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Recursion
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Recursion
Recursion is the process of repeating items 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.
The term has a variety of meanings specific to a variety of disciplines
ranging from linguistics to logic. The most common application of
recursion is in mathematics and computer science, in which it refers to
a method of defining functions in which the function being defined is
applied within its own definition. Specifically this defines an infinite
number of instances (function values), using a finite expression that for
some instances may refer to other instances, but in such a way that no
loop or infinite chain of references can occur. The term is also used
more generally to describe a process of repeating objects in a
self-similar way.
A visual form of recursion known as the Droste
effect. The woman in this image is holding an
object which contains a smaller image of her
holding the same object, which in turn contains a
smaller image of herself holding the same object,
and so forth.
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:
1. A simple base case (or cases), and
2. 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]
Recursion in a screen recording program, where
the smaller window contains a snapshot of the
entire screen.
• For all integers n > 1: Fib(n) is (Fib(n-1) + Fib(n-2)) [recursive definition]
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
Recursion
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 (as Chomsky suggests) 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. Andrew Nevins, David Pesetsky
and Cilene Rodrigues provide a debate against this proposal.[1] Indirect proof that Everett's ideas are wrong comes
from works in neurolinguistics where it appears that all human beings are endowed with the very same
neurobiological structures to manage with all and only recursive languages. For a review, see Kaan et al. (2002)
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 invoking the
procedure itself. A procedure that goes through recursion is said to be 'recursive'.
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 cookbook in that it is the
possible steps, while running a procedure is actually preparing the meal.
Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of
some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in
turn requires heating water, and so forth. However, a recursive procedure is special in that (at least) one of its steps
calls for a new instance of the very same procedure. This of course immediately creates the danger of an endless
loop; recursion can only be properly used in a definition if the step in question is skipped in certain cases so that the
procedure can complete. Even if properly defined, a recursive procedure is not easy for humans to perform, as it
requires distinguishing the new from the old (partially executed) invocation of the procedure; this requires some
administration of how far various simultaneous instances of the procedures have progressed. For this reason
recursive definitions are very rare in everyday situations. An example could be the following procedure to find a way
through a maze. Proceed forward until reaching either an exit or a branching point (a dead end is considered a
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Recursion
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branching point with 0 branches). If the point reached is an exit, terminate. Otherwise try each branch in turn, using
the procedure recursively; if every trial fails by reaching only dead ends, return on the path that led to this branching
point and report failure. Whether this actually defines a terminating procedure depends on the nature of the maze: it
must not allow loops. In any case, executing the procedure requires carefully recording all currently explored
branching points, and which of their branches have already been exhaustively tried.
Recursive humor
A common joke is the following "definition" of recursion.[2]
Recursion
See "Recursion".
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 some editions of Kernighan and Ritchie's book "The C
Programming Language":
recursion 86, 139, 141, 182, 202, 269
The earliest version of this joke was in "Software Tools" by Kernighan and Plauger, and also appears in "The UNIX
Programming Environment" by Kernighan and Pike. It did not appear in the first edition of The C Programming
Language.
A Google search for Recursion suggests Did you mean: Recursion [3]
Other examples are recursive acronyms, such as GNU, PHP, YAML, HURD or WINE.
Recursion in mathematics
Recursively defined sets
Example: the natural numbers
The canonical example of a recursively defined set is
given by the natural numbers:
1 is in
if n is in
, then n + 1 is in
The set of natural numbers 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.
A Sierpinski triangle—a confined recursion of triangles to form a
geometric lattice.
• 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 propositions satisfying these conditions.
Recursion
This set is called 'true reachable propositions' because in non-constructive approaches to the foundations of
mathematics, the set of true propositions may be larger than the set recursively constructed from the axioms and
rules of inference. See also Gödel's incompleteness theorems.
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 easily be
expressed without recursion.
Proofs involving recursive definitions
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.
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.
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Recursion
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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
, the theorem states that there is a unique function
(where
denotes
the set of natural numbers including zero) such that
for any natural number n.
Proof of uniqueness
Take two functions
and
such that:
where a is an element of X.
It can be proved by mathematical induction that
Base Case:
Inductive
for all natural numbers n:
so the equality holds for
Step:
Suppose
for
Hence F(k) = G(k) implies F(k+1) = G(k+1).
By Induction,
for all
.
.
Examples
Some common recurrence relations are:
•
Factorial:
•
Fibonacci numbers:
•
Catalan numbers:
•
Computing compound interest
•
The Tower of Hanoi
•
Ackermann function
,
some
.
Then
Recursion
Bibliography
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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.
Barwise, Jon; Moss, Lawrence S. (1996). Vicious Circles. Stanford Univ Center for the Study of Language and
Information. ISBN 0-19-850050-5. - 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.
Kernighan, B.; Ritchie, D. (1988). The C programming Language. Prentice Hall. ISBN 0-13-110362-8.
Stokey, Nancy,; Robert Lucas; Edward Prescott (1989). Recursive Methods in Economic Dynamics. Harvard
University Press. ISBN 0674750969.
Hungerford (1980). Algebra. Springer. ISBN 978-0387905181., first chapter on set theory.
References
[1] Nevins, Andrew; Pesetsky, David; Rodrigues, Cilene (2009). "Evidence and argumentation: A reply to Everett (2009)" (http:/ / web. mit. edu/
linguistics/ people/ faculty/ pesetsky/ Nevins_Pesetsky_Rodrigues_2_Evidence_and_Argumentation_Reply_to_Everett. pdf) (PDF).
Language 85 (3): 671–681. doi:10.1353/lan.0.0140. .
[2] "recursion" (http:/ / catb. org/ ~esr/ jargon/ html/ R/ recursion. html). Catb.org. . Retrieved 2010-04-07.
[3] Google Search for word recursion (http:/ / www. google. com/ search?q=recursion)
External links
• Recursion (http://www.freenetpages.co.uk/hp/alan.gauld/tutrecur.htm) - tutorial by Alan Gauld
• A Primer on Recursion (http://amitksaha.files.wordpress.com/2009/05/recursion-primer.pdf)- contains
pointers to recursion in Formal Languages, Linguistics, Math and Computer Science
• Zip Files All The Way Down (http://research.swtch.com/2010/03/zip-files-all-way-down.html)
• Nevins, Andrew and David Pesetsky and Cilene Rodrigues. Evidence and Argumentation: A Reply to Everett
(2009). Language 85.3: 671--681 (2009) (http://www.ucl.ac.uk/psychlangsci/staff/linguistics-staff/
nevins-publications/npr09b)
• Kaan, E. – Swaab, T. Y. (2002) “The brain circuitry of syntactic comprehension”, Trends in Cognitive Sciences,
vol. 6, Issue 8, 350-356. (http://faculty.washington.edu/losterho/kaan_and_swaab.pdf)
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Image Sources, Licenses and Contributors
Image:Droste.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Droste.jpg License: Public Domain Contributors: Jan (Johannes) Musset?
File:Screenshot Recursion via vlc.png Source: http://en.wikipedia.org/w/index.php?title=File:Screenshot_Recursion_via_vlc.png License: GNU General Public License Contributors: vlc
team, ubuntu, Hidro (talk)
Image:Sierpinski Triangle.svg Source: http://en.wikipedia.org/w/index.php?title=File:Sierpinski_Triangle.svg License: Public Domain Contributors: Marco Polo, 2 anonymous edits
License
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