Variables and Instantiation

1
Variables and Instantiation

• Either a variable or a constant may be used to
  express a predicate
• Gt_zero(x) \/ EQ_zero(x) x is a
  variable
• Carnivorous(y) -gt eats(y, meat) y is a
  variable meat is a constant
• Instantiation
• Let an expression, E, contain a variable, y.
  Then substituting a specific term ( or constant),
  t, for the variable y in E will create a new
  Expression E. E is the instantiation of E and t
  is an instant of y
• Manager(x,y) substitute Bob for x and Sam for
  y, we get Manager(Bob, Sam) to create a specific
  instance of this predicate which states Bob is
  the manager of Sam.
• Eats(Tom, w) substitute apple for w and we get
  Tom eats apple.
• tested(z) -gt runs(z) substitute MS-Word for z,
  we get the statement tested MS-Word implies that
  MS-Word runs.

2
Quantifiers

• With the concept of instantiation,
  ------quantification becomes easier to discuss
• Consider the statements
• All computers have cpus
• Some computers are defective
• Every program is tested at Microsoft
• At least one student remembers Pascal
• Note that these statements indicate how many of
  the specific items in the universe of discourse
  are True. This notion of instances of truth is
  captured by Quantifiers
• Some
• Every
• All
• At least one

3
Fuzzy Logic Temporal Logic

• Fuzzy Logic the notion of quantification (all,
  some, etc.) is different from the notion of
  degree of truth which asserts the probability
  of truth, P(x is True). But the truth may be
  measured in other than the binary T/F.
• Joe is sick a lot can be looked upon as a
  predicate, sick(Joe), which has a probability of
  70 truth.
• Temporal Logic quantifiers here are also
  different from another kind of quantification
  that addresses how often P(x) is True
• It is sometimes the case that Joe is sick can
  be looked upon as a predicate, sick(Joe), that is
  True at some instance of time and False at other
  instance of time.
• We will not pursue Fuzzy logic or Temporal logic
  in this course even though they have applications
  to many real world problems.

4
Notion of Quantification

• Quantification A mechanism to express the whole
  universe of discourse or a part of the universe
  of discourse
• Existential Quantifier part of
• often expressed with
• Universal Quantifier whole
• often expressed with

5
Existential Quantifier

• The Existential Quantifier, , asserts that at
  least one object in a collection of objects (or
  the universe of discourse) has the stated
  property or relationship.
• The general Existential Quantifier expression
• identifier class list predicate
• e.g. z N zgt 1000
  where
• z is the identifier
  or variable
• N is the class list
  or universe of discourse
• z gt 1000 is the
  predicate
• e.g. some programs are written in
  JAVA
• p programs
  prog_lang(p, Java)

We often short hand the expression and do not
include the class list or the
6
Universal Quantifier

• The Universal Quantifier, , asserts that all
  objects in a collection of items or in the
  universe of discourse have the stated property or
  relationship.
• The general Universal Quantifier expression
• identifier class list predicate
• e.g. X integers x2 gt 0 \/ x2 0
• (square of integers are greater
  than or equal to zero)
• p programs run(p) \/
  run(p)
• ( every program either runs or
  does not run)

We often short hand the expression and do not
include the class list or the
7
Bound and Free Variables

• In either Existential or Universal
  quantification, the variable in the statement and
  the variable associated with the quantifier is
  treated as a unit.
• (z) S(z,y) - the variable z is
  treated as a unit with the statement S.
• the variable, z, is considered bounded and the
  statement S is considered as the scope of the
  quantifier while the variable, y, is considered
  as free

This notion of a variable that is bound by the
quantifier within the scope of the statement is
similar to variables bound within the scope of a
program routine or subroutine.
8
Quantifiers are Unary Operators

• Quantifiers must be treated just as the unary
  operator, just like the operator.
• Consider S1(x) to be x is alive, and s2(x) to
  be x is dead. then
• (x) S1(x) \/ S2(x) says everyone
  is alive or x is dead
• (x) ( S1(x) \/ S2(x) ) says everyone is
  alive or dead
• Note that in the first case, x in S1(x) is bound
  and S1 is the scope of the quantifier. The x in
  S2(x) is a free variable. One may view the x in
  S2(x) as a different variable from the x in
  S1(x). You may want to add parenthesis for
  clarity.
• In the second case the x in both S1 and S2 are
  bound by the quantifier and S1 and S2 jointly
  form the scope.

9
Collision in Bound/Free Scoping

• Consider x M(x,y)
• x is a bound variable and local to the predicate
  M. y on the other hand is a free variable and
  different from x.
• We can replace the variable x with z and have
  essentially the same statement, z M(z,y).
• Can we replace x with y and have y M(y,y) ?
• This may cause a PROBLEM.
• The previously free variable y becomes bound and
  the two previously different variable becomes the
  same.
• Consider the predicate M to mean mother-Of
• x M(x,y) says y has a mother
• y M(y,y) says y is ys own mother

10
Phrasing Universal Quantifier

• Consider the statement
• All Apple hardware are computers.
• How would you phrase this using predicate
  calculus ?

11
Phrasing Universal Quantifier

• Phrasing All Apple hardware are computers as
  follows
• If x is an Apple hardware, then it is a
  computer.
• (x) (apple_hardware(x) -gt computer(x) )
• Note we used for all quantifier to capture
  the all Apple hardware concept.

12
Phrasing Existential Quantifier

• Consider the statement
• Some Apple hardware are computers.
• How would you phrase this in predicate calculus?

13
Phrasing Existential Quantifier

• Phrasing Some Apple hardware are computers as
  follows
• There exists an Apple hardware which is a
  computer.
• There exists an Apple hardware and that hardware
  is a computer
• (x) (apple_hardware(x) /\ computer(x) )
• Can this use an implies operator?

14
Phrasing with Quantifiers

• How may one phrase the following?
• Only MS programs crash.
• (x) ( MS_program(x) -gt crash (x) )
• OR is the following better?
• (x) ( crash(x) -gt MS_program(x) )

15
Composite or Nesting of Quantifiers

• Consider the statement
• every program calls another program.
• How can we express this ?
• Program x calls program y Calls (x, y)
• There exists a program called by x y
  Calls(x,y)
• Every program calls another program x y
  Calls(x,y)

How would you state the following all programs
are recursive Or all programs call itself ?