# Ken Iverson

Ken Iverson in 1989

Kenneth Eugene Iverson (17 December 1920 – 19 October 2004) was a Canadian mathematician and computer scientist, noted for the development of APL. He was honored with the Turing Award in 1979 "for his pioneering effort in programming languages and mathematical notation resulting in what the computing field now knows as APL; for his contributions to the implementation of interactive systems, to educational uses of APL, and to programming language theory and practice".

## History

Main article: Kenneth E. Iverson

While teaching mathematics, Iverson developed an alternative mathematical notation and later described, in his book A Programming Language, how this could be used for instructing computers. It was based on this book that Larry Breed and Philip Abrams implemented the first interpreter, and it eventually was named A.P.L. acronymising the book title.

On 17 December 2020, BAA organised an Iverson Centernary event, celebrating the 100th anniversary of Iverson's birth.

### Floor and Ceiling

Carl Friedrich Gauss introduced the square bracket notation ${\displaystyle [x]}$ in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Iverson introduced the names "floor" and "ceiling" and the corresponding notations ${\displaystyle \lfloor x\rfloor }$ and ${\displaystyle \lceil x\rceil }$ in his 1962 book A Programming Language. The names and notations have gained widespread use in mathematics.

### Iverson bracket

In mathematics, the Iverson bracket generalises the Kronecker delta. It converts any logical proposition into a number that is 1 if the proposition is satisfied, and 0 otherwise, and is generally written by putting the proposition inside square brackets:

${\displaystyle [P]={\begin{cases}1&{\text{if }}P{\text{ is true;}}\\0&{\text{otherwise,}}\end{cases}}}$

where ${\displaystyle P}$ is a statement that can be true or false.

In the context of summation, the notation can be used to write any sum as an infinite sum without limits: If ${\displaystyle P(k)}$ is any property of the integer ${\displaystyle k}$,

${\displaystyle \sum _{k}f(k)\,[P(k)]=\sum _{P(k)}f(k).}$

Note that by this convention, a summand ${\displaystyle f(k)[{\textbf {false}}]}$ must evaluate to 0 regardless of whether ${\displaystyle f(k)}$ is defined. Likewise for products:

${\displaystyle \prod _{k}f(k)^{[P(k)]}=\prod _{P(k)}f(k).}$

While the Iverson bracket was adopted into mainstream mathematics, it's use has not been very widespread. Donald Knuth has argued strongly for its wider use.

In APL, the Iverson bracket is implied in all logical propositions, for example 4≥3 evaluates to 1 and 'hello'≡'world' evaluates to 0.

## Naming things

Iverson's work on programming languages sometimes crossed over with his studies of the English language. Roger Hui recounts that "Ken was deeply interested in words, their use and their etymology. He indeed did read the dictionary, and kept a copy of the American Heritage Dictionary (along with other dictionaries) by his easy chair for ready reference."[1] Iverson also remarked on an interest in etymology among early APLers at IBM.[2] He chose names for APL concepts carefully and sometimes relied on obscure meanings, for example "ravel" in the sense of unmaking a knitted fabric[3] and nub meaning the essential part. A suggestion of "coadunate representation" regarding reference counting in SHARP APL[4] was changed to "joint representation" by the time the feature was implemented.[5] In A Dictionary of APL, Iverson began to describe programming syntax in terms of English grammar, a choice that carried through to J—implementer Hui shared and encouraged Iverson's interest in English.

The name APL was chosen by Adin Falkoff and not Iverson, albeit based on his A Programming Language. However, some of Iverson's names have become accepted in mainstream mathematics or programming, such as "floor" and "ceiling" as mentioned above, and reduction, which may have been named in part for how it reduces the argument's rank by one. It's possible that Iverson coined the term "bubble sort", as A Programming Language is the first publication known to use it.[6][7]