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.
An example of a recursive image
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" 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.
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:
- 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.
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):
f(0) = 1
f(n) = n * f(n - 1) for any natural number n > 0
Given this definition, also called a recurrence relation, we work out f(3) as follows:
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
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.
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
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 Recursion Theorem
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))
for any natural number n.
Proof of Uniqueness
Take two functions f and g of domain N and codomain A such that:
f(0) = a
g(0) = a
f(n + 1) = F(f(n))
g(n + 1) = F(g(n))
where a is an element of A. We want to prove that 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 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!
Proof of Existence
[See Hungerford, "Algebra", first chapter on set theory]
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.
It is important to note that 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.
A common geeky joke (for example ) is the following "definition" of 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.
Another example is recursive acronyms, such as GNU.
Further reading and references
- Richard Johnsonbaugh : Discrete Mathematics, Prentice Hall, ISBN 0131176862
Douglas Hofstadter: Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, ISBN 0465026567
- Joseph R. Shoenfield : Recursion Theory, A K Peters, ISBN 1568811497
- Robert L. Causey : Logic, Sets, and Recursion, Jones & Bartlett Publishers, ISBN 0763716952
- Rene Cori , Daniel Lascar , Donald H. Pelletier : Recursion Theory, Godel's Theorems, Set Theory, Model Theory, Oxford University Press, ISBN 0198500505