1. Lecture 9 CS 1813 – Discrete Mathematics
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
Lecture 9 CS 1813 – Discrete Mathematics
Predicate Calculus
Propositions Plus Plus
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
What is a Predicate?
Predicate
Parameterized collection of propositions
P(x)
Typically a different proposition for each x
Universe of discourse
Values that x may take
Must be specified
Otherwise, all bets off — muchas contradicciónes
Non-empty
Empty universe calls for special handling
Default assumption: non-empty universe
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. Predicate on a Program for Sums
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
Predicate on a Program for Sums
sum::[Rational] -> Rational function-type arrow, not implication
Input (argument)
Sequence of Rational values (ratios of whole numbers)
Type: [Rational] — square brackets indicate sequence-type
Output (value delivered): Rational — not a sequence (so, no brackets)
Predicate
S(n)  sum[x1, x2, …, xn] = x1 + x2 + … + xn
S is a predicate — each equation S(n) is a different proposition
Each S(n) is either True or False
S(3) is the (atomic) proposition sum[x1, x2, x3] = x1 + x2 + x3
Universe of discourse (collection of values that parameterize S)
Natural numbers N = {0, 1, 2, … } — a collection of infinite size
Why not T([x1, x2, … xn]) instead of S(n)?
Nothing wrong with T([x1, x2, … xn])
Universe of discourse for T: finite sequences of numbers
Universe of discourse for S: counting numbers {0, 1, 2, …}
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. Predicate on a Program for Maximums
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
Predicate on a Program for Maximums
12/10/2018
maximum :: [String] -> String
Argument
Sequence of strings (String means sequence of characters)
Type: [String] — square brackets indicate sequence-type
Value: String — not a sequence of strings (so, no brackets)
Predicate
B(n, k)  maximum[s1, s2, …, sn]  sk
B is a predicate — each B(n, k) is a proposition (True or False)
B(4,2) is the (atomic) proposition maximum[s1, s2, s3 , s4]  s2
Universe of discourse (collection of values that parameterize B)
Pairs of non-zero natural numbers {(1,1), (2,1), (2,2), (3,1), … } where second number in pair does not exceed first
Why not C([s1, s2, … sn], sk) instead of B(n, k)?
C is ok, but depends on the strings si — not just on n and k
Proofs involving B must take care to encompass arbitrary si’s
Note: Proofs will work for any type of class Ord, not just String
What is the universe of discourse for C([s1, s2, … sn], sk)?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Another Predicate about Maximums
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
Another Predicate about Maximums
12/10/2018
maximum :: [String] -> String
Argument
Sequence of strings (String means sequence of characters)
Type: [String] — square brackets indicate sequence-type
Value: String — not a sequence of strings (so, no brackets)
Predicate
E(n, k)  maximum[s1, s2, …, sn] = sk
E is a predicate —each equation E(n, k) is an atomic proposition
Each E(n, k) is either True or False
E(4,1) is the proposition maximum[s1, s2, s3 , s4] = s1
Universe of discourse (collection of values that parameterize E)
Pairs of non-zero natural numbers {(1,1), (2,1), (2,2), (3,1), … } where second number in pair does not exceed first
Is E(n, k) usually True or usually False?
Would you expect E(n, k) to be True for some pairs (n, k)?
Given a sequence of strings [s1, s2, … sn], would you expect to be able to find any values of k for which E(n, k) is true?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Predicates about a Sorting Program
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
Predicates about a Sorting Program
qsort :: Ord a => [a] -> [a]
Argument
Sequence of type [a] — where a is a type of class Ord
Value — Also a sequence of type [a]
Predicates
L(n)  length(qsort[a1, a2, …, an] ) = n
L is a predicate — each equation L(n) is a proposition
Universe of discourse: Natural numbers N = {0, 1, 2, … }
What does [a1, a2, …, an] mean when n = 0
Would you expect L(n) to be True, regardless of the ai’s?
I(n, k)  (qsort[a1, a2, …, an] = [b1, b2, …, bn] )  (bk  bk+1 )
I is a predicate — each I(n, k) is a proposition (non-atomic, btw)
Universe of discourse: Pairs of non-zero natural numbers where first number exceeds second {(2,1), (3,1), (3,2), (4,1), … }
For what pairs would you expect I(n, k) to be True?
R(n, j, k)  (qsort[a1, a2, …, an] = [b1, b2, …, bn] )  (aj = bk)
Universe of discourse? Places in universe where R(n, j, k) is True?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. Predicates about Concatenation
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
Predicates about Concatenation
(++) :: [a] -> [a] -> [a]
Two arguments
Sequences of type [a] — where a can be any type
Value — Also a sequence of type [a]
Predicates
C(n, m)  length( [a1, a2, …, an] ++ [b1, b2, …, bm] ) = n + m
C is a predicate — each equation C(n, m) is a proposition
Universe of discourse: Pairs of natural numbers {(0,0), (1,0), … }
Would you expect C(n, m) to be True?
A(xs, ys, zs)  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs
A is a predicate — each equation A(xs, ys, zs) is a proposition
Universe of discourse: Triples of sequences of type [a]
For what sequences would you expect A(xs, ys, zs) to be True?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8.  — the Universal Quantifier, Forall
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
 — the Universal Quantifier, Forall
x.P(x)
This formula is a WFF of predicate calculus whenever P(x) is a WFF of predicate calculus
True if the proposition P(x) is True for every value of x in the universe of discourse
False if there is some value x in the universe of discourse for which P(x) is False
Equivalent to forming the Logical And of all P(x)’s
Example – S predicate about sum
S(n)  sum[x1, x2, …, xn] = x1 + x2 + … + xn
n.S(n)
Universe of discourse: natural numbers N = {0, 1, 2, … }
n.S(n) means S(0)  S(1)  S(2)  …
So, “” provides a way to write formulas that would contain an infinite number of symbols if written in propositional calculus notation (but infinitely long formulas aren’t WFFs)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. Another Example with Forall  — the Universal Quantifier
Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
Another Example with Forall  — the Universal Quantifier
B — predicate about maximum
B(n, k)  maximum[s1, s2, …, sn]  sk
Universe of discourse
Pairs of non-zero natural numbers where second number in pair does not exceed first
U = {(n, k)  N N | 0  k  n} = {(1,1), (2,1), (2,2), (3,1), … }
OK, so we haven’t gotten to set theory yet — Sabe, verdad?
(n, k).B(n, k)
A predicate calculus formula because each B(n, k) is a proposition, which is a portion of predicate calculus
Do you expect (n, k).B(n, k) to be True?
Other ways to write (n, k).B(n, k)
nN . kN . (0  k)  (k  n)  B(n, k)
nN, n  0 . kN, 0  k  n . B(n, k)
Note nesting and direct specification of universe of discourse
Technically, U = N N
in this case, but values of B(n, k) outside the old U are immaterial.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. Lecture 9 - CS 1813 Discrete Math, University of Oklahoma
12/10/2018
End of Lecture 9
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page