1. CSCE 355 Foundations of Computation
Lecture 2 Overview
Topics
What I forgot from last lecture
Proof techniques continued
Alphabets, strings, languages
Automata
Aug 27, 2007

2. What I forgot from last lecture
Slides skipped or not well covered:
3 – Course outcomes
8 – Examples of Relations
9 – POSETS
10 – Equivalence relations
15 – Equality of Sets
17 – If and only if
25 – Structural Induction Proof

4. 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
G = E + F
G = E * F
G = ( E )

5. Graphs – Visual representation of relations (binary)
a R b if and only if ab in the graph

6. 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.

7. Mutual Inductions
Example On-Off pushbutton automaton

8. Languages Alphabet – a finite set of symbols
String – finite sequence of characters from an alphabet
Empty string, length of string
Language (over an alphabet)

9. Operations on Strings

10. Examples of Languages

11. Operations on Languages
Suppose S and T are languages (sets of strings)
Union, intersection, complement
concatenation

12. Powers, Kleene Closure S1 = S Sn = S Sn-1
How would you prove Sn Sm = Sn+m ?
What is S0? S*

13. Some Special Languages

14. Finite Automata - Informally

15. Finite Automata - formally
A Deterministic Finite Automata (DFA) is a 5-tuple

16. Transition Diagrams; Transition Tables
Conventions
Notes
Number of out-arcs
Dead state

17. Path determined by a string

18. Language accepted by a DFA

19. Example L(M) for DFA M

20. Given L find DFA for it

21. Important application of Pigeon Hole Principle to DFAs

22. Regular Expressions

23. Homework
What’s wrong with Ullman’s proof of “if a complete binary tree has n leaves then it has 2n-1 nodes.” (extra credit)
Operations on Strings Given strings s = abc and t=12
What are st, s2, and s3?
Is ε (empty string) in every language?
Is ϕ a language? It is a sublanguage of every language?
Operations on Languages. Given S = {a, b, ab} and T = {1, 22}
What is ST?
What is TS?
What is T2? T3?

24. 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

25. References– Mathematical Foundations
- The website for the textbook
Extended “Proof” techniques
Fair Use Books Online
Books
Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills