1. Lecture 2 Overview Topics What I forgot from last lecture Proof techniques continued Alphabets, strings, languages Automata June 2, 2015 CSCE 355 Foundations of Computation

2. – 2 – CSCE 355 Summer 2015 http://infolab.stanford.edu/~ullm an/ialc/slides/slides1.pdf

3. – 3 – CSCE 355 Summer 2015 Recursive Def of Arithmetic Expressions  Basis: a number or a variable is an expression.  If E and F are expressions then a new expression G can be formed by applying one of the three rules 1.G = E + F 2.G = E * F 3.G = ( E )

4. – 4 – CSCE 355 Summer 2015 Graphs – Visual representation of relations (binary)  a R b if and only if a  b in the graph

5. – 5 – CSCE 355 Summer 2015 The Pigeon Hole Principle  Proof techniques Continued  If you have n boxes and more than n balls to put in the boxes then you must be two balls (at least two) in the same box.  Formally If A and B are sets with |A| > |B| then there is no 1-1 function from A to B.

6. – 6 – CSCE 355 Summer 2015 Mutual Inductions  Example 1.23 On-Off pushbutton automaton

7. – 7 – CSCE 355 Summer 2015 Languages  Alphabet – a finite set of symbols  String – finite sequence of characters from an alphabet  Empty string, length of string  Language (over an alphabet)

8. – 8 – CSCE 355 Summer 2015 Operations on Strings

9. – 9 – CSCE 355 Summer 2015 Examples of Languages

10. – 10 – CSCE 355 Summer 2015 Operations on Languages Suppose S and T are languages (sets of strings)  Union, intersection, complement  concatenation

11. – 11 – CSCE 355 Summer 2015 Powers, Kleene Closure  S 1 = S  S n = S S n-1  How would you prove S n S m = S n+m ?  What is S 0 ?  S *

12. – 12 – CSCE 355 Summer 2015 Some Special Languages

13. – 13 – CSCE 355 Summer 2015 Finite Automata - Informally

14. – 14 – CSCE 355 Summer 2015 Finite Automata - formally  A Deterministic Finite Automata (DFA) is a 5-tuple

15. – 15 – CSCE 355 Summer 2015 Transition Diagrams; Transition Tables ConventionsNotes 1.Number of out-arcs 2.Dead state

16. – 16 – CSCE 355 Summer 2015 Path determined by a string

17. – 17 – CSCE 355 Summer 2015 Language accepted by a DFA

18. – 18 – CSCE 355 Summer 2015 Example L(M) for DFA M

19. – 19 – CSCE 355 Summer 2015 Given L find DFA for it

20. – 20 – CSCE 355 Summer 2015 Important application of Pigeon Hole Principle to DFAs

21. – 21 – CSCE 355 Summer 2015 Regular Expressions

22. – 22 – CSCE 355 Summer 2015 Homework 1.What’s wrong with Ullman’s proof of “if a complete binary tree has n leaves then it has 2n- 1 nodes.” (extra credit) 2.Operations on Strings Given strings s = abc and t=12  What are st, s2, and s3?  Is ε (empty string) in every language? c. Is ϕ a language? It is a sublanguage of every language? 3.Operations on Languages. Given S = {a, b, ab} and T = {1, 22} a. What is ST? b. What is TS? c. What is T 2 ? T 3 ?

23. – 23 – CSCE 355 Summer 2015 4.DFA recognizing the following languages  {w in {a,b}* | each a is immediately preceded by a b}  {w in {a,b}* | neither aa nor bb is a substring of w } 5. What language does the DFA below recognize

24. – 24 – CSCE 355 Summer 2015 References– Mathematical Foundations  http://infolab.stanford.edu/~ullman/ialc.html - The website for the textbook http://infolab.stanford.edu/~ullman/ialc.html Extended “Proof” techniques  http://www.maths.uwa.edu.au/~berwin/humour/invalid. proofs.html http://www.maths.uwa.edu.au/~berwin/humour/invalid. proofs.html http://www.maths.uwa.edu.au/~berwin/humour/invalid. proofs.html Fair Use Books Online  http://fair-use.org/bertrand-russell/the-principles-of- mathematics/ http://fair-use.org/bertrand-russell/the-principles-of- mathematics/ http://fair-use.org/bertrand-russell/the-principles-of- mathematics/Books Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills