1. Lecture 5 NFAs Topics: Induction review/PopQuiz NFAs Delta, delta hat Strings accepted, languages accepted Subset Construction June 5, 2015 CSCE 355 Foundations of Computation

2. – 2 – CSCE 355 Summer 2015  Last Time: Readings through 2.2 Induction Proofs format, Examples from HW, Pseudo Pop Quiz Review of Previous Lecture Ruby Simulation - dfa0.rb More Examples given DFA find L (Example 2.4) given L find DFA (Exercise 2.2.4) Given L1 and L2 and Machines M1 and M2 with L1 = L(M1) and L2 = L(M2) construct the DFA for L1 Union L2  New: Readings section 2.3 HW Hints and Induction example Review Induction DFA for Union, revisit Example 2.4 Pop Quiz Uses of Finite automata NFA Delta-hat, a string accepted by an NFA, the language accepted Subset construction converting NFA  equivalent DFA

3. – 3 – CSCE 355 Summer 2015 Induction Revisited

4. – 4 – CSCE 355 Summer 2015 Induction HW

5. – 5 – CSCE 355 Summer 2015 Induction Pseudo Pop Quiz

6. – 6 – CSCE 355 Summer 2015 Dfa.rb implementation glitch def delta(state, inputChar) case case when state == 0 when state == 0 case case when inputChar == 'a' then return 1; when inputChar == 'a' then return 1; when inputChar == 'b' then return 3 when inputChar == 'b' then return 3 end end when state == 1 when state == 1 case case when inputChar == 'a' then return 2 when inputChar == 'a' then return 2 when inputChar == 'b' then return 0 when inputChar == 'b' then return 0 end end... else return 999 else return 999 end end pacolet> ruby dfa0.rb Enter the input string: ababxgaaa line=ababxgaaa delta(=0, char=a) = 1 delta(=1, char=b) = 0 delta(=0, char=a) = 1 delta(=1, char=b) = 0 delta(=0, char=x) = delta(=, char=g) = 999 delta(=999, char=a) = 999 Reject

7. – 7 – CSCE 355 Summer 2015 # foreach x in inp inp.each { |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

8. – 8 – CSCE 355 Summer 2015 Example 2.4 revisited

9. – 9 – CSCE 355 Summer 2015 Uses of Finite Automata Recognize strings: lexical anlayzers Text processing Unix tools: lex, flex, grep

10. – 10 – CSCE 355 Summer 2015 Nondeterministic Finite Automata (NFAs)  Nondeterminism – nextstate function not “determined” δ could map a state and an input into several states or none These could be viewed as “choices” for the nextstateFormally  N = (Q, Σ, δ, q0, F) with everything defined as before except δ : Q x Σ  2 Q,

11. – 11 – CSCE 355 Summer 2015 Example: Tenth symbol from the right is ‘1’

12. – 12 – CSCE 355 Summer 2015 Example from Author’s website Process input 211

13. – 13 – CSCE 355 Summer 2015 Delta hat for NFAs Recursive definition Basis δ(q, ε ) = { q } Recursive assume w = xa Assume δ(q, x ) = { p 1, p 2, p 3 … p k } Then δ(q, x ) =

14. – 14 – CSCE 355 Summer 2015 String accepted by an NFA

15. – 15 – CSCE 355 Summer 2015 Language accepted by an NFA

16. – 16 – CSCE 355 Summer 2015 Mutual Induction proof L(N) = L L = { Third symbol from right is 1.} M Show L contains L(M) and show L(M) contains L Mutual Induction

17. – 17 – CSCE 355 Summer 2015 Subset Construction  Constructing an equivalent DFA from and NFA  What does equivalent mean?  Why?

18. – 18 – CSCE 355 Summer 2015 Subset example from Figure 2.9

19. – 19 – CSCE 355 Summer 2015 Subset Example from Author’s website slides2.pdf