snick

snack
CPSC 121: Models of Computation
2010 Winter Term 2
Describing the World
with Predicate Logic
Steve Wolfman, based on notes by
Patrice Belleville and others
1
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Prelude: Scope and Predicate Definition
• Problems and Discussion
– Liszt Etudes
– Sorted Lists
– Comparing Algorithms
• Next Lecture Notes
2
Learning Goals: Pre-Class
By the start of class, you should be able to:
– Evaluate the truth of predicates applied to
particular values.
– Show predicate logic statements are true by
enumerating examples (i.e., all examples in the
domain for a universal or one for an existential).
– Show predicate logic statements are false by
enumerating counterexamples (i.e., one
counterexample for universals or all in the domain
for existentials).
– Translate between statements in formal predicate
logic notation and equivalent statements in
closely matching informal language (i.e., informal
statements with clear and explicitly stated
quantifiers).
6
Learning Goals: In-Class
By the end of this unit, you should be able
to:
– Build statements about the relationships
between properties of various objects—which
may be real-world like “every candidate got
votes from at least two people in every
province” or computing related like “on the ith
repetition of this algorithm, the variable min
contains the smallest element in the list
between element 0 and element i”)—using
predicate logic.
7
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Prelude: Motivation, Scope & Defining Predicates
• Problems and Discussion
– Liszt Etudes
– Sorted Lists
– Comparing Algorithms
• Next Lecture Notes
8
Limitations of Propositional
Logic as a Model
Which of the following can propositional logic model
effectively?
a. Relationships among factory production lines
(e.g., “wheel assembly” and “frame welding” both
feed the undercarriage line).
b. Defining what it means for a number to be prime.
c. Generalizing from examples to abstract patterns
like “everyone takes off their shoes at airport
security”.
d. Prop logic can model all of these effectively.
e. Prop logic cannot model any of these effectively.
9
What Does Predicate Logic
Model?
Propositional logic is a good (but not perfect)
model of combinational circuits.
What is predicate logic good for modeling?
• Relationships among real-world objects
• Generalizations about patterns
• Infinite domains
• Generally, problems where the properties
of different concepts, ideas, parts, or
entities depend on each other.
10
But... Would You Ever Really
Use Pred Logic Like This?
• Data Structures Example: “...every key is less than
or equal to all of its children’s keys...”
• AI example: “...let h' be a “heuristic” function
evaluating game states and h be the true value of
the state. For all nodes n, h'(n)  h(n)...”
• Java example: “...there is no path via references
from any variable in scope to any memory location
available for garbage collection...”
• Economics/elections example: “...for any distinct
pair of candidates c1 and c2, if all voters prefer c1
to c2, then society must rank c1 above c2...”
11
Quantifier Scope
A quantifier applies to everything to its right
until a closing parenthesis stops it.
12
Quantifier Scope
A quantifier applies to everything to its right
until a closing parenthesis stops it.
x  D, (y  E, Q(x,y)  z  F, R(y,z))  P(x).
13
Quantifier Scope
A quantifier applies to everything to its right
until a closing parenthesis stops it.
x  D, (y  E, Q(x,y)  z  F, R(y,z))  P(x).
14
Quantifier Scope
A quantifier applies to everything to its right
until a closing parenthesis stops it.
x  D, (y  E, Q(x,y)  z  F, R(y,z))  P(x).
15
A Bit of Syntax:
Quantifier Scope
Which of the following placements of
parentheses yields the same meaning as:
x  Z, y  Z, x < y  Even(y).
a. ()x  Z, y  Z, x < y  Even(y).
b. (x)  Z, y  Z, x < y  Even(y).
c. (x  Z), y  Z, x < y  Even(y).
d. (x  Z, y  Z, x < y)  Even(y).
e. (x  Z, y  Z, x < y  Even(y)).
