Formal Methods

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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  Hobbies:  Reading, writing  Enjoying ethnic cuisine  DJing  Lockpicking

  Office:Esbenshade 284B  Phone:(717)  Office hours: MWF 11:00am – 12:00pm MF 3:30 – 5:00pm T 1:00 – 3:00pm And by appointment  Website:

 Susanna S. Epp  Discrete Mathematics with Applications  4 th Edition, 2010, Brooks Cole  ISBN-10:  ISBN-13:

 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

 It’s mostly discrete math  Meaning math that is not continuous  It is not discreet math  Meaning math that won’t tell people what you did  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

 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

 For more information, visit the webpage:  The webpage will contain:  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:

 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

 Homework assignments must be turned in by saving them in your class folder ( J:\SP \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  Students are encouraged to ask questions  There is no better way to learn material than by teaching it  Polishing public speaking skills is never a bad thing

 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

 There will be three equally weighted in-class exams totaling 45% of your final grade  Exam 1:2/08/2016  Exam 2:3/11/2016  Exam 3:4/11/2016  The final exam will be worth 15% of your grade  Final:2:30 – 5:30pm 5/05/2016

WeekStartingTopicsChapters 101/11/16Propositional logic2 201/18/16Predicate logic3 301/25/16Proofs4 402/01/16Number theory4 502/08/16Mathematical induction5 602/15/16Set theory6 702/22/16Functions7 02/29/16Spring Break 803/07/16Relations8 903/14/16Counting and probability9 1003/21/16Counting and probability continued9 1103/28/16Graphs and trees /04/16Running time /11/16Regular languages /18/16Higher level languagesHandouts 1504/25/16Review Handouts

30% Ten homeworks 5% Impromptu student lectures 5% Quizzes 45% Three equally weighted midterm exams 15% Final exam

 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  Second offense:  I will fail you for the course and try to kick you out of Elizabethtown College  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 ( ) or 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? The gold is not here. The gold is in Box B. ABC

 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?

 Politicians lie.  True statement!  Cast iron sinks.  True statement!  Politicians lie in cast iron sinks.  Absurd statement!

 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

 "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 OperationSymbolExample AND  p  q OR  p  q NOT~~p~p

 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 pq p  qp  q TTT TFT FTT FFF

 What’s the truth table for q  p?  Consider: (p  q)  ~(p  q)  What’s its truth table?  What’s its meaning?  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?

 Two different statements can be written differently and yet be logically equivalent  p  ~(~p)  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

 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

 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 pq p  qp  q TTT TFF FTT FFT

 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  What other combination of p and q is logically equivalent to p  q ?

 More on implications  Validity of arguments  Tautologies and contradictions  Rules of inference  Examples with digital circuits

 Read Chapter 2