Context-Free GrammarsContext-Free Grammars

Section 2.1

CSC 4170

Theory of Computation

What is a CFG2.1.a

A BA B 0A1

Terminals: 0,1

Variables: A,B

Start variable: A

Productions:

Derivation: A B 0A1 0B1 00A11 0011

Parse tree:

A

B

A

B

A

0 1

0 1

Our grammar simplified2.1.b

A 0A1 A

A 0A1 00A11 000A111 0000A1111 00001111

What language does this grammar produce?

A more complex CFG2.1.c1

S N’_V_N’ N’ N | N_who_V_N’N men | women | childrenV like | hate | respect

SN’_V_N’N’_like_N’N_like_N’N_like_Nwomen_like_Nwomen_like_children

S

N’ _ V _ N’

N like N

women children

A more complex CFG2.1.c2

S N’_V_N’ N’ N | N_who_V_N’N men | women | childrenV like | hate | respect

S

N’ _ V _ N’

N respect N _ who _ V _ N’

children men hate N

women

Formal definitions2.1.d

A context-free grammar is a 4-tuple (V,,R,S), where 1. V is a finite set called the variables; 2. is a finite set, disjoint from V, called the terminals; 3. R is a finite set of rules, with each rule being a pair of a variable and a string of variables and terminals; 4. S is an element of V called the start variable.

If u,v, and w are strings of variables and terminals and A w is a rule,we say that uAv yields uwv, written uAv uwv.

The language produced (defined, described) by the grammar is {w | S * w and w is a string of (only) terminals}.

A context-free language is a language produced by some CFG.

x * y means that x=y, or x y, or there are z1,…,zn such that x z1 … zn y.

Ambiguity: An informal example 2.1.e

the girl touches the boy with the flower

the girl touches

the boy with the flower

with the flower

the girl touches the boy

Does this mean

the girl touches (the boy with the flower)

or

(the girl touches the boy) with the flower

?

An example of an ambiguous CFG 2.1.f

<EXPR> <EXPR> + <EXPR> | <EXPR> <EXPR> | a

<EXPR>

<EXPR> + <EXPR>

a <EXPR> <EXPR>

a a

<EXPR>

<EXPR> <EXPR>

<EXPR> + <EXPR>

a a

a

a + a a

A grammar is ambiguous iff it has two different parse trees for the same string

An equivalent but unambiguous grammar2.1.g

<EXPR> <EXPR> + <TERM> | <TERM><TERM> <TERM> a | a

<EXPR>

<EXPR> + <TERM>

<TERM>

a

<TERM> a

a

a + a a

A more complex unambiguous grammar2.1.h

<EXPR> <EXPR> + <TERM> | <TERM><TERM> <TERM> <FACTOR> | <FACTOR><FACTOR> (<EXPR>) | a

<EXPR>

<EXPR> + <TERM>

<TERM>

<FACTOR>

a

<TERM> <FACTOR>

<FACTOR>

a

a

<EXPR>

<TERM>

<TERM> <FACTOR>

<FACTOR>

( <EXPR> )

<EXPR> + <TERM>

<TERM>

<FACTOR>

a

<FACTOR>

a

a

a + a a (a + a) a

Designing context-free grammars2.1.i

Design a CFG that produces all regular expressions over the alphabet {0,1}:

<RE>

Design a CFG G that produces the union of the languages produced by two given CFGs G1 and G2.

G1:A1 w1

…An wn

G2:B1 u1

…Bm um

Converting a DFA into a CFG2.1.j

Q1

0

Q2

0

1

1

Variables: The states of the DFA

Start variable: The start state of the DFA

Productions: 1. Qi a Qj, whenever there is an a-arrow from Qi to Qj; 2. Qi , whenever Qi is an accept state.

Testing in work2.1.j

Q1

0

Q2

0

1

1

Q1 0 Q1

Q1 1 Q2

Q2 0 Q2

Q2 1 Q1

Q2

011001Q1

0Q1

01Q2

011Q1

0110Q1

01100Q1

011001Q2

011001