Formal Language & Automata Theory Shyamanta M Hazarika Computer Sc. & Engineering Tezpur University http://www.tezu.ernet.in/~smh

Introduction Finite Automata accept all regular languages and only regular languages Even very simple languages are non regular ( = {a,b}): - {anbn : n = 0, 1, 2, …} - {w : w is palindrome word} We are going to define a new class of languages, called context-free languages that contain all regular languages and many more (including the 2 above)

Context-Free Grammar A context-free grammar is a kind of program Languages that are generated by context-free grammars are called context-free languages Context-free grammars are more expressive than finite automata: if a language L is accepted by a finite automata then L can be generated by a context-free grammar

Context-Free Grammar (CFG) Definition. A context-free grammar (CFG) is a 4-tuple (V, , R, S), where:  is an alphabet (characters  are called terminals) V is a set (elements in NT are called variables) R is a subset of NT  (  NT)* S, the start variable, is one of the variables in NT V   =  If (,)  R, we write       is called a rule

Derivations Definition. u yields v in one-step, written u  v, if: for some u,v in (V  )* the following 3 conditions hold: u = xz v = xz    in R Definition. u derives v, written u * v, if: There is a chain of one-step yields of the form: u  u1  u2  …  v

Example = {a,b} V = {S} R = { S aSb, S  e }

Context-Free Languages Definition. Given a context-free grammar G = (V, , R, S), the language generated or derived from G is the set:   L(G) = {w  *: } S * w Definition. A language L is context-free if there is a context-free grammar G = (, NT, R, S), such that L is generated from G

Example = {a,b} NT = {S} R = { S aS, S  Sb, S  e}

Example = {a,b} NT = {S} R = { S aSa, S  bSb, S  e}

Parse Tree A parse tree of a derivation u  u1  u2  …  v is a tree in which: Each internal node is labeled with a variable If a rule A  A1A2…An occurs in the derivation then A is a parent node of nodes labeled A1, A2, …, An S a e b

Leftmost, Rightmost Derivations Definition. A leftmost derivation of a sentential form is one in which rules transforming the left-most nonterminal are always applied Definition. A rightmost derivation of a sentential form is one in which rules transforming the right-most nonterminal are always applied

Ambiguous Grammar Definition. A grammar G is ambiguous if there is a word w  L(G) having are least two different leftmost derivations S  A S  B S  AB A  aA B  bB A  e B  e Notice that the word a has at least two left-most derivations Some ambiguous grammars G can be disambiguated: find an unambiguous grammar G’ such that L(G) = L(G’) Some languages cannot be disambiguated