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In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of the class.

File:Safari browser self-portrait (small).png
An example of a recursive image

An example

For example, the following is a recursive definition of person's ancestors:

  • One's parents are one's ancestors (base case);
  • The parents of any ancestor are also ancestors of the person under consideration (recursion step).

For instance, your ancestors are:

  • your parents, and
  • your parents' parents (= grandparents), and
  • your grandparents' parents, and
  • everyone else you get by successively adding ancestors

It is convenient to think that a recursive definition defines objects in terms of "previously defined" objects of the class to define.

Definitions such as these are ubiquitous in mathematics. In fact, the formal definition of natural numbers is very similar: 0 is a natural number, and each natural number has a successor, which is also a natural number.

For visualizing recursion, it can be helpful to consider recursively-defined geometric figures, such as the Koch curve, the Sierpinski triangle, or the Cantor set.

Also, examples of recursion abound in natural language, often appearing as, or transforming into jokes. A consideration of these jokes is useful in developing an intuition for recursion.

For example, the process produced by responding to the question, "What do you mean what do I mean?" with "What do you mean 'What do you mean what do I mean?'?", can clearly go on forever, although it is unclear how many iterations can be meaningful...

Although artist- and poet-types often make jokes of this kind, studying recursion in the context of mathematics and programming languages can help to understand the meaning and philosophy of language in this sense, and a careful observation of language in this sense can help to understand recursion.

Recursion in mathematics

Mathematical recursion involves a function calling on itself over and over until reaching an end state. Each iteration increases the depth of the call. Once the end state is achieved, the function then backs all the way out, step by step. The key points of this kind of recursion are two fold:

  1. You have to have a function that calls itself with a smaller subset of the values with which it began.
  2. The function must be aware that an end state exists which terminates the recursive process.

Recursion in computing

Recursion in computer programming is where a function is defined in terms of itself. The commonly used example (using the Euphoria programming language, in this case) is the function used to calculate the factorial of an Integer: 

function Factorial ( integer X )
   if X < 0 then return "Invalid argument" end if
   if X = 0 then return 1 end if
   return Factorial(X-1) * X
end function

Although the recursive factorial function is calculating the same value as the iterative function below, depending on language implementation, recursive function may consume additional memory for each call. I.e. factorial(20) may use ten times more memory than factorial(10). The Scheme language is especially good at doing recursion efficiently, but naive recursive implementations in earlier versions of C were very inefficient.

Recursive functions are often regarded as more elegant than alternative implementations and they are usually shorter and simpler. For example, above function coded in the same language without recursion would be as follows: 

function Factorial ( integer X )
   integer temp_result
   if X < 0 then return "Invalid argument" end if
   temp_result = 1
   for i = 1 to X do
       temp_result *= i
   end for
   return temp_result
end function

Recursion in language

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.

Niels K. Jerne, the 1984 Nobel Prize laureate in Medicine and Physiology, 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:

  1. Are we done yet? If so, return the results. Without such a termination condition a recursion would go on forever.
  2. 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.

A more humorous illustration goes: "In order to understand recursion, one must first understand recursion." 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, sets, and especially fractals.

Recurrence relations or algorithms

Recurrence relations are equations to define one or more sequences recursively. Some specific kinds of recurrence relation can be "solved" to obtain a nonrecursive definition.

Recursively defined sets

Example: the natural numbers

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

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

Here's an alternative recursive definition of N:

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.

Example: The set of true reachable propositions

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.

This set is called 'true reachable propositions' because: in nonconstructive approaches to the foundations of mathematics, the set of true propositions is larger than the set recursively constructed from the axioms and rules of inference. See also Gödel's incompleteness theorems.

(Note that determining whether a certain object is in a recursively defined set is not an algorithmic task.)

Recursively defined functions

Functions whose domains can be recursively defined can be given recursive definitions patterned after the recursive definition of their domain.

The canonical example of a recursively defined function is the following definition of the factorial function f ( n ) {\displaystyle f(n)} : 


  
    
      
        f
        (
        0
        )
        =
        1
      
    
    {\displaystyle f(0)=1}
  


  
    
      
        f
        (
        n
        )
        =
        n
        ∗

        f
        (
        n
        −

        1
        )
      
    
    {\displaystyle f(n)=n*f(n-1)}
  
  for any natural number 
  
    
      
        n
        >
        0
      
    
    {\displaystyle n>0}

Given this definition, also called a recurrence relation, we work out f ( 3 ) {\displaystyle f(3)} as follows: 


  
    
      
        f
        (
        3
        )
        =
        3
        ∗

        f
        (
        3
        −

        1
        )
      
    
    {\displaystyle f(3)=3*f(3-1)}
  


  
    
      
        =
        3
        ∗

        f
        (
        2
        )
      
    
    {\displaystyle =3*f(2)}
  


  
    
      
        =
        3
        ∗

        2
        ∗

        f
        (
        2
        −

        1
        )
      
