Grammars I

1. CSE340 - Principles of
Programming Languages
Lecture 09:
Grammars 1
Javier Gonzalez-Sanchez
javiergs@asu.edu
BYENG M1-38
Office Hours: By appointment

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Ambiguity
• A grammar is said to be ambiguous if there is at
least one string with two or more parse trees.
• Note that ambiguity is a property of grammars, not
languages.

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Ambiguity

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Solution
• If a grammar can be made unambiguous at all, it is
usually made unambiguous through layering.
• Have exactly one way to build each piece of the
string.
• Have exactly one way of combining pieces back
together.
• Recursive constructions

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Layering
Rule 1
Rule 2
Rule 3
Rule 4
Rule 5
. . . . .
Leaf (Terminals, i.e., Tokens)
Root
Start symbol
Layers

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Layering
1
2
4
5
3
inputs:
§ 1 + 2 + 3 < 4 * 5
§ 1 * 2 + 3 + 4 < 5
§ 1 < 2 + 3 + 4 * 5
§ 1 + 2 < 3 * 4 + 5
§ 1 + 2 * 3 < 4 + 5

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Exercise
Provide a Grammar that is Not ambiguous
for arithmetic expressions
10 * 20 + 15
Precedence of operators and Associativity
þ (10 * 20) + 15
ý 10 * (20 + 15)

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Exercise | Precedence
New Grammar:
<E> à <__> + <__>
< > à <__> - <__>
< > à <__> * <__>
< > à <__> / <__>
< > à - <__>
Original Grammar:
E à E OP E
E à integer
OPà + | - | * | /

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Exercise | Hand written notes

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Exercise | Hand written notes

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Exercise | Precedence
<E> à <A> | <A> {'+' <A>} | <A> {'-' <A>}
<A> à <B> | <B> {'*' <B>} | <B> {'/' <B>}
<B> à '-'<C> | <C>
<C> à integer

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Exercise | Precedence
<E> à <A> {(’+'|'-’) <A>}
<A> à <B> {('*'|'/') <B>}
<B> à ['-'] <C>
<C> à integer

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Syntax Diagrams
E
A
B
C

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Exercise 1
Include rules for handling parenthesis into the previous
grammar. The grammar should accept as correct the
following expressions:
10 * 20 + 15
(10 * 20) + 15
10 * (20 + 15)
(10) * (20) + (15)
(10 * 20 + 15)
10 * (20) + 15

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Exercise 1 | Hand written notes

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Syntax Diagrams
E
A
B
C

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Exercise 2 | Hand written notes

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Exercise 2 | Hand written notes

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Exercise 2
Include rules to accept variables names (identifiers) in
expressions. The grammar should accept as correct the
following expressions:
A * 20 + time
(x * y) + 15
10 * (ASU + cse340)
(10) * (20) + (15)
(hello * world + Arizona)
10 * (counter) + 15

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Syntax Diagrams
E
A
B
C

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Exercise 3
Provide a Grammar that is
Not ambiguous
with Precedence of operators and Associativity for this:
10 + 20 > 15 & -10 != 1 | 20 / ( 10 + 1) < 5

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Exercise 3 | Note
Precedence of operators
|
&
!
< > == != <= >=
+ -
* /
-
( )

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Exercise 3 | Hand written notes
...

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Exercise 3 | Hand written notes
{ }
{ }
{ } { } { } { }...
{ } { }
{ } { }

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Exercise 3
R
EXPRESSION
X
Y

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Exercise 3
E
A
B
C

27. CSE340 - Principles of Programming Languages
Javier Gonzalez-Sanchez
javiergs@asu.edu
Summer 2015
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