# Ambiguous Grammar

You can also read our previously discussed article on Classification of Context Free Grammars.

**C**ontext **F**ree **G**rammars(CFGs) are classified based on:

- Number of Derivation trees
- Number of strings

Depending on Number of Derivation trees, CFGs are sub-divided into 2 types:

- Ambiguous grammars
- Unambiguous grammars

**Ambiguous grammar:**

A CFG is said to ambiguous if there exists more than one derivation tree for the given input string i.e., more than one **L**eft**M**ost **D**erivation **T**ree (LMDT) or **R**ight**M**ost **D**erivation **T**ree (RMDT).

**Definition:** G = (V,T,P,S) is a CFG is said to be ambiguous if and only if there exist a string in T* that has more than on parse tree.

where V is a finite set of variables.

T is a finite set of terminals.

P is a finite set of productions of the form, A -> α, where A is a variable and α ∈ (V ∪ T)* S is a designated variable called the start symbol.

**For Example:**

1. Let us consider this grammar : **E -> E+E|id**

We can create 2 parse tree from this grammar to obtain a string **id+id+id ** :

The following are the 2 parse trees generated by left most derivation:

Both the above parse trees are derived from same grammar rules but both parse trees are different. Hence the grammar is ambiguous.

2. Let us now consider the following grammar:

Set of alphabets ∑ = {0,…,9, +, *, (, )} E -> I E -> E + E E -> E * E E -> (E) I -> ε | 0 | 1 | … | 9

From the above grammar String **3*2+5** can be derived in 2 ways:

I) First leftmost derivation II) Second leftmost derivation E=>E*E E=>E+E =>I*E =>E*E+E =>3*E+E =>I*E+E =>3*I+E =>3*E+E =>3*2+E =>3*I+E =>3*2+I =>3*2+I =>3*2+5 =>3*2+5

Following are some examples of ambiguous grammars:

- S-> aS |Sa| Є
- E-> E +E | E*E| id
- A -> AA | (A) | a
- S -> SS|AB , A -> Aa|a , B -> Bb|b

Whereas following grammars are unambiguous:

- S -> (L) | a, L -> LS | S
- S -> AA , A -> aA , A -> b

**Inherently ambiguous Language:**

Let L be a Context Free Language (CFL). If every Context Free Grammar G with Language L = L(G) is ambiguous, then L is said to be inherently ambiguous Language. Ambiguity is a property of grammar not languages. Ambiguous grammar is unlikely to be useful for a programming language, because two parse trees structures(or more) for the same string(program) implies two different meanings (executable programs) for the program.

An inherently ambiguous language would be absolutely unsuitable as a programming language, because we would not have any way of fixing a unique structure for all its programs.

For example,

L = {a^{n}b^{n}c^{m}} ∪ {a^{n}b^{m}c^{m}}

**Note :** Ambiguity of a grammar is undecidable, i.e. there is no particular algorithm for removing the ambiguity of a grammar, but we can remove ambiguity by:

**Disambiguate the grammar** i.e., rewriting the grammar such that there is only one derivation or parse tree possible for a string of the language which the grammar represents.

This article is compiled by **Saikiran Goud Burra**.

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