Context free grammars (CFGs) are formal systems that describe the structure of languages. A CFG consists of variables, terminals, production rules, and a start variable. Production rules take the form of a single variable producing a string of terminals and/or variables. CFGs can capture the recursive structure of natural languages while ignoring agreement and reference. They are used to define context-free languages and generate parse trees. Ambiguous grammars have sentences with multiple parse trees, and disambiguation aims to impose an ordering on derivations. While ambiguity cannot always be eliminated, simplifying and restricting grammars has theoretical and practical benefits.

1. Context Free Grammars
Ronak Thakkar
Roll no 32
M.Sc. Computer Science

2. What are Context Free Grammars?
In Formal Language Theory , a Context free Grammar(CFG)
is a formal grammar in which every production rule is of the
form
V w
Where V is a single nonterminal symbol and w is a string of
terminals and/or nonterminals (w can be empty)
The languages generated by context free grammars are
knows as the context free languages

3. What does CFG do?
A CFG provides a simple and mathematically precise
mechanism for describing the methods by which phrases in
some natural language are built from smaller blocks,
capturing the “block structure” of sentences in a natural way.
Important features of natural language syntax such as
agreement and reference is are not the part of context free
grammar , but the basic recursive structure of sentences , the
way in which clauses nest inside other clauses, and the way in
which list of adjectives and adverbs are swallowed by nouns
and verbs is described exactly.

4. Formal Definition of CFG
A context-free grammar G is a 4-tuple (V, ∑, R, S), where:
 V is a finite set; each element v ∈ V is called a non-terminal character or
a variable.
 ∑ is a finite set of terminals, disjoint from , which make up the actual
content of the sentence.
 R is a finite relation from V to (V U ∑)* , where the asterisk
represents the Kleene star operation.
If (α,β) ∈ R, we write production α  β
β is called a sentential form
• S, the start symbol, used to represent the whole sentence (or
program). It must be an element of V.

5. Production rule notation
A production rule in R is formalized mathematically as a pair
(α,β) , where α is a non-terminal and β is a string of
variables and nonterminals; rather than using ordered pair
notation, production rules are usually written using an arrow
operator with α as its left hand side and β as its right hand
side: α  β.
It is allowed for β to be the empty string, and in this case it is
customary to denote it by ε. The form α  ε is called an ε-
production.

6. Context-Free Languages
•Given a context-free grammar
G = (V,∑,R, S), the language generated or derived from
G is the set
L(G) = {w :S ⇒* w}
A language L is context-free if there is a context-free
grammar G = (V,∑, R, S), such that L is generated from G.

7. Example :Well-formed
parentheses
The canonical example of a context free grammar is
parenthesis matching, which is representative of the general
case. There are two terminal symbols "(" and ")" and one
nonterminal symbol S. The production rules are
S → S
SS → (S)
S → ()
The first rule allows Ss to multiply; the second rule allows Ss
to become enclosed by matching parentheses; and the third
rule terminates the recursion.

8. Parse Tree
A parse tree of a derivation is a tree in which:
• Each internal node is labeled with a nonterminal
• If a rule A  A1A2…An occurs in the derivation then A is
a parent node of nodes labeled A1, A2, …, An
S
a S
a S
b
S
e

9. Leftmost, Rightmost Derivations
A left-most derivation of a sentential form is one in
which rules transforming the left-most nonterminal are
always applied
A right-most derivation of a sentential form is one in
which rules transforming the right-most nonterminal
are always applied

10. Ambiguous Grammar
. A grammar G is ambiguous if there is a word w ∈
L(G) having are least two different parse trees
SA
SB
S  AB
A  aA
B  bB
Ae
Be
Notice that a has at least two left-most derivations

11. Ambiguity & Disambiguation
Given an ambiguous grammar, would like an equivalent
unambiguous grammar.
Allows you to know more about structure of a given
derivation.
Simplifies inductive proofs on derivations.
Can lead to more efficient parsing algorithms.
In programming languages, want to impose a canonical
structure on derivations. E.g., for 1+2×3.
Strategy: Force an ordering on all derivations.

12. CFG Simplification
Can’t always eliminate ambiguity.
But, CFG simplification & restriction still useful
theoretically & pragmatically.
Simpler grammars are easier to understand.
Simpler grammars can lead to faster parsing.
Restricted forms useful for some parsing algorithms.
Restricted forms can give you more knowledge about
derivations.