Understanding Context-Free Languages and Grammars: A Comprehensive Overview

1. 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)

2. Context-Free Grammar (preliminaries) • 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

3. My First Context-Free Grammar S  bA A  aA A  b •  = {a,b} • Elements in  are called terminals • S and A are called variables

4. 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

5. 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

6. Example (2) • = {a,b} • V = {S} • R = { S aSb, • S e }

7. 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

8. Example (3) • = {a,b} • NT = {S} • R = { S aS, • S  Sb, • S e}

9. Example (4) • = {a,b} NT = {S} R = { S aSa, S  bSb, S e}

10. S a S S a b 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

11. 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

12. 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

13. Chomsky Normal Form • Definition: A grammar is in Chomsky Normal Form if every rule is of the form: • A  BC (A, B, C variables; B and C are not the start variable) • A  a • S  e (S is the start variable) • Theorem: Any CFG G can be converted into a grammar G’ in Chomsky Normal Form such that L(G) = L(G’) • Add new rule S0 S (S0 is the new start variable) • Remove rules of the form A  e, and for every rule B  <…>A<…> add a new rule: B <…><…> • Remove rules of the form A  B and for every rule B  <…> add a new rule: A  <…> • Remove rules A  <C1|c1> …<Cn|cn> with n > 2 and add rules: A  <C1|c1> A1, A1  <C2|c2>, …, An-1  <Cn-1|cn-1> <Cn|cn> • Replace any rule: A  cAi with A  UAi, U  c See example 2.10