16
A Bit of Syntax:
Negation Scope
Which of the following placements of
parentheses yields the same meaning as:
~x  Z+, y  Z+, x < y  Even(y).
a. (~)x  Z+, y  Z+, x < y  Even(y).
b. (~(x))  Z+, y  Z+, x < y  Even(y).
c. (~(x  Z+)), y  Z+, x < y  Even(y).
d. (~(x  Z+, y  Z+, x < y))  Even(y).
e. (~(x  Z+, y  Z+, x < y  Even(y))).
17
A Bit of Semantics:
Unbound Variables
What is the truth value of the following formula?
x  Z, x*x = y.
a. True, because (for example) 5*5=25.
b. False, because (for example) no integer
multiplied by itself equals 3.
c. It depends on y, but given a value for y, we
could calculate a truth value.
d. It depends on y, but we may also need
5
additional information.
e. None of the above.
5
18
Defining a Predicate Using Expressions
with “Unbound” Variables
A pred. logic formula with only bound variables is a
proposition, something that is either true or false:
x  Z, x*x = 25.
true
x  Z, x*x = 3.
false
y  Z, x  Z, x*x = y. false
5
5
19
Defining a Predicate Using Expressions
with “Unbound” Variables
A pred. logic formula with unbound variables is itself a
predicate, something whose truth depends on its
unbound variables’ values:
PerfectSquare(y)  x  Z, x*x = y.
PerfectSquare(25).
PerfectSquare(3).
y  Z, PerfectSquare(y).
true
false
false
5
5
20
Unbound Variables Check
Which variable(s) does this formula’s truth
depend on?
i  Z+, (i > n)  ~v  Z0, Elt(a, i, v).
a.
b.
c.
d.
e.
i and v
a and n
n and v
i and n
None of these is correct.
21
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Prelude: Scope and Predicate Definition
• Problems and Discussion
– Liszt Etudes
– Sorted Lists
– Comparing Algorithms
• Next Lecture Notes
22
Defining Lists (aka Arrays)
Let Elt(a, i, v) be a predicate indicating that
list a has the integer value v at index i,
where indexes must be  1.
mylist
1
2
3
4
5
6
2
4
5
7
6
10
Elt(mylist, 3, 5) is true.
Elt(mylist, 2, 1) and Elt(mylist, 7, 2) are false.
23
List Element Warmup
Elt(a, i, v)  list a has value v at index i.
Which of the following could describe a valid
list (assume all other Elt(x, y, z) are false)?
a. Elt(list, 1, 7), Elt(list, 2, 4), Elt(list, 1, 3)
b. Elt(list, 1, 7), Elt(list, 2, 4), Elt(list, 4, 3)
c. Elt(list, -1, 7), Elt(list, 0, 4), Elt(list, 1, 3)
d. Nothing. (i  Z+, v  Z, ~Elt(list, i, v).)
e. None of these is a valid list.
24
List Element Warmup
Elt(a, i, v)  a has value v at index i. Let A be the set of all lists.
Which of these means: a list cannot have more than one
element at any index.
a. a  A, i  Z+, v1  Z,
Elt(a,i,v1)  ~v2  Z, Elt(a,i,v2).
b. a  A, i  Z+, v1  Z, v2  Z,
Elt(a,i,v1)  Elt(a,i,v2).
c. a  A, i  Z+, v1  Z, v2  Z,
Elt(a,i,v1)  Elt(a,i,v2)  v1 = v2.
d.
e.
Both a and c.
All of these
We will assume henceforth that a list cannot
25
have more than one element at any index.
List Length Warmup
Elt(a, i, v)  a has value v at index i.
Let Length(a, n) mean list a is n items long.
What does a  A, n  Z0, Length(a, n)
mean?
a. Every list is n items long.
b. There are many lists that are n items long.
c. Every list has a length.
d. No list has more than one length.
e. None of these.
26
We will assume henceforth that this statement is true.
List Length Properties
Which of the following should not be true of the
length of a list?
a. Every list should have a length.
b. No list should have more than one length.
c. No list should have elements at any index
larger than its length.
d. A list should have an element at every index
up to its length.
e. All of these should be true.
