1. CS314 – Section 5 Recitation 3
Long Zhao
DFA, NFA
Context Free Grammars
Slides available at

2. Finite-State Automaton
A Finite-State Automaton is a quadruple:
<S, s, F, T>
S is a set of states
s is the start state
F is a set of final states
T is a set of labeled transitions, of the form (state, input) => state

3. DFA & NFA
DFA - Determistic Finite Automaton: At most one transition for a state and an input symbol. (Recognizers should be a DFA.)
NFA - Nondeterministic Finite Automaton: More than one transition possible for a state and an input symbol.

4. Constructing a DFA from a RE
Regular Expression (RE) →NFA with ϵ moves
NFA with ϵ moves to NFA
NFA→DFA

5. NFA with ϵ moves (5 rules)B AB

6. NFA with ϵ moves (5 rules)
A|B B ϵ ϵ ϵ ϵ A ϵ A* ϵ

7. NFA with ϵ moves (Examples)
(A|B)C A ϵ ϵ ϵ C B ϵ ϵ

8. NFA with ϵ moves (Examples)
A|BC A ϵ ϵ ϵ B C ϵ ϵ

9. NFA with ϵ moves (Examples)
((A|B)C)* ϵ A ϵ ϵ ϵ ϵ C ϵ ϵ B ϵ ϵ

10. NFA with ϵ moves (Examples)
((A|B)C)* ϵ A C B ϵ

11. NFA with ϵ moves to NFA ((A|B)C)* 1 2 3 A B C ϵ* (->)1 2 - 1,3 3

12. NFA with ϵ moves to NFA ((A|B)C)* 1 2 3 ϵ*Aϵ* ϵ*Bϵ* ϵ*Cϵ* (->|O)1 2
Since ϵ*(1) is {1, 3} which contains the final state 3, state 1 should be a final state in the new NFA as well.
((A|B)C)*
ϵ*Aϵ*
ϵ*Bϵ*
ϵ*Cϵ*
(->|O)1 2 -
1,3
(O)3 ϵ
A,B C 1 2 3 ϵ

13. NFA with ϵ moves to NFA ((A|B)C)* 1 2 3 ϵ*Aϵ* ϵ*Bϵ* ϵ*Cϵ* (->|O)1 2
1,3
(O)3 C
A,B
A,B 1 2 3 C

14. NFA to DFA ϵ*Aϵ* ϵ*Bϵ* ϵ*Cϵ* 2 - 1,3 (O)3 A B C 2 - 1,3 (O)1,3

15. NFA to DFA
((A|B)C)* A B C
(->|O)1 2 -
1,3
(O)1,3
A,B
A,B 1 2
1,3 C

16. NFA to DFA All strings contain {a, b} and end with an ‘a’. RE: (a|b)*a1 2
a,b

17. NFA to DFA
(a|b)*a a b
(->)1
1,2 1
(O)2 - a 1 2
a,b

18. NFA to DFA
(a|b)*a a b
(->)1
1,2 1
(O)2 - a b
(->)1
1,2 1
(O)1,2

19. NFA to DFA
(a|b)*a a b
(->)1
1,2 1
(O)1,2 a 1
1,2 b b a

20. NFA to DFA (Example) 2 ϵ a a b
NFA without ϵ
move? DFA? 1 3 c c 4 ϵ

21. NFA to DFA (Example) 2 1 3 4 a b c ϵ* (->)1 2 - 4 1 3 1,2 (O)3 3,4

22. NFA to DFA (Example) 2 1 3 4 ϵ*aϵ* ϵ*bϵ* ϵ*cϵ* (->)1 1,2 - 3,4 2 3

23. NFA to DFA (Example) 2 1 3 4 a ϵ*aϵ* ϵ*bϵ* ϵ*cϵ* (->)1 1,2 - 3,4 2
NFA without ϵ move
NFA to DFA (Example) a 2
ϵ*aϵ*
ϵ*bϵ*
ϵ*cϵ*
(->)1
1,2 -
3,4 2 3
(O)3
(O)4 a a
b,c c a a c 1 3 a a a c c 4

24. NFA to DFA (Example) a b c (->)1 1,2 - 3,4 2 3 (O)3 (O)4 a b c

25. NFA to DFA (Example) 1 3 a b c (->)1 1,2 - 3,4 3 (O)3,4 (O)3 a a a

26. Context-Free Grammars
A formalism for describing languages
CFGs are a quadruple < T,N,P,S >:
A set T of terminal symbols (tokens)
A set N of nonterminal symbols
A set P production rules
A special start symbol S
BNF is a notation for describing CFGs.

27. Simple BNF Grammar Terminals letters, digits, :=
Nonterminals <letter> <digit> <stmt> <identifier>
Productions
<letter> ::= A | B | C | ...| Z
<digit> ::= 0 | 1 | 2 | ...| 9
<identifier> ::= <letter> |<identifier> <letter> |<identifier> <digit>
<stmt> ::= <identifier> := 0
Start Symbol <stmt>

28. Regular Grammars
CFGs with restrictions on the shapes of production rules.
Left-linear:
<N> ::= <X> a b
<X> ::= a | <X> b
Right-linear:
N ::= b | b <Y>
Y ::= a b | a b <Y>

29. Context-Free Grammars
Give the context-free grammar in BNF notation that generates the following language:
0 𝑛 1 𝑛 𝑛≥0}, with alphabet {0, 1}
𝑎 𝑚 𝑏 𝑐 𝑛 𝑚,𝑛≥0}, with alphabet {a, b, c}
All strings of the form 0 𝑎 1 𝑏 0 𝑐 such that a + c = b.

30. Context-Free Grammars
0 𝑛 1 𝑛 𝑛≥0}, with alphabet {0, 1}
Terminals 0, 1, ϵ
Nonterminals <S>
Productions
<S> ::= 0<S>1
<S> ::= ϵ
Start Symbol <S>

31. Context-Free Grammars
𝑎 𝑚 𝑏 𝑐 𝑛 𝑚,𝑛≥0}, with alphabet {a, b, c}
The regular expression a*bc*
Terminals a, b, c, ϵ
Nonterminals <S> <A> <C>
Productions
<S> ::= <A>
<A> ::= a<A>|b<C>
<C> ::= c<C>|ϵ
Start Symbol <S>

32. Context-Free Grammars
All strings of the form 0 𝑎 1 𝑏 0 𝑐 such that a + c = b.
Terminals 0, 1, ϵ
Nonterminals <S> <T> <U>
Productions
<S> ::= <T><U>
<T> ::= 0<T>1|ϵ
<U> ::= 1<U>0|ϵ
Start Symbol <S>