**Recursion** is a way of specifying a process by means of itself. More precisely (and to dispel the appearance of circularity in the definition), "complicated" instances of the process are defined in terms of "simpler" instances, and the "simplest" instances are given explicitly.

Mathematical linguist Noam Chomsky produced evidence that unlimited extension of a language such as English is possible only by the recursive device of embedding sentences in sentences. Thus, a talky little girl may say, "Dorothy, who met the wicked Witch of the West in Munchkin Land where her wicked Witch sister was killed, liquidated her with a pail of water." Clearly, two simple sentences -- "Dorothy met the Wicked Witch of the West in Munchkin Land", and "Her sister was killed in Munchkin Land" -- can be embedded in a third sentence, "Dorothy liquidated her with a pail of water", to obtain a very talky sentence.

The 1984 Nobel Prize laureate in Medicine and Physiology, Niels K. Jerne, used Chomsky's transformational-generative grammar model to explain the human immune system, equating "components of a generative grammar ... with various features of protein structures". The title of Jerne's Stockholm Nobel lecture was "The Generative Grammar of the Immune System".

Here is another, perhaps simpler way to understand recursive processes:

- Are we done yet? If so, return the results. Without such a
*termination condition*a recursion would go on forever. - If not,
*simplify*the problem, solve those simpler problem(s), and assemble the results into a solution for the original problem. Then return that solution.

Or perhaps more accurate is the following due to Andrew Plotkin - "If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to Douglas Hofstadter than you are; then ask him or her what recursion is."

Examples of mathematical objects often defined recursively are functions and sets.

The canonical example of a recursively defined set is the natural numbers:

- 0 is in
**N** - if
*n*is in**N**, then*n*+1 is in**N** - The natural numbers is the smallest set satisfying the previous two properties.

- 0, 1 are in
**N**; - if
*n*and*n*+1 are in**N**, then*n*+2 is in**N**; **N**is the smallest set satisfying the previous two properties.

Another interesting example is the set of all true "reachable" propositions in an axiomatic system.

- if a proposition is an axiom, it is a true reachable proposition.
- if a proposition can be obtained from true reachable propositions by means of inference rules, it is a true reachable proposition.
- The set of true reachable propositions is the smallest set of reachable propositions satisfying these conditions.

(*It needs to be pointed out that determining whether a certain object is in a recursively defined set is not an algorithmic task.*)

(*Insert definition of recursively defined set here*)

The canonical example of a recursively defined function is
the following definition of the factorial function *f*(*n*):

*f*(0) = 1*f*(*n*) =*n*·*f*(*n*-1) for any natural number*n*> 0

Another example is the definition of Fibonacci numbers.f(3) = 3 ·f(3-1) = 3 ·f(2) = 3 · 2 ·f(2-1) = 3 · 2 ·f(1) = 3 · 2 · 1 ·f(1-1) = 3 · 2 · 1 ·f(0) = 3 · 2 · 1 · 1 = 6

A common method of simplification is to divide a problem into subproblems of the same type. As a programming technique is called divide and conquer and is key to the design of many important algorithms, as well as being a fundamental part of dynamic programming.

Virtually all programming languages in use today allow the direct specification of recursive functions and procedures. When such a function is called, the computer keeps track of the various instances of the function by using a stack. Conversely, every recursive function can be transformed into an iterative function by using a stack.

Any function that can be evaluated by a computer can be expressed in terms of recursive functions, without use of iteration, and conversely.

Indeed, some languages designed for logic programming and functional programming provide recursion as the only means of repetition *directly* available to the programmer. Such languages generally make tail recursion as efficient as iteration, letting programmers express other repetition structures (such as Scheme's `map`

and `for`

) in terms of recursion.

Recursion is deeply embedded in the theory of computation, with the theoretical equivalence of recursive functions and Turing machines at the foundation of ideas about the universality of the modern computer.

John McCarthy's function, McCarthy's 91 is another example of a recursive function.

In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set *X*, an element *a* of *X* and a function *f* : *X* `->` *X*, the theorem states that there is a unique function *F* : **N** `->` *X* (where **N** denotes the set of natural numbers) such that

*F*(0) =*a**F*(*n*+1) =*f*(*F*(*n*))

Take two functions *f* and *g* of domain **N** and codomain *A* such that:

*i*. have equal domains/codomains;*ii*. have the same graphic.*i*. Done!*ii*. Mathematical induction: for all*n*in**N**,*f*(*n*) =*g*(*n*)? (We shall call this condition, say, Eq(*n*)):- 1.:Eq(0) iff
*f*(0) =*g*(0) iff a = a. Done! - 2.:Let
*n*be an element of**N**. Assuming that Eq(*n*) holds, we want to show that Eq(*n*+1) holds as well, which is easy because:*f*(*n*+1) =*F*(*f*(*n*)) =*F*(*g*(*n*)) =*g*(*n*+1). Done!

- 1.:Eq(0) iff

We define *F* on zero, then one and so on. Define *F*(0) = *a*;

Now assuming that *F* has been defined on all numbers less than equal to *n* we define *F*(*N*+1) = *f*( *F*(*N*) ). Note that since *F*(*N*) has been defined by hypothesis and that *f* is a function *X* to *X* so the right hand side is indeed a member of *X* and so this a correct definition.

Alternatively, and informally, *F*(*N*) is the result of applying *f* to *a* *N* times.

Some common recurrence relations are:

- Factorial --
*n*! =*n*· (*n*-1) · ... · 1 - Fibonacci numbers --
*f*(*n*) =*f*(*n*-1) +*f*(*n*-2) - Catalan numbers --
*C*(2*n*,*n*)/(*n*+1) - Computing compound interests
- The tower of Hanoi
- Ackermann function
- Population growth

- Recursion
- Self-reference
- Strange loop
- Recursive function
- Primitive recursive function
- Recursive acronym
- Decidable language
- Decidable set

- Richard Johnsonbaugh,
*Discrete Mathematics*5th edition. 1990 Macmillan