27
No More than One Length
Which of the following means that a list should not have
more than one length?
a. a  A, n1  Z0, Length(a,n1) 
~n2  Z0, n1  n2  Length(a,n2).
b. a  A, n1  Z0, n2  Z0,
n1  n2  Length(a,n1)  Length(a,n2).
c. a  A, n1  Z0, n2  Z0,
Length(a,n1)  Length(a,n2)  n1 = n2.
d. Both a and c.
e. All of these
28
Defining Length(a, n)
Consider: Length(a,n)  i  Z+,
(v  Z0, Elt(a, i, v))  (i  n).
In English: There’s an element at i if and only if i is a valid index
(no greater than the list’s length).
Why does this guarantee that no list has more than one length?
a. Because it states that no list has more than one length.
b. Because a list with two lengths would now both have and not
have a value at some entry.
c. Because i now cannot be greater than the length of the list.
d. None of these, but it is guaranteed.
e. None of these, and it’s not guaranteed.
29
We will assume henceforth that this statement is true.
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Prelude: Scope and Predicate Definition
• Problems and Discussion
– Liszt Etudes
– Sorted Lists
– Comparing Algorithms
• Next Lecture Notes
30
Problem: Sorted Lists
Problem: Give a definition for the predicate
Sorted(a) in terms of Elt(a, i, v).
Assume lists cannot have more than one
element at an index,
?
every list has a length,
and lists are “filled” to
their length.
31
?
Problem: Sorted Lists
Problem: Give a definition for the predicate
Sorted(a) in terms of Elt(a, i, v).
Which of the following is a problem with this definition?
Sorted(a)  i  Z+, Elt(a,i,v1) < Elt(a,i+1,v2).
a.
b.
c.
d.
e.
a isn’t quantified.
v1 and v2 aren’t quantified.
We can’t use < on Elt (or any other predicate)
a and b
b and c
32
?
Problem: Sorted Lists
Problem: Give a definition for the predicate
Sorted(a) in terms of Elt(a, i, v).
What’s wrong with the following definition?
Sorted(a)  i  Z+, v1  Z, v2  Z,
Elt(a,i,v1)  Elt(a,i+1,v2)  v1 < v2.
a.
b.
c.
d.
e.
It’s missing quantifiers.
It’s too restrictive (e.g., for equal values).
It doesn’t handle the “boundary case” when i=length.
a and b
b and c
33
But... Would You Ever Really
Use Pred Logic Like This?
• Data Structures Example: “...every key is less than
or equal to all of its children’s keys...”
• AI example: “...let h' be a “heuristic” function
evaluating game states and h be the true value of
the state. For all nodes n, h'(n)  h(n)...”
• Java example: “...there is no path via references
from any variable in scope to any memory location
available for garbage collection...”
• Economics/elections example: “...for any distinct
pair of candidates c1 and c2, if all voters prefer c1
to c2, then society must rank c1 above c2...”
34
Predicate Logic Patterns
“There exists” means “there’s at least one”. We
often want “there’s exactly one” (e.g., lists have
exactly one element at each valid index).
Common problems like this lead to common patterns
to solve them.
• “At least one” plus “at most one” means “exactly one”.
• “At most one” is: if any two items have this property,
then they’re really the same item.
• “Two distinct” or “at least two” is: exists one, exists
another, such that the first is not equal to the second
35
Intuition ♥ Formality
“...when we become comfortable with formal
manipulations, we can use them to check our
intuition, and then we can use our intuition to
check our formal manipulations.”
- Epp (3rd ed), p. 106-107
We’ll often use predicate logic informally in the
future, but the ability to express and reason
about ideas formally keeps us honest and helps
us discover points we may overlook otherwise.
36
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Prelude: Scope and Predicate Definition
• Problems and Discussion
– Liszt Etudes
– Sorted Lists
– Comparing Algorithms
• Next Lecture Notes
37
Efficiency of Algorithms
Let’s say each student is in a “MUG” (1st
year orientation group). For each of their
MUG-mates, each student has a list of all
of their classes.
Assume each MUG has 13 students and
each student is taking 5 classes.
I want to determine how many
students in my class have a
MUG-mate in my class.
38
Which algorithm is
generally faster?
(a) Ask each student for the list of their MUGmates’ classes, and check for each class
whether it is CPSC 121. If the answer is ever
yes, include the student in my count.
(b) For each student s1 in the class, ask the
student for each other student s2 in the class
whether s2 is a MUG-mate. If the answer is
ever yes, include s1 in my count.
(c) Neither.
39
Concrete Examples:
10 students
Say checking if a class on a list is CPSC
121 takes 1 second and checking if a
classmate is in your MUG takes 1 second.
Algorithm (a) takes ~10*12*5 seconds = 10 minutes.
Algorithm (b) takes ~10*10 seconds < 2 minutes.
40
Concrete Examples:
100 students
Say checking if a class on a list is CPSC
121 takes 1 second and checking if a
classmate is in your MUG takes 1 second.
Algorithm (a) takes ~100*12*5 seconds = 100 minutes.
Algorithm (b) takes ~100*100 seconds  167 minutes.
41
Concrete Examples:
400 students
Say checking if a class on a list is CPSC
121 takes 1 second and checking if a
classmate is in your MUG takes 1 second.
Algorithm (a) takes ~400*12*5 seconds  7 hours.
Algorithm (b) takes ~400*400 seconds  44 hours.
42
Which algorithm is
generally faster?
(a) Ask each student for the list of their MUGmates’ classes, and check for each class
whether it is CPSC 121. If the answer is ever
yes, include the student in my count.
(b) For each student s1 in the class, ask the
student for each other student s2 in the class
whether s2 is a MUG-mate. If the answer is
ever yes, include s1 in my count.
(c) Neither.
43
Comparing at One Input Size
Let the predicate Faster(a1, a2, n)
mean algorithm a1 is faster than algorithm
a2 on a problem of size n, where n is a
positive integer.
time
Alg A
Alg B
problem size
44
We’ll assume the Faster predicate is given to us.
How Faster Works (1 of 3)
a1 is faster than a2 at size n.
Which of the following means “no algorithm is
ever faster than itself”?
Faster(a1,a2,n):
a. n  Z+, a  A, ~Faster(a,a,n).
b. n  Z+, a1  A, a2  A,
Faster(a1,a2,n)  ~Faster(a2,a1,n).
c. n  Z+, a1  A, a2  A, a3  A,
Faster(a1,a2,n)  Faster(a2,a3,n) 
Faster(a1,a3,n).
d.
None
of
these.
time
Alg A
Alg B
problem size
45
We will assume this statement is true.
How Faster Works (2 of 3)
Faster(a1,a2,n):
a1 is faster than a2 at size n.
Which of the following means “two algorithms cannot be
faster than each other”?
a. n  Z+, a1  A,
Faster(a1,a2,n)
b. n  Z+, a1  A,
Faster(a1,a2,n)
c. n  Z+, a1  A,
Faster(a1,a2,n)
a2  A,
 ~Faster(a2,a1,n).
a2  A,
 ~Faster(a2,a1,n).
a2  A,
 Faster(a2,a1,n).
d. b and c.
e. All of these.
time
Alg A
Alg B
problem size
46
We will assume this statement is true.
How Faster Works (3 of 3)
Faster(a1,a2,n):
a1 is faster than a2 at size n.
What does the following statement mean?
n  Z+, a1  A, a2  A, a3  A,
Faster(a1,a2,n)  Faster(a2,a3,n) 
Faster(a1,a3,n).
a.
b.
c.
d.
e.
time
Three algorithms cannot be faster than each other.
Three algorithms are the same “speed”.
Of any three algorithms, one is the fastest.
An algorithm cannot be faster than itself.
None of these.
Alg A
Alg B
problem size
47
We will assume this statement is true.
How Faster Works (4 of 3, bonus!)
Faster(a1,a2,n):
a1 is faster than a2 at size n.
We assume Faster is:
• “Anti-reflexive”: No algorithm is faster than itself.
• “Anti-symmetric”: No two algorithms are faster
than each other.
• “Transitive”: If one algorithm is faster than a
second, which is faster than a third, then the first
algorithm is faster than the third. (We can “chain”
fasters together, analagously to the transitivity
rule for propositional logic.)
time
Alg A
Alg B
problem size
48
General Efficiency of Algorithms
Faster(a1,a2,n): a1 is faster than a2 at size n.
Problem: Create a definition of
GenerallyFaster(a1, a2) in terms of
Faster(a1, a2, n) that you can live with.
time
Alg A
Alg B
problem size
49
Desirable Properties of
Generally Faster
Which of these properties should Generally
Faster not share with Faster?
a. Anti-reflexivity
b. Anti-symmetry
c. Transitivity
d. Should share all of these.
e. Should share none of these.
time
Alg A
Alg B
problem size
50
English-Language
Generally Faster Definitions
Which one do you want? [Your definitions here.]
a.
b.
c.
time
d. None of these.
Alg A
51
Alg B
Formal
Generally Faster Definitions
Which one do you want? [Your definitions here.]
a.
b.
c.
time
d. None of these.
Alg A
52
Alg B
Which algorithm is
generally faster?
(a) Ask each student for the list of their MUGmates’ classes, and check for each class
whether it is CPSC 121. If the answer is ever
yes, include the student in my count.
(b) For each student s1 in the class, ask the
student for each other student s2 in the class
whether s2 is a MUG-mate. If the answer is
ever yes, include s1 in my count.
(c) Neither.
time
Alg A
53
Alg B
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Prelude: Scope and Predicate Definition
• Problems and Discussion
– Liszt Etudes
– Sorted Lists
– Comparing Algorithms
• Next Lecture Notes
54
Learning Goals: In-Class
By the start of class, you should be able to:
– Build statements about the relationships
between properties of various objects—which
may be real-world like “every candidate got
votes from at least two people in every
province” or computing related like “on the ith
repetition of this algorithm, the variable min
contains the smallest element in the list
between element 0 and element i”) using
predicate logic.
55
Next Lecture Learning Goals:
Pre-Class
By the start of class, you should be able to:
– Determine the negation of a quantified statement in logical
notation as well as English. (But, feel free to entirely or
partially translate English to logic in the process!)
– Given a quantified statement and an equivalence rule,
apply the rule to create an equivalent statement
(particularly the De Morgan’s and contrapositive rules).
– Prove and disprove quantified statements using the
“challenge” method (Epp, 3rd ed 98-99, Epp, 4th ed 118119).
– Apply universal instantiation, universal modus ponens, and
universal modus tollens to predicate logic statements that
correspond to the rules’ premises to infer statements
implied by the premises.
56
Next Lecture Prerequisites
Reread Sections 3.1 and 3.3 (including the
negation part that we skipped previously).
Read Sections 3.2 and 3.4.
(See course website for other texts’ sections.)
(You needn’t learn the “diagram” technique, but
it may make more sense than other
explanations!)
Complete the open-book, untimed quiz on Vista
that’s due before the next class.
57
snick

snack
More problems to solve...
(on your own or if we have time)
58
Problem: Java Collections
Problem: Translate the following text from
the Java 1.6.0 API page for the Collection
interface into predicate logic.
[T]he specification for the
contains(Object o) method says: "returns
true if and only if this collection
contains at least one element e such that
(o==null ? e==null : o.equals(e))."
c ? a : b acts essentially like a multiplexer.
59
If c is true, it evaluates to a; otherwise, it evaluates to b.
