1. Lecture 6
Parsing trees and Ambiguity
Dr. Hussien Sharaf
Computer Science Department
dr.sharaf@from-masr.com

2. Parsing
CFG grammar is about categorizing
the statements of a language.
Parsing using CFG means
categorizing a certain statements
into categories defined in the CFG.
Parsing can be expressed using a
special type of graph called Trees
where no cycles exist.
A parse tree is the graph
representation of a derivation.
Dr. Hussien M. Sharaf 2

3. Parse tree
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(1)A vertex with a label which is a
Non-terminal symbol is a parse tree.
(2) If A → y1 y2 … yn is a rule in R,
then the tree
A
y1 y2 yn. . .
is a parse tree.

4. Ambiguity
A grammar can generate the same
string in different ways.
Ambiguity occurs when a string has
two or more leftmost derivations for
the same CFG.
There are ways to eliminate
ambiguity such as using Chomsky
Normal Form (CNF) which does n’t
use Λ.
Λ cause ambiguity.
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5. Ex 1
 i.e. {Λ, ab, abbaabba,… }
 S → aSa| bSb| Λ Derive abaaba
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S
S
Λ
aa
S bb
S aa

6. Example2
Deduce CFG of a addition and parse
the following expression 2+3+5
1] S→S+S|N
2] N→1|2|3|4|5|6|7|8|9|0
Dr. Hussien M. Sharaf 6
S
S+N
S
+
N
5
S
+
3
N
2
N
Can u make
another parsing
tree ?

7. Example3
Deduce CFG of a
addition/multiplication and parse the
following expression 2+3*5
1] S→S+S|S*S|N
2] N→1|2|3|4|5|6|7|8|9|0| N
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S
S*S
S
*
N
5
S
+
3
N
2
N
Can u make
another parsing
tree ?

8. Ex4 CFG without ambiguity
 Deduce CFG of a addition/multiplication
and parse the following expression 2*3+5
1] S→ Term|Term + S
2] Term → N|N * Term
3] N→1|2|3|4|5|6|7|8|9|0
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S
S+N
S
+
N
5
S
*
3
N
2
N
Can u make
another parsing
tree ?

9. Example 5
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S A | A B
A Λ| a | A b | A A
B b | b c | B c | b B
Sample derivations:
S AB AAB
aAB aaB aabB aabb
S AB AbB Abb AAbb
Aabb aabb
S
A B
AA Bb
a a b
S
BA
b
A
b
AA
a a

10. Example 6
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S A | A B
A Λ | a | A b | A A
B b | b c | B c | b B
S
A B
AA b
a
a
bA
S
A
A A
AA bA
a
a
bA
S
A B
AA Bb
a a b

11. Dr. Hussien M. Sharaf 11
Eliminate ambiguity.
Eliminate “useless” variables.
Eliminate Λ-productions: A Λ.
Eliminate unit productions: A B.
Eliminate redundant productions.
Avoid left recursion by trading left-
and right-recursion.

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13. PDA
 A Pushdown Automata (PDA ) is an NFA
equipped with a single stack.
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1
4
3
2

14. PDA as a deterministic machine
Its inputs are
1. The current state (of its “NFA”),
2. The current input symbol (or
Λ), and
3. The current symbol on top of its
stack.
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1
4
3
2
Its actions are:
1. Change state.
2. Change stack if needed.

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