1. CSCE 355 Foundations of Computation
Lecture 4 DFAs II
Topics:
Induction review
DFAs again
Delta, delta hat
Strings accepted, languages accepted
NFAs
June 4, 2015

2. Text site .

3. DFA Deterministic finite automata
Transition function δ(q, a) – really a function
δ : States x ∑  States,
henceforth Q= set of States and δ : Q x ∑  Q
To be explicit in a DFA for each state there is one and only one transition for each input
∑* =

4. Set Formers – Defining languages
{w ἐ ∑* | some restriction on the string w} or just
{w | some restriction on the string w} if ∑ is understood
Examples .
Variation with variables on the left
{x01y |
Examples that we will return to
{0n1n | n >= 1}
{0i1j | 0 <= i <= j}

5. Induction Pseudo Pop Quiz

6. Review DFA Delta hat Path determined by input DFA accepting a string
Language accepted by DFA

7. Pigeon Hole Principle application

8. DFA Simulation in Ruby
# !/usr/bin/ruby #MMM 9/5/2008 # Simulation/Implementation of a DFA in Ruby # the DFA M=(Q, Sigma, delta, q0, F) numstates = 4 # Q = [0, 1, 2, 3] alphabet = ['a', 'b'] # Sigma = {a, b} # delta implemented as function q0 = 0 # Start state is state 0 acceptingState = Array.new(4, 0) acceptingState[1] = 1 # F = { 1 }

9. def delta(state, inputChar) case when state == 0 when inputChar == 'a' then return 1; when inputChar == 'b' then return 3 end when state == 1 when inputChar == 'a' then return 2 when inputChar == 'b' then return 0 . else return 999

10. # Start Simulation state = 0 print "alphabet = #{alphabet}\n" print "acceptingState = #{acceptingState}\n" # print "Enter the input string: " line = gets puts "line=#{line}" inp = line.chomp.split(//)

11. # foreach x in inp inp.each { |x| break x if x =="\n" ###break x if x =='X' nextstate = delta(state, x) print "delta(=#{state}, char=#{x}) = #{nextstate}\n" state = nextstate } if acceptingState[state] == 1 then puts "Accept" else puts "Reject" end

12. DFA that accepts Union

13. Example 2.4
L = { w | w has both an even number of 0’s and an even number of 1’s}

14. DFA that accepts the Intersection

15. Exercise 2.2.4 A “*” on a exercise, what does this mean?
2.2.4 *a { w | w ends in 00}
Homework
2.2.4 b,c
2.2.2

16. Nondeterministic Finite Automata