    
    {\displaystyle =3*2*f(2-1)}
  


  
    
      
        =
        3
        ∗

        2
        ∗

        f
        (
        1
        )
      
    
    {\displaystyle =3*2*f(1)}
  


  
    
      
        =
        3
        ∗

        2
        ∗

        1
        ∗

        f
        (
        1
        −

        1
        )
      
    
    {\displaystyle =3*2*1*f(1-1)}
  


  
    
      
        =
        3
        ∗

        2
        ∗

        1
        ∗

        f
        (
        0
        )
      
    
    {\displaystyle =3*2*1*f(0)}
  


  
    
      
        =
        3
        ∗

        2
        ∗

        1
        ∗

        1
      
    
    {\displaystyle =3*2*1*1}
  


  
    
      
        =
        6
      
    
    {\displaystyle =6}

Recursive algorithms

A common method of simplification is to divide a problem into subproblems of the same type. As a computer programming technique, this 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 (for most languages on most stack-based architectures) or the language implementation keeps track of the various instances of the function (on many architectures, by using a stack, although other methods may be used). 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.

To make a very literal example out of this: If an unknown word is seen in a book, the reader can make a note of the current page number and put the note on a stack (which is empty so far). The reader can then look the new word up and, while reading on the subject, may find yet another unknown word. The page number of this word is also written down and put on top of the stack. At some point an article is read that does not require any explanation. The reader then returns to the previous page number and continues reading from there. This is repeated, sequentially removing the topmost note from the stack. Finally, the reader returns to the original book. This is a recursive approach.

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 91 function is another example of a recursively defined function.

The Quicksort and Mergesort algorithms are also commonly done using recursion, which allows them to run in an average of O(n log n) time.

Many operations involving tree data structures also use recursion, as it is often quite difficult to iteratively traverse a tree of unknown length.

In addition, some numerical methods for finding approximate solutions to mathematical equations use recursion. In Newton's method, for example, an approximate root of a function is provided as initial input to the method. The calculated result (output) is then used as input to the method, with the process repeated until a sufficiently accurate value is obtained.

The recursion theorem

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

F ( 0 ) = a {\displaystyle F(0)=a}
F ( n + 1 ) = f ( F ( n ) ) {\displaystyle F(n+1)=f(F(n))}

for any natural number n {\displaystyle n} .

Proof of uniqueness

Take two functions f {\displaystyle f} and g {\displaystyle g} of domain N {\displaystyle N} and codomain A {\displaystyle A} such that:

f ( 0 ) = a {\displaystyle f(0)=a}
g ( 0 ) = a {\displaystyle g(0)=a}
f ( n + 1 ) = F ( f ( n ) ) {\displaystyle f(n+1)=F(f(n))}
g ( n + 1 ) = F ( g ( n ) ) {\displaystyle g(n+1)=F(g(n))}

where a {\displaystyle a} is an element of A {\displaystyle A} . We want to prove that f = g {\displaystyle f=g} . Two functions are equal if they:

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


you should consider N union {0} as a domain of F.

Proof of existence

Some common recurrence relations are:

Recursion in plain English

Recursion is the process a procedure goes through when one of the steps of the procedure involves rerunning the entire same procedure. A procedure that goes through recursion is said to be recursive. Something is also said to be recursive when it is the result of a recursive procedure.

To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps that are to be taken based on a set of rules. The running of a procedure involves actually following the rules and performing the steps. An analogy might be that a procedure is like a menu in that it is the possible steps, while running a procedure is actually choosing the courses for the meal from the menu.

A procedure is recursive if one of the steps that makes up the procedure calls for a new running of the procedure. Therefore a recursive four course meal would be a meal in which one of the choices of appetizer, salad, entrée, or dessert was an entire meal unto itself. So a recursive meal might be potato skins, baby greens salad, chicken parmesan, and for dessert, a four course meal, consisting of crab cakes, Caesar salad, for an entrée, a four course meal, and chocolate cake for dessert, so on until each of the meals within the meals is completed.

A recursive procedure must complete every one of its steps. Even if a new running is called in one of its steps, each running must run through the remaining steps. What this means is that even if the salad is an entire four course meal unto itself, you still have to eat your entrée and dessert.

Recursive humour

A common geeky joke (for example ) is the following "definition" of recursion.

Recursion
See "Recursion".

This is a parody on references in dictionaries, which in some careless cases may lead to circular definitions; in fact the above is the shortest possible one. Every joke has an element of wisdom, and also an element of misunderstanding. This one is also the second-shortest possible example of an erroneous recursive definition of an object, the error being the absence of the termination condition (or lack of the initial state, if to look at it from an opposite point of view). Newcomers to recursion are often bewildered by its apparent circularity, until they learn to appreciate that a termination condition is key.

Other examples are recursive acronyms, such as GNU, PHP or TTP (Dilbert; "The TTP Project").

See also

References

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  • . ISBN 0072930330. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)
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External links

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