Recursion

Recursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition; specifically it is defining an infinite statement using finite components. The term is also used more generally to describe a process of repeating objects in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion.

Formal definitions of recursion

In mathematics and computer science, a class of objects or methods exhibit recursive behavior when they can be defined by two properties:

A simple base case (or cases), and
A set of rules which reduce all other cases toward the base case.

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

One's parents are one's ancestors (base case).
The parents of one's ancestors are also one's ancestors (recursion step).

The Fibonacci sequence is a classic example of recursion:

Fib(0) is 0 [base case]
Fib(1) is 1 [base case]
For all integers n > 1: Fib(n) is (Fib(n-1) + Fib(n-2)) [recursive definition]

A convenient mental model of recursion defines the recursive object (whether that object is an equation, an algorithm, an image, or a rule) in terms of "previously defined" objects of the same class. For example: How do you move a stack of 100 boxes? Answer: you move one box, remember where you put it, and then solve the smaller problem: how do you move a stack of 99 boxes? Eventually, you're left with the problem of how to move a single box, which you know how to do.

Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers in set theory follows: 1 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers

A more humorous illustration goes: "To understand recursion, you must first understand recursion." Or perhaps more accurate is the following, from 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."

Recursively defined mathematical objects include functions, sets, and especially fractals.

Recursion in language

Linguist Noam Chomsky theorizes that unlimited extension of a language such as English is possible using the recursive device of embedding phrases within sentences. Thus, a chatty person 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 verbose sentence.

The idea that recursion is an essential property of human language is challenged by linguist Daniel Everett in his work Cultural Constraints on Grammar and Cognition in Pirahã: Another Look at the Design Features of Human Language in which he hypothesizes that cultural factors made recursion unnecessary in the development of the Pirahã language. This concept, which challenges Chomsky's idea that recursion is the only trait which differentiates human and animal communication, is currently under debate.

Recursion in linguistics enables 'discrete infinity' by embedding phrases within phrases of the same type in a hierarchical structure. Without recursion, language does not have 'discrete infinity' and cannot embed sentences into infinity (with a 'Russian nesting doll' effect). Everett contests that language must have discrete infinity, and that the Pirahã language - which he claims lacks recursion - is in fact finite. He likens it to the finite game of chess, which has a finite number of moves but is nevertheless very productive, with novel moves being discovered throughout history.

Recursion in plain English

Recursion is the process a procedure goes through when one of the steps of the procedure involves rerunning the 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 in reference to the analogy of the menu is that even if the salad is an entire four-course meal unto itself, you still have to eat your entrée and dessert.

Recursive humor

A common joke is the following "definition" of recursion.^[1]

Recursion

See "Recursion".

This is a parody on references in dictionaries, which in some cases may lead to circular definitions among related words. Jokes often have an element of wisdom, and also an element of misunderstanding. This one is also the 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 looked at 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.

An example of this can be found by searching Google for the term.^[2] Google puts the searcher in an endless cycle, of suggesting "Recursion" as the word they were trying to spell, even though this is the input they just used.

A variation on this joke is:

Recursion

If you still don't get it, see: "Recursion".

which actually does terminate, as soon as the reader "gets it".

Another example occurs in an index entry on page 269 of Kernighan and Ritchie's book "The C Programming Language":

recursion 86, 139, 141, 182, 202, 269

Other examples are recursive acronyms, such as GNU, PHP, YAML or HURD.

Recursion in mathematics

File:Sierpinski Triangle.svg

A Sierpinski triangle—a confined recursion of triangles to form a geometric lattice.

Recursively defined sets

Example: the natural numbers

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

1 is in

\mathbb {N}

if n is in

\mathbb {N}

, then n + 1 is in

\mathbb {N}

The set of natural numbers is the smallest set of real numbers satisfying the previous two properties.

(The doubt with this definition is that we assume: 1.we understand the "+" operation and 2.n + 1 is not in current $\mathbb {N}$ . These two assumptions mean that before we understand the natural numbers, we already know the "+" operation on them.)

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 non-constructive 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.)

Functional recursion

A function may be partly defined in terms of itself. A familiar example is the Fibonacci number sequence: F(n) = F(n − 1) + F(n − 2). For such a definition to be useful, it must lead to values which are non-recursively defined, in this case F(0) = 0 and F(1) = 1.

A famous recursive function is the Ackermann function which, unlike the Fibonacci sequence, cannot be expressed without recursion.

Recursive proofs

Applying the standard technique of proof by cases to recursively-defined sets or functions as in the preceding sections yields structural induction, a powerful generalization of mathematical induction which is widely used to derive proofs in mathematical logic and computer science.

For instance, the standard way to define new systems of mathematics or logic is to define objects (such as "true" and "false", or "all natural numbers"), then define operations on these. These are the base cases. After this, all valid computations in the system are defined with rules for assembling these. In this way, if the base cases and rules are all proven to be calculable, then any formula in the mathematical system will also be calculable.

