How do we check whether a word is in the language of the grammar? We start with the start symbol and use one production to replace the nonterminal symbol by the until we have no nonterminal symbols left - this is called a derivation. I.e. in the example :
Given any grammar we define the relation derives in one step
We now say that the language of a grammar is given by all words (over ) which can be derived in any number of steps, i.e.
A language which can be given by a context-free grammar is called a context-free language (CFL).