Lecture 4: NFAs in Foundations of Computation

1. Lecture 4NFAs CSCE 355 Foundations of Computation Topics: • Induction review • DFAs again • NFAs • Delta, delta hat • Strings accepted, languages accepted Sept 8, 2008

2. Induction Revisited

3. Induction HW

4. Common Mistakes in HW

5. Induction Pseudo Pop Quiz

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

9. Pigeon Hole Principle application

10. Homework from Email

11. 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 }

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

13. # 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(//)

14. # 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

15. Example 2.4

16. Exercise 2.2.4

17. DFA that accepts Union

18. Nondeterministic Finite Automata