unambiguous grammar

Let’s see the solution of following question:
Problem:

Consider the grammar:
S -> aS | aSbS | epsilon
where S is the only non-terminal, and epsilon is the null string.

a) Show that the grammar is ambiguous, by giving two parse trees for the string aab.
b) Find an unambiguous grammar that generates these strings.

Solution:

Solution-(a)

The ambiguity is easy to show: you can derive the string aab as follows:
(at every step we expand the leftmost non-terminal);

S -> aSbS -> aaSbS -> aabS -> aab
S -> aS -> aaSbS -> aabS -> aab

These two parses correspond to associating the b with the first or the
second a.

Solution-(b)

We can disambiguate by using a similar approach to the dangling else, and
decide that each b should be associated with the nearest a. This means that
the expansion within an ab pair should always be balanced. This leads to
the following grammar:

S -> a S | S1 S | epsilon
S1 -> a S1 S1 b | epsilon

It is easy to verify that this generates the same strings as the original
grammar, and the parse tree is always unique, because one b is always associated
with the most recent a.

Note that the answer is not necessarily unique. If the grammar is ambiguous,
it means that we get to choose between possible parses, and each choice is
in a sense a different language. For example, given the ambiguous grammar
for expressions:

E -> E + E | E * E | id

We say that the unambiguous grammar we want is:

E -> E + T | T, T -> T * T | id

because it gives us the proper precedence between the two operators. But that
choice is in no way mandated by the grammar. We could just as well choose:

E -> E * T | T , T -> T + T | id

which generates the same strings, but gives the opposite precedence to
operators.

Dangling Else Problem

In this post, we will see dangling else problem. It is the best example of ambiguous grammar.

Dangling Else Problem:
The dangling else is a well-known problem in computer programming in which a seemingly well defined grammar can become ambiguous. In many programming languages you can write code like this:

if a then if b then s1 else s2

which can be understood in two ways. Either as

if a then
if b then
s1
else
s2
or as
if a then
if b then
s1
else
s2

It can be solved either at the implementation level, by telling the parse the right way to solve the ambiguity, or at the grammar level by using a parsing expression grammar or equivalent.

In next post , I will give an example which will cover all topics related to ambiguous grammar