Aho states the motivation of formalizing IGs in his original publication [1] that part of this interest in larger classes of language (i.e., larger than context-free languages) stems from the inadequacy of context-free grammars in specifying all of the syntactic structures found in many modern-day algorithmic programming languages, such as ALGOL. The class of indexed languages assumes a natural position in the Chomsky hierarchy of languages, which has been showed to includes all context-free languages and some context-sensitive languages, and can be generated using a nested stack automaton. We give the definition of an indexed grammar from Aho [1] below.
Many other definitions such as the reflexive and transitive closures are identical to the ones defined in context-free grammars. Following the IG definition, it is not hard to see all context-free languages can be generated by IGs, and thus belongs to indexed languages. Clearly, if the F set is empty, then an IG is a context-free grammar. Aho [1] proved that indexed languages are more closely resemble context-free languages than they are to context-sensitive languages under the full abstract family of languages (full AFL), where indexed and context-free languages are full AFLs, and context-sensitive languages are not. It is well known that a full AFL is a class of languages closed under the operations of union, concatenation, Kleene closure, homomorphism, inverse homomorphism, and intersection with regular sets. It is also showed [1] that indexed languages are closure under word reversal and substitution. We do not get into full details of the proofs here. Instead, an examples is given below from [1] to get some taste of IGs comparing with context-free and context-sensitive grammars.
Example 1. 
. The language is
well-known not context-free. However, the following IG generates
the language. Let 
, where P consists of the productions 
and 
, and 
.
Applying the first production once, the second production n - 1
times, 
, and then the third production once, we have:
.
Expanding B by consuming indices, we have:
.
It is easy to show that the only strings in terminals which can be
derived from S in 
are all of the form 
. Thus,
. The language can be easily
generated with a context-sensitive grammar too. A formal proof of
the containment within context-sensitive languages for indexed
languages is provided in [1].