# Recursive definition of languages in Automata

In the Theory of automata, languages can be defined with different techniques. Some of these are mentioned below;

## What is Recursive definition of languages?

Three steps are required in recursive definition of a language.

Step 1: Base Case: Need to define some basic words in the language.

Step 2: Recursive Step:

Defines how to build new elements of the language from already defined elements.

Step 3: Closure: The strings constructed in step 1 and step 2 are valid strings, all other string are invalid and not to be a part of language.

### Example of Recursive definition of languages

Recursive definition of EVEN length strings

Step 1: 2 is in EVEN.

Step 2: If x is in EVEN, then x+2 and x-2 are EVEN. For example, if 2 is even then, 4 is also even  number.

Step 3:

The strings constructed in step 1 and step 2 are valid strings (for example 2, 4, 6, 8,….)

All other string are invalid and not to be a part of language. (for example 1, 3, 5, 7, 9,….)