Problem: Java Collections
Problem: The API goes on to say:
This specification should not be construed
to imply that invoking Collection.contains
with a non-null argument o will cause
o.equals(e) to be invoked for any element
e.
Explain whether and how this is consistent
with your definition.
60
More Quantifier Examples
Someone is in charge.
Everyone except the person in charge
reports to someone else.
61
More Quantifier Examples
n is a prime number.
Note: we use x|y as a predicate meaning x divides y
62
(i.e., x “goes into” y with no remainder).
More Quantifier Examples
n is a prime number.
Let’s define a new predicate P(x) in terms
of this “clause”. Then, let’s express…
There’s some prime number larger than 10.
There’s some prime number larger than
every natural number.
63
Yet More Examples
Eating food causes
Alice to grow or
shrink.
F = set of all foods
E(x): Alice eats x
g: Alice grows
s: Alice shrinks
Solution:
x  F, E(x)  g  s.
64
Yet More Examples
Alice shrank when she
ate some food.
F = set of all foods
E(x): Alice eats x
g: Alice grows
s: Alice shrinks
Solution:
x  F, E(x)  s.
65
Yet More Examples
All lions are fierce.
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
Domain for all is the set of
all creatures.
Solution:
x  F, L(x)  F(x).
66
Yet More Examples
Some lions do not drink
coffee.
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
Domain for all is the set of
all creatures.
67
Yet More Examples
All fierce creatures are
not lions.
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
Domain for all is the set of
all creatures.
68
Is that English sentence ambiguous?
Yet More Examples
Is x, K(x, y) a
proposition?
Why or why not?
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
K(x, y): x has been in
y’s kitchen
Domain for all is the set of
all creatures (with
kitchens?).
69
Yet More Examples
Every creature has
been in its own
kitchen.
Some creature has not
been in its own
kitchen.
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
K(x, y): x has been in
y’s kitchen
Domain for all is the set of
all creatures (with
kitchens?).
70
Yet More Examples
There is a creature that
has been in every
creature’s kitchen.
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
K(x, y): x has been in
y’s kitchen
Every creature’s kitchen
has had some
Domain for all is the set of
creature in it.
all creatures (with
kitchens?).
71
Are these the same?
Yet More Examples
Every creature has
been in every other
creature’s kitchen.
F(x): x is a fierce creature
L(x): x is a lion
C(x): x drinks coffee
K(x, y): x has been in
Every creature’s kitchen
y’s kitchen
has had every
creature in it.
Domain for all is the set of
all creatures (with
kitchens?).
Are these the same?
72
What if we removed the word “other”?
Problem: Voting Database
Consider a database that tracks the votes in
an election. In the database, the predicate
Tally(d, c, n) means that district d reported
that candidate c received n votes, where n
is an integer  0.
Problem: Define a predicate GotVote(c) in
terms of Tally whose truth set is the set of
all candidates who received at least one
vote.
73
Problem: Voting Database
Problem: Define a predicate whose truth set
is the set of all candidates who won at
least one district.
Why work so hard on defining predicates?
74
This is essentially how we query databases.
Problem: Voting Database
Let’s assume that every candidate has
exactly one vote total for every district.
That is, there’s no missing and no
duplicate data.
Problem: Write a logical statement that
describes this constraint.
Predicates are clumsy for expressing this common idea.
75
Later, we’ll use functions to do a better job.
Problem: Voting Database
Let Winner(c) indicate that candidate c is the
winner of the election.
Problem: Write a logical statement that
means that the winner of the election must
have received at least one vote.
76
Problem: Voting Database
Let D be the set of all districts and C be the
set of all candidates.
Problem: Determine what the following
statement means, whether it is necessarily
true (no matter what the actual vote tallies
are), and justify your stance:
c1C, dD, c2C, Winner(c1)  c1c2 
n1Z, n2Z, Tally(d,n1,c1)  Tally(d,n2,c2)  n1>n2
77