Context-free Grammars

Example :

S Shortened notation :

S aSa S | aSa | bSb

S bSb

Which strings can be generated from S ?

[Section 6.1]

Context-free Grammars

Def: A context-free grammar (CFG) is a 4-tuple G = (V,,S,P) where

- V is a finite set of variables (nonterminal symbols)

- is a finite set of terminal symbols (terminals)

- S 2 V is the start symbol

- P is the finite set of grammar rules (productions) of the form A , where A 2 V and 2 (V [ )*

(V and are disjoint.)

For example from previous slide :

[Section 6.1]

Context-free Grammars

A derivation of the string “abaaba” :

Def :

Let G=(V,,S,P) be a CFG. For , 2 (V [ )* we write )G if there is a production C in P such that = 1C2 and = 12. We say that is derived from .

If 1, …, n 2 (V [ )* and k ) k+1 for k=1,…,n-1, we write 1 )G

* n.

Language generated by G, denoted L(G), is { x 2 * | ______ }

[Section 6.1]

Context-free Grammars

Examples :

{ akbk | k ¸ 0 }

{ x 2 {a,b}* | x has at least 2 b’s }

{ x 2 {a,b}* | |x| is even }

[Section 6.1]

Context-free Grammars

Example :

S | abS | baS | aSb | bSa | Sab | Sba

What is the language generated by this grammar ?

(We’ve seen this before…)

[Section 6.1]

Context-free Grammars

Our G : S | abS | baS | aSb | bSa | Sab | Sba | SS

Recall that we proved (using structural induction) that every string generated by G contains the same number of a’s and b’s.

Moreover, we can generate every string with equal number of a’s and b’s by the rules of G.

[Section 6.1]

Closure Properties of CFL’s

Given are two CFG’s G1 = (V1,,S1,P1) and G2 = (V2,,S2,P2).

- Give a CFG G such that L(G) = L(G1) [ L(G2).

- Give a CFG G such that L(G) = L(G1)L(G2).

- Give a CFG G such that L(G) = L(G1)*.

Thus, context-free languages (CFL’s) are closed under union, concatenation, and Kleene’s star.

[Section 6.2]

Derivation Trees & Ambiguity

Consider G: S S + S | S * S | (S) | x

Describe the language generated by G:

Give all derivations of x + x * x :

[Section 6.4]

Derivation Trees

We will draw derivation trees of x + x * x :

A leftmost derivation is a derivation in which each derivation step uses a rule for the leftmost nonterminal symbol.

Derivation trees are in 1-1 correspondence with leftmost derivations.

[Section 6.4]

Ambiguity

A CFG G is ambiguous iff there exists a string x 2 L(G) such that x has more than one distinct derivation trees (or, equivalently, more than one leftmost derivation).

Is S S + S | S * S | (S) | x ambiguous ?

Can you give an unambiguous grammar for mathematical expressions ?

[Section 6.4]

Ambiguity in Programming Languages

Consider the following production rule :

<statement>

if ( <expression> ) <statement> |

if ( <expression> ) <statement> else <statement> |

<otherstatement>

What are the terminals and nonterminals in the above ?

How to fix the problem ?

[Section 6.4]

CFL’s vs. Regular Languages

Is every context-free language regular ?

Is every regular language context-free ?

(Recall the definition of regular languages.)

[Section 6.5]

Parsing & Normal Forms

How to determine if a string is in a language generated by a CFG ? (Algorithms that do this are called parsers.)

It helps to have the grammar in a normal form (more restrictions than the original definition of CFG’s).

Chomsky normal form :

Every rule is of the form A or A A1A2 where A,A1,A2 2 V and 2 .

Example : S S + S | S * S | (S) | x

Convert this grammar into Chomsky normal form.

[Section 6.6]

Parsing & Normal Forms

How to determine if a string is in a language generated by a CFG ? (Algorithms that do this are called parsers.)

It helps to have the grammar in a normal form (more restrictions than the original definition of CFG’s).

Chomsky normal form :

Every rule is of the form A or A A1A2 where A,A1,A2 2 V and 2 .

Example : S S + S | S * S | (S) | x

Convert this grammar into Chomsky normal form.

[Section 6.6]

Parsing & Normal Forms

Greibach normal form :

Every rule is of the form A where 2 and 2 V*.

Example : S S + S | S * S | (S) | x

Convert this grammar into Greibach normal form.

[Section 6.6]

Machines for CFG ?

Deterministic or nondeterministic ?

Should be able to recognize { akbk | k ¸ 0 },

{ w 2 {a,b} | w is a palindrome },

but not recognize { ww | w 2 {a,b}* }.