Recursion and Dorothy Height: Difference between pages

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'''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.

{{Infobox Biography

|subject_name=Dorothy I. Height|120px

|image_name= DorothyHeight_Book_Nordstrom_VA_15feb97.jpg

|image_caption= Dorothy Irene Height

|date_of_birth=[[March 24]], [[1912]]

|place_of_birth=[[Richmond, Virginia]] {{USA}}

|date_of_death=

|place_of_death=

}}

'''Dorothy Irene Height''' (born [[March 24]], [[1912]]) is an [[African American]] [[Public administration|administrator]], [[educator]], social [[Activism|activist]], and a recipient of the [[Congressional Gold Medal]].

Height was born in [[Richmond, Virginia]]. At an early age, she moved with her family to [[Rankin, Pennsylvania]]. While in high school, Height was awarded a scholarship to [[Barnard College]] for her oratory skills, but upon arrival was denied entrance. (At the time, Barnard admitted only two African Americans per academic year and Dorothy had arrived after the other two admittees.) Years later, at its [[1980]] commencement ceremonies, the college awarded Height its highest honor, the [[List_of_Barnard_College_people#Recipients_of_the_Medal_of_Distinction|

Examples of mathematical objects often defined recursively are [[function]]s and [[set]]s.

Barnard Medal of Distinction]]. She later pursued studies at [[New York University]], where she earned her Master's Degree in psychology.

Height started working as a caseworker with the New York City Welfare Department, but at the age of twenty-five, she began her [[civil rights]] activist's career when she joined the [[National Council of Negro Women]]. She fought for equal rights for both African Americans and women, and in [[1944]] she joined the national staff of the [[YWCA]]. She served as National President of [[Delta Sigma Theta]], Incorporated from 1946-1957.

The canonical example of a recursively defined function is

the following definition of the [[factorial]] function:

She remained active with the organization until [[1977]], and while there she developed leadership training programs and interracial and ecumenical education programs. In [[1957]], Height was named president of the National Council of Negro Women, a position she held until [[1997]]. During the height of the [[American Civil Rights Movement (1955-1968)|civil rights movement]] of the 1960s, Height organized "[[Wednesdays in Mississippi]]", which brought together black and white women from the North and South to create a dialogue of understanding. American leaders regularly took her counsel, including [[First Lady]] [[Eleanor Roosevelt]], and Height also encouraged [[President of the United States|President]] [[Dwight D. Eisenhower]] to desegregate schools and President [[Lyndon B. Johnson]] to appoint African American women to positions in government.

:0! = 1

:''n''! = ''n'' * (''n''-1)!   for any [[natural number]] ''n''>0

[[Image:dorothyheight.jpg|left|thumb|225px|Dorothy Height]] Height has served on a number of committees, including as a consultant on African affairs to the Secretary of State, the President's Committee on the Employment of the Handicapped, and the President's Committee on the Status of Women. She has received the [[Presidential Medal of Freedom]], the ''[[Franklin Delano Roosevelt]] [[Freedom From Want Award]]'' and the [[Spingarn Medal]] from the [[NAACP]]. She has also been inducted into the [[National Women's Hall of Fame]].

Therefore, given this definition, we work out 3! as follows:

In [[2004]], she was awarded the [[Congressional Gold Medal]] by President [[George W. Bush]] on behalf of the [[United States Congress]].

3! = 3 * (3-1)!

= 3 * 2!

= 3 * 2 * (2-1)!

= 3 * 2 * 1!

= 3 * 2 * 1 * (1 - 1)!

= 3 * 2 * 1 * 1

= 6

Dr. Height is currently, at age 95, the Chairperson of the Executive Committee of the [[Leadership Conference on Civil Rights]], the largest civil rights organization in the USA.

Another example is the definition of [[Fibonacci number|Fibonacci numbers]].

The musical stageplay ''If This Hat Could Talk'', based on her memoirs "Open Wide The Freedom Gates", opened in the summer of 2005 and is currently on tour. It showcases her unique perspective on the civil rights movement and details many of the behind-the-scenes figures/mentors who shaped her life, including [[Mary McLeod Bethune]] and [[Eleanor Roosevelt]].

In [[set theory]] there is a theorem guaranteeing that such functions exist.

==References==

'''The recursion theorem.''' Given a set ''X'', an element ''a'' of ''X'' and a function ''f'':''X''->''X'', then there is a unique function ''F'':'''N'''->''X'' such that

*Height, Dorothy. ''Open Wide the Freedom Gates: A Memoir.''

:''F''(0)=''a'', and

:''F''(''n''+1)=''f''(''F''(''n''))   for any natural number ''n''>0.

[[Category:1912 births|Height, Dorothy]]

''[A proof of the recursion theorem from set theory is needed]''

[[Category:Living people|Height, Dorothy]]

[[Category:African Americans|Height, Dorothy]]

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

[[Category:Congressional Gold Medal recipients|Height, Dorothy]]

:0 is in '''N'''

:if ''n'' is in '''N''', then ''n''+1 is in '''N'''

The natural numbers can be defined as the smallest set satisfying the definition.

Another interesting example is the set of all true propositions in an [[axiomatic system]].

:if a proposition is an axiom, it is true.

:if a proposition can be obtained from true propositions by means of inference rules, it is true.

''[It needs to be pointed out that determining whether a certain object is in a recursively defigned set is not an algorithmic task]''

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

#Are we done yet? If so, return the results.

#If not, ''simplify'' the problem and send it to 1.

A common method of simplification is to divide the problem into subproblems. Such a programming technique is called ''divide-et-impera'' or ''divide and conquer'' and is a fundamental part of [[dynamic programming]].

Virtually all [[programming language|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 program can be transformed into an interative program 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]].

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 programming language|Scheme's]] <code>map</code> and <code>for</code>) in terms of recursion.

Recursion is deeply embedded in the [[theory of computation]], with the theoretical equivalence of [[recursive function]]s and [[Turing machine]]s at the foundation of ideas about the universality of the modern computer.

See also:

* [[Recursion]]

* [[Self-reference]]

* [[Primitive recursive function]]