Context-free Grammars[Section 2.1]

- more powerful than regular languages

- originally developed by linguists

- important for compilation of programming languages

Context-free Grammars[Section 2.1]

Example: A -> 0A1

A -> B

B -> #

Terminology:

- substitution rules (productions)

- variables (including the start variable) – typically upper-case

- terminals – typically lower-case, other symbols

- derivation, parse tree

Context-free Grammars[Section 2.1]

Def 2.2: A context-free grammar is a 4-tuple (V,§,R,S), where

- V is a finite set of variables

- § is a finite set of terminals, V Å § = ;

- R is a finite set of rules, each rule is of the form A -> w where A 2 V and w 2 (V [ §)*

- S 2 V is the start variable

If A -> w 2 P, then we write uAv => uwv (read “uAv yields uwv”), and we write u =>* v (read “u derives v”) if u=v or if there exists a sequence u1,u2,…,uk such that

u => u1 => u2 => … uk => v

The language of the grammar is { w2§* | S=>*w }.

Context-free Grammars[Section 2.1]

Examples: give context-free grammars for the following languages:

- { aibjck | i=k, i,j,k ¸ 0 }

- { aibjck | i=j, i,j,k ¸ 0 }

- strings over { ( , ) } that are well-parenthesized

- strings over { 0,1 } that contain equal number of 0’s and 1’s

Ambiguity[Section 2.1]

Example:

[EXPR] ->[EXPR] + [EXPR] | [EXPR] x [EXPR] |

( [EXPR] ) | a

Give a derivation (and parse trees) for the string a+axa.

Notice: for every parse tree there is a unique left-most derivation.

Ambiguity[Section 2.1]

Def 2.7: A context-free grammar is called ambiguous if there exists a string that can be generated by two different left-most derivations.

Note: Some context-free languages do not have unambigous grammars (e.g { aibjck | i=j or j=k } ). These are called inherently ambiguous.

Example: give an unambigous CFG for the language of arithmetic expressions over { +, x, a }

Ambiguity[Section 2.1]

Def 2.7: A context-free grammar is called ambiguous if there exists a string that can be generated by two different left-most derivations.

For every x2\Sigma let y… L = {a^ib^jc^k | i,j,k\geq 0}.

Example: give an unambigous CFG for the language of arithmetic expressions over { +, x, a }

Chomsky Normal Form[Section 2.1]

Def 2.8: A CFG is in Chomsky normal form if every rule is of the form A -> BC or A -> a, where B,C 2 V-{S}, and a 2 §.

Thm 2.9: Any CFL can be generated by a CFG in Chomsky normal form.

Note: what about CFL that contains ε ?

Why a normal form ?

Chomsky Normal Form[Section 2.1]

Thm 2.9: Any CFL can be generated by a CFG in Chomsky normal form.

“Proof” by example:

S -> ASA | aB Things to fix: 1.

A -> B | S 2.

B -> b | ε 3.

4.