Introduction to Languages, Grammars, and Automata, Study notes of Design and Analysis of Algorithms

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Three Basic Concepts

• Languages

• Grammars

• Automata

Languages

• (^) Alphabet: a finite and nonempty set of symbols  = {a, b}
• (^) String: finite sequence of symbols from  w = abaaa : empty string : the set of all strings on  (+^ = ^  {})

Languages

• (^) Example 1:  = {a, b} *^ = {, a, b, aa, ab, ba, aaa, ...} L 1 = {a, aa, aab} (finite language) L 2 = {anbn^ | n  0} = {, ab, aabb, ...}

Languages

• (^) Language concatenation: L 1 L 2 = {xy | xL 1 , yL 2 } Ln^ = L L ... L (n times) L^0 = {}

Languages

• (^) Example 2: L = {anbn^ | n  0} L^2 = {anbnambm^ | n  0, m  0}

Languages

• (^) Star-closure: L*^ = L^0  L^1  L^2 ...
• (^) Positive closure: L+^ = L^1  L^2 ...

Grammars

• (^) Formal grammar: G = (V, T, S, P) V: finite set of variables T: finite set of terminal symbols SV: start variable P: finite set of productions

Grammars

• (^) Productions: x  y x(VT)+^ y(VT)*
• (^) w = uxv derives z = uyv w  z w 1 *^ wn (w 1  w 2  ...  wn | w 1 = wn) w 1 +^ wn

Grammars

• (^) Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S   S  aSb  aaSbb  aabb aabb: sentence aaSbb: sentential form

Grammars

• (^) Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S  

Grammars

• (^) Example 4: G 1 = ({A, S}, {a, b}, S, P 1 ) P 1 : S  aAb |  A  aAb | 

Grammars

• (^) Example 4: G 1 = ({A, S}, {a, b}, S, P 1 ) P 1 : S  aAb |  A  aAb |  L(G 1 ) = {anbn^ | n  0} G and G 1 are equivalent

Grammars

• (^) Example 5: G 2 = ({S}, {a, b}, S, P 2 ) P 2 : S  SS S   S  aSb S  bSa L(G 2 ) = {w | na(w) = nb(w)}

Automata

An abstract model of digital computer: Control unit Input file Output Storage