CS314 – Section 5 Recitation 3 Long Zhao (lz311@rutgers.edu) DFA, NFA Context Free Grammars Slides available at http://www.ilab.rutgers.edu/~lz311/CS314

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

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.

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

NFA with ϵ moves (5 rules) B AB

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

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

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

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

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

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

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

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

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

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

NFA to DFA All strings contain {a, b} and end with an ‘a’. RE: (a|b)*a 1 2 a,b

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

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

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

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

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

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

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

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

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

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.

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>

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>

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.

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>

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>

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>