While the above example may seem unexciting, this type of proof is the normal way to prove that a calculation is impossible. This can often save a lot of time. For example, this type of proof was used to prove that the area of a circle is not a simple ratio of its diameter, and that no angle can be trisected with compass and straightedge—both puzzles that fascinated the ancients.

Recursive optimization

Dynamic programming is an approach to optimization which restates a multiperiod or multistep optimization problem in recursive form. The key result in dynamic programming is the Bellman equation, which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step).

Recursion in computer science

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. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach is dynamic programming. This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached.

A classic example of recursion is the definition of the factorial function, given here in C code:

 unsigned int factorial(unsigned int n) 
 {
     if (n <= 1) 
         return 1;
     else
         return n * factorial(n-1);
 }

The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n, until reaching the base case, analogously to the mathematical definition of factorial.

Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of itself. The solution to the problem is then devised by combining the solutions obtained from the simpler versions of the problem. One example application of recursion is in parsers for programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program.

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

Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually simplicity. The main disadvantage is often that the algorithm may require large amounts of memory if the depth of the recursion is very large.

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 $f:X\rightarrow X$ , the theorem states that there is a unique function $F:\mathbb {N} \rightarrow X$ (where $\mathbb {N}$ denotes the set of natural numbers including zero) such that

F(0)=a

F(n+1)=f(F(n))

for any natural number n.

Proof of uniqueness

Take two functions $F:\mathbb {N} \rightarrow X$ and $G:\mathbb {N} \rightarrow X$ 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 X.

It can be proved by mathematical induction that $F(n)=G(n)$ for all natural numbers n:

Base Case:

F(0)=a=G(0)

so the equality holds for

n=0

.

Inductive Step: Suppose

F(k)=G(k)

for some

k\in \mathbb {N}

. Then

F(k+1)=f(F(k))=f(G(k))=G(k+1).

Hence F(k) = G(k) implies F(k+1) = G(k+1).

By Induction, $F(n)=G(n)$ for all $n\in \mathbb {N}$ .

Examples

Some common recurrence relations are:

Factorial: $n!=n(n-1)!=n(n-1)\cdots 1$
Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$
Catalan numbers: $C_{0}=1$ , $C_{n+1}=(4n+2)C_{n}/(n+2)$
Computing compound interest
The Tower of Hanoi
Ackermann function
Population growth rate
Odds of a shared birthday among a group of people.

References

^ "recursion". Catb.org. Retrieved 2010-04-07.
^ "A Google search for the word "Recursion"". Google.com. 2010-03-03. Retrieved 2010-04-07.

External links

Johnsonbaugh, Richard (2004). Discrete Mathematics. Prentice Hall. ISBN 0-13-117686-2.
Hofstadter, Douglas (1999). Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books. ISBN 0-465-02656-7.
Shoenfield, Joseph R. (2000). Recursion Theory. A K Peters Ltd. ISBN 1-56881-149-7.
Causey, Robert L. (2001). Logic, Sets, and Recursion. Jones & Bartlett. ISBN 0-7637-1695-2.
Cori, Rene; Lascar, Daniel; Pelletier, Donald H. (2001). Recursion Theory, Godel's Theorems, Set Theory, Model Theory. Oxford University Press. ISBN 0-19-850050-5.{{cite book}}: CS1 maint: multiple names: authors list (link)
Barwise, Jon; Moss, Lawrence S. (1996). Vicious Circles. Stanford Univ Center for the Study of Language and Information. ISBN 0-19-850050-5.{{cite book}}: CS1 maint: multiple names: authors list (link) - offers a treatment of corecursion.
Rosen, Kenneth H. (2002). Discrete Mathematics and Its Applications. McGraw-Hill College. ISBN 0-07-293033-0.
Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, Clifford Stein (2001). Introduction to Algorithms. Mit Pr. ISBN 0-262-03293-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
Kernighan, B.; Ritchie, D. (1988). The C programming Language. Prentice Hall. ISBN 0-13-110362-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
Stokey, Nancy,; Robert Lucas; Edward Prescott (1989). Recursive Methods in Economic Dynamics. Harvard University Press. ISBN 0674750969.{{cite book}}: CS1 maint: multiple names: authors list (link)
Hungerford; title=Algebra (1980). Springer. ISBN 978-0387905181. {{cite book}}: Missing or empty |title= (help); Missing pipe in: |author= (help)CS1 maint: multiple names: authors list (link), first chapter on set theory.
Recursion - tutorial by Alan Gauld
A Primer on Recursion- contains pointers to recursion in Formal Languages, Linguistics, Math and Computer Science
Zip Files All The Way Down

[1] "recursion". Catb.org. Retrieved 2010-04-07.

[2] "A Google search for the word "Recursion"". Google.com. 2010-03-03. Retrieved 2010-04-07.

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