Context-Free Grammars: Description, Natural Language Examples, and Programming Languages, Slides of Theory of Automata

Download Context-Free Grammars: Description, Natural Language Examples, and Programming Languages and more Slides Theory of Automata in PDF only on Docsity!

Context-free languages

Context-free grammar

• This is an a different model for describing

languages

• The language is specified by productions

(substitution rules) that tell how strings can be

obtained, e.g.

• Using these rules, we can derive strings like this:

A → 0A

A → B

B →

A, B are variables 0, 1, # are terminals A is the start variable

A ⇒ 0A1 ⇒ 00A11 ⇒ 000A111 ⇒ 000B111 ⇒ 000#

Natural languages

• We can describe (some fragments) of the

English language by a context-free grammar:

SENTENCE → NOUN-PHRASE VERB-PHRASE NOUN-PHRASE → CMPLX-NOUN NOUN-PHRASE → CMPLX-NOUN PREP-PHRASE VERB-PHRASE → CMPLX-VERB VERB-PHRASE → CMPLX-VERB PREP-PHRASE PREP-PHRASE → PREP CMPLX-NOUN CMPLX-NOUN → ARTICLE NOUN CMPLX-VERB → VERB NOUN-PHRASE CMPLX-VERB → VERB

ARTICLE → a ARTICLE → the NOUN → boy NOUN → girl NOUN → flower VERB → likes VERB → touches VERB → sees PREP → with

variables: SENTENCE, NOUN-PHRASE, … terminals: a, the, boy, girl, flower, likes, touches, sees, with start variable: SENTENCE

Programming languages

• Context-free grammars are also used to

describe (parts of) programming languages

• For instance, expressions like (2 + 3) * 5 or

3 + 8 + 2 * 7 can be described by the CFG

→ + → * → () → 0 → 1 … → 9

Variables: Terminals: +, *, (, ), 0, 1, …, 9

Definition of context-free grammar

• A context-free grammar (CFG) is a 4-tuple

( V , T , P , S) where

• V is a finite set of variables or non-terminals
• T is a finite set of terminals ( V ∩ T = ∅)
• P is a set of productions or substitution rules of the

form

where A is a symbol in V and α is a string over V ∪ T

• S is a variable in V called the start variable

A → α

Shorthand notation for productions

• When we have multiple productions with the

same variable on the left like

we can write this in shorthand as

E → E + E E → E * E E → (E) E → N

E → E + E | E * E | (E) | 0 | 1 N → 0N | 1N | 0 | 1

Variables: E, N Terminals: +, *, (, ), 0, 1 Start variable: E

N → 0N N → 1N N → 0 N → 1

Language of a CFG

• The language of a CFG ( V , T , P , S) is the set of

all strings containing only terminals that can be

derived from the start variable S

• This is a language over the alphabet T

• A language L is context-free if it is the

language of some CFG

L = {ω | ω ∈ T* and S ⇒* ω }

Example 1

• Is the string 00#11 in L?

• How about 00#111, 00#0#1#11?

• What is the language of this CFG?

A → 0A1 | B

B →

variables: A, B terminals: 0, 1, # start variable: A

L = {0 n #1 n : n ≥ 0}

Examples: Designing CFGs

• Write a CFG for the following languages

• Linear equations over x, y, z, like: x + 5y – z = 9 11x – y = 2
• Numbers without leading zeros, e.g., 109, 0 but not 019
• The language L = {a n b n c m d m^ | n ≥ 0, m ≥ 0}

n m m n Docsity.com

Context-free versus regular

• Write a CFG for the language (0 + 1)*

• Can you do so for every regular language?

• Proof:

S → A

A → ε | 0A | 1A

Every regular language is context-free

regular expression NFA DFA

Context-free versus regular

• Is every context-free language regular?

• No! We already saw some examples:

• This language is context-free but not regular

A → 0A1 | B

B →

L = {0 n #1 n : n ≥ 0}

Parse tree

• Derivations can also be represented using

parse trees

E ⇒ E + E ⇒ V + E ⇒ x + E ⇒ x + (E) ⇒ x + (E − E) ⇒ x + (V − E) ⇒ x + (y − E) ⇒ x + (y − V) ⇒ x + (y − z)

E

E + E

V ( E )

E −^ E

V V

x

y z

E → E + E | E - E | (E) | V V → x | y | z

Left derivation

• Always derive the leftmost variable first:

• Corresponds to a left-to-right traversal of parse

tree

E ⇒ E + E ⇒ V + E ⇒ x + E ⇒ x + (E) ⇒ x + (E − E) ⇒ x + (V − E) ⇒ x + (y − E) ⇒ x + (y − V) ⇒ x + (y − z)

E

E + E

V ( E )

E −^ E

V V

x

y z

Ambiguity

• A grammar is ambiguous if some strings have

more than one parse tree

• Example: E^ →^ E + E | E^ −^ E | (E) | V

V → x | y | z

x + y + z

E

E + E

V E + E

x V V y z

E

E + E

E + E V

V V

x y

z