Formal Methods
DAVID BOWIE • 1947 - 2016
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Dr. Barry Wittman
Not Dr. Barry Whitman
Education:
 PhD and MS in Computer Science, Purdue University
 BS in Computer Science, Morehouse College
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Hobbies:
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Reading, writing
Enjoying ethnic cuisine
DJing
Lockpicking
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E-mail:
wittmanb@etown.edu
Office:
Esbenshade 284B
Phone:
(717) 361-4761
Office hours: MWF 11:00am – 12:00pm
MF 3:30 – 5:00pm
T
1:00 – 3:00pm
And by appointment
Website:
http://users.etown.edu/w/wittmanb/
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Susanna S. Epp
Discrete Mathematics with Applications
4th Edition, 2010, Brooks Cole
ISBN-10: 0495391328
ISBN-13: 978-0495391326
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You are expected to read the material before
class
If you're not prepared, you will be asked to
leave
 You will forfeit the opportunity to take quizzes
 Much more importantly, you will forfeit the
education you have paid around $100 per class
meeting to get
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It’s mostly discrete math
 Meaning math that is not continuous
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It is not discreet math
 Meaning math that won’t tell people what you did
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There are certain kinds of math that are really
beneficial to CS
We have collected a big chunk of these and
put them in this course
It’s a grab bag
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Logic
Proofs
Basic number theory
Mathematical induction
Set theory
Functions and relations
Counting and probability
Graphs and trees
Regular expressions and finite automata
Running time
Formal languages and grammars
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For more information, visit the webpage:
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The webpage will contain:
http://users.etown.edu/w/wittmanb/cs322
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The most current schedule
Notes available for download
Reminders about exams and homework
Syllabus (you can request a printed copy if you like)
Detailed policies and guidelines
Piazza will allow for discussion and questions about
assignments:
https://piazza.com/etown/spring2016/cs322/
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30% of your grade will be ten equally
weighted homework assignments
Each will focus on a different set of topics
from the course
All homework is to be done individually
I am available for assistance during office
hours, through Piazza, and through e-mail
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Homework assignments must be turned in by
saving them in your class folder (J:\SP20152016\CS322A) before the deadline
Do not put assignments in your public
directories
Late homework will not be accepted
Paper copies of homework will not be accepted
Each homework done in LaTeX will earn 0.5%
extra credit toward the final semester grade
Doing every homework in LaTeX will raise your
final grade by 5% (half a letter grade)
5% of your grade will be lectures that you give
You will give roughly two of these lectures at any
time during the semester, without any warning
 You will have 3 to 5 minutes in which you must
present the material to be read for that day
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 Students are encouraged to ask questions
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There is no better way to learn material than by
teaching it
Polishing public speaking skills is never a bad
thing
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5% of your grade will be pop quizzes
These quizzes will be based on material
covered in the previous one or two lectures
They will be graded leniently
They are useful for these reasons:
1. Informing me of your understanding
2. Feedback to you about your understanding
3. Easy points for you
4. Attendance
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There will be three equally weighted in-class
exams totaling 45% of your final grade
 Exam 1:
 Exam 2:
 Exam 3:
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2/08/2016
3/11/2016
4/11/2016
The final exam will be worth 15% of your
grade
 Final:
2:30 – 5:30pm
5/05/2016
Week
Starting
Topics
Chapters
1
01/11/16
Propositional logic
2
2
01/18/16
Predicate logic
3
3
01/25/16
Proofs
4
4
02/01/16
Number theory
4
5
02/08/16
Mathematical induction
5
6
02/15/16
Set theory
6
7
02/22/16
Functions
7
02/29/16
Spring Break
8
03/07/16
Relations
8
9
03/14/16
Counting and probability
9
10
03/21/16
Counting and probability continued
9
11
03/28/16
Graphs and trees
10
12
04/04/16
Running time
11
13
04/11/16
Regular languages
12
14
04/18/16
Higher level languages
Handouts
15
04/25/16
Review
2 - 12 + Handouts
30%
• Ten homeworks
5%
• Impromptu student lectures
5%
• Quizzes
45%
• Three equally weighted midterm exams
15%
• Final exam
A
93-100
B-
80-82
D+
67-69
A-
90-92
C+
77-79
D
63-66
B+
87-89
C
73-76
D-
60-62
B
83-86
C-
70-72
F
0-59
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Don’t cheat
First offense:
 I will give you a zero for the assignment, then lower
your final letter grade for the course by one full grade
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Second offense:
 I will fail you for the course and try to kick you out of
Elizabethtown College
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Refer to the Student Handbook for the official
policy
Ask me if you have questions or concerns
Elizabethtown College welcomes otherwise qualified students with
disabilities to participate in all of its courses, programs, services, and
activities. If you have a documented disability and would like to
request accommodations in order to access course material, activities,
or requirements, please contact the Director of Disability Services,
Lynne Davies, by phone (361-1227) or e-mail daviesl@etown.edu. If
your documentation meets the college’s documentation guidelines,
you will be given a letter from Disability Services for each of your
professors. Students experiencing certain documented temporary
conditions, such as post-concussive symptoms, may also qualify for
temporary academic accommodations and adjustments. As early as
possible in the semester, set up an appointment to meet with me, the
instructor, to discuss the academic adjustments specified in your
accommodations letter as they pertain to my class.
