftOf(x, c))
• ?x Cube(x) ? BackOf(x, c)
• ?x (Small(x) ? Cube(x)) ? BackOf(x, c)

Some Tet is to the left of c.
Some large Tet is to the left of c.
Every cube is in back of c.
Every small cube is in back of c.
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Translation

• all x (Student(x) ? Smart(x)).
• Every student is smart.
• Pick anything at all. Either its not a
  student or its smart (or both!)
• Pick anything at all. It cannot be a student
  who is not smart.
• Pick anything at all. If its a student,
  its smart.

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Whats wrong with this translation?

• Every student is smart
• Bad translation
• all x (Student(x) Smart(x))
• Good translation
• all x (Student(x) ? Smart(x))

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Whats wrong with this translation?

• Some student is smart
• Bad translation
• exists x (student(x) ? smart(x))
• Good translation
• exists x (student(x) smart(x))

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Translation

• ?x and ?
• ?x and ?

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Quantifiers and Functions

• Everyone is taller than Maxs father.
• Someones father is taller than Max.
• e is the leftmost object in the row of something.

?x (Person(x) (x father(max))) ? Taller(x,
father(max))
?x (Person(x) Taller(father(x), Max))
?x (lm(x) e)
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Quantifiers can make things interestingWhich are
valid?
Yup!

• all x (Student(x) ? Smart(x))
• all x (Student(x))
• all x (Smart(x))

• exist x (Student(x) ? Smart(x))
• exist x (Student(x))
• exist x (Smart(x))

Nope!
Yup!
Nope!
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Quantifiers and Tautology

• When is a sentence involving quantifiers a
  tautology? Example ?x Cube(x) ? ?x ?Cube(x)
• Example ?x Small(x) ? ?x Small(x)

No!
Yes!
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Tautologies and Quantifiers

• One way to arrive at tautologies is to substitute
  complex sentences into known tautologies
• P v P is a tautology. Thus, so is
• (all x (Doctor(x) ? Rich(x))) v
• (all x (Doctor(x) ? Rich(x)))

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Truth-Functional Form Algorithm

• Start at the beginning of sentence S and proceed
  to the right. If you encounter a quantifier,
  underline the quantifier and the stuff in its
  scope. If you encounter an atomic sentence simply
  underline it. When you come to the end of an
  underline assign a letter (A, B, C, ). If an
  underlined sentence is identical to a previous
  one (character for character), use the same
  letter. If the sentence formed from the letters
  is a tautology, then the original is too.Note
  Small(a) is not the same as Small(b). Similarly,
  Small(x) is not the same as Small(y).

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When is a sentence a tautology?

• A quantified sentence S is a tautology if and
  only if its truth-functional form is a tautology.
• Example
• (exist y (P(y) v R(y)) ? all x (P(x) Q(x))) ?
• (all x (P(x) Q(x)) ? exist y (P(y) v R(y)))

TFF (A ? B) ? (B ? A)
Tautology? Yes
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Another Example

• ?x Cube(x) ? ?x ?Cube(x)

Since A ? B is not a tautology, then neither is
the sentence above.
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Another Example

• ?x Cube(x) ? ??x Cube(x)

Since A ? ?A is a tautology, then so is the
sentence above.
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Another Example

• ?(Tet(d) ? ?x Small(x)) ? (?Tet(d) ? ?y Small(y))

?(A ? B) ? (?A ? C)
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Tautological Valid?

• Is the following argument valid or invalid?
  

?x Cube(x)?x Small(x) ?x (Cube(x) ? Small(x))
ABC
Not tautologically valid!!!In fact, it is not
logically valid.
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Tautological Valid?

• Is the following argument valid or invalid?
  

?x Cube(x) ? ?x Small(x) ?x Cube(x) ?x
Small(x)
Is tautologically valid!!!
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Note

• An argument that is not tautologically valid may
  still be logically valid. In this case, we
  cannot recognize it is valid by checking all the
  rows of the truth table.

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Tautological Valid?

• Is the following argument valid or invalid?
  

?x Cube(x) ? Small(x)?x Cube(x) ?x Small(x)
Is not tautologically valid.It is logically
valid!