Recursive language

Recursive language is a special kind of language studied in mathematics and theoretical computer science. It refers to a set of strings made from an alphabet that can be completely decided by a Turing machine. This means a computer, following a set of rules, can always tell whether a given string belongs to the language or not.

In simpler terms, a recursive language is one where a computer can always give a clear yes or no answer about whether a phrase fits the language's rules. These languages are important because they help us understand the limits of what computers can solve.

All common types of languages, like regular languages used in searching and context-free languages used in programming, are examples of recursive languages. This shows how recursive languages include many of the languages we use every day in computing and logic. For more about how computers use self-repeating functions, see Recursion (computer science).

The idea of recursive languages helps scientists and mathematicians explore what problems can be solved by computers, guiding the development of new algorithms and understanding of computation.

Definitions

There are two main ways to think about a recursive language:

1. A recursive language is a special kind of group made from all possible combinations of symbols from an alphabet that have a limited length. These combinations are called words.
2. A recursive language is also a type of language for which a special kind of machine, called a Turing machine, can tell us whether a word belongs to the language or not.

We can prove that a problem can be solved by showing a Turing machine that runs a algorithm and stops for every possible input. If a problem cannot be solved this way, it is called an undecidable problem.

Examples

Every language that follows a special pattern is recursive. For example, consider the set of words made from the letters a, b, and c, where each word has the same number of a's, b's, and c's in that order, like "abc", "aabbcc", or "aaabbbccc". This type of language is recursive because we can always tell if a word fits this pattern.

Another example of a recursive language is harder to explain simply. It involves a special kind of math that deals with numbers and addition but not multiplication. While this math can be written in a way that computers can understand, checking whether a statement in this math is true can take a very long time. This shows that some languages can be decided by a computer but are not as simple as the examples above.

Closure properties

Recursive languages stay recursive when you combine them in certain ways. If L and P are two recursive languages, then combining them using these operations will also give you recursive languages:

• The Kleene star L ∗
• The image φ(L) under an e-free homomorphism φ
• The concatenation L ∘ P
• The union L ∪ P
• The intersection L ∩ P
• The complement of L
• The set difference L − P

The last property works because set difference can be shown using intersection and complement.