Consider three boxes A, B, and C
One contains gold, but the other two are empty
Each box has a message printed on it
Two of the messages are lies, and one is telling the
truth
 Which box has the gold?
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The gold is not here.
A
The gold is not here.
B
The gold is in Box B.
C
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On a given island, everyone is either a Knight
or a Knave
Knights always tell the truth, and Knaves
always lie
Imagine that I meet two inhabitants of this
island and ask, "Is either of you a Knight?"
Given his response, I know the answer to my
question
Are the inhabitants in question Knights or
Knaves?
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Politicians lie.
 True statement!
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Cast iron sinks.
 True statement!
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Politicians lie in cast iron sinks.
 Absurd statement!
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Propositional logic is the logic that governs
statements
A statement is either true or false (but
nothing else!)
We want to combine them, infer things about
them, prove them true or false
First, we have to learn their rules
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"The moon is made of green cheese" is a statement,
that is, something that is either true or false
It takes a long time to write "The moon is made of
green cheese"
Mathematicians are lazy, and so they let a variable
represent this statement
p and q are common choices for propositional logic
So, we can use p in place of "The moon is made of
green cheese"
Similarly, we can use q in place of "The earth is made
of rye bread"
Like programming,
combining two values
with AND will be true only
if both values are true
 Using OR makes the result
true if either is true
 NOT changes a true to
false and a false to true
 Mathematicians use their
own symbols for AND, OR,
and NOT
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Operation
Symbol
Example
AND

pq
OR

pq
NOT
~
~p
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To better understand an operation, we can
make a truth table, giving all possible input
values and the corresponding output values
This truth table is for p  q
p
T
T
F
F
q
T
F
T
F
pq
T
T
T
F
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What’s the truth table for q  p?
Consider: (p  q)  ~(p  q)
 What’s its truth table?
 What’s its meaning?
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What’s the truth table for p  q  r?
How many lines are in a truth table with n
symbols?
How many different truth tables are possible
for two input symbols?
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Two different statements can be written
differently and yet be logically equivalent
 p  ~(~p)
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Make a truth table
 If all outputs match up, the statements are
logically equivalent
 If even one output doesn’t match, the statements
are not equivalent
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What’s an expression that logically equivalent
to ~(p  q) ?
What about logically equivalent to ~(p  q) ?
De Morgan’s Laws state:
 ~(p  q)  ~p  ~q
 ~(p  q)  ~p  ~q
 Essentially, the negation flips an AND to an OR
and vice versa
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You can construct all possible outputs using
combinations of AND, OR, and NOT
But, sometimes it’s useful to introduce notation
for common operations
This truth table is for p  q
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
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We use  to represent an if-then statement
Let p be "The moon is made of green cheese"
Let q be "The earth is made of rye bread"
Thus, p  q is how a logician would write:
 If the moon is made of green cheese, then the
earth is made of rye bread
 Here, p is called the hypothesis and q is called the
conclusion
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What other combination of p and q is logically
equivalent to p  q ?
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More on implications
Validity of arguments
 Tautologies and contradictions
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Rules of inference
Examples with digital circuits
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Read Chapter 2