A Programming Language: Difference between revisions

'''''A Programming Language''''' is a book published in 1962 by [[Kenneth E. Iverson]] to describe one iteration of his [[Iverson notation]]. The book's title later was used to form the acronym APL. At the time of writing Iverson notation was used for mathematics and description of IBM's hardware, and its purely human purposes are reflected in the loose conventions (relative to APL) and two-dimensional structure of the notation presented in ''A Programming Language''.

''A Programming Language'' does not feature a full multidimensional [[array model]]. Rather, operations are defined on [[scalars]], [[vectors]], and [[matrices]] and higher-[[rank]] arrays are not discussed. Nonetheless, it features many of the array conveniences that became characteristics of APL:

* The [[shape]] vector is not used, but dimension functions are present: <math>ν</math> gives the length of a vector, and <math>ν</math> and <math>μ</math> give the row length and column length of a matrix.

* The arithmetic [[scalar functions]] [[Sum]] (<math>+</math>), [[Difference]] (<math>-</math>), [[Product]] (<math>×</math>), and [[Quotient]] (<math>÷</math>) are defined as in mathematics.

* The logical functions [[And]] (<math>∧</math>) and [[Or]] (<math>∨</math>) also match mathematical usage. [[Logical negation]] is denoted with an overbar.

* The functions [[Absolute Value]], [[Ceiling]], and [[Floor]] use paired symbols (<math>|a|</math>, <math>⌈a⌉</math>, and <math>⌊a⌋</math>). The paired symbols for ceiling and floor were adopted by mathematicians (absolute value was already in use), but APL implementations dropped the closing symbol for consistency with monadic function syntax.

* [[Iota]] is called "Interval" and uses the letter <math>\iota</math> with an optional [[index origin]] subscript.

* [[Reduction]] (<math>/</math>, or <math>//</math> instead of <source lang=apl inline>⌿</source>) starts from the left rather than the right. For reductions of [[empty]] arrays, the [[identity element]] is returned.

* [[Rotate]] is written with arrows: <math>↑</math> for left rotation and <math>↓</math> for right rotation.

* [[Indexing]] is written with a subscript, or <math>∫_j</math> to allow [[index origin]] specification.

* [[Grade]] is called "ordering", and the Grade of <math>x</math> with [[index origin]] <math>j</math> is written <math>0_j/x</math>

* [[Base]] (<math>⊥</math>) on vectors works like in APL. On matrices, rows are paired up, or columns with a doubled base symbol.

* The [[Intersection]] and [[Union]] are written with <math>∩</math> and <math>∪</math>, and the [[Set Difference]] with <math>\Delta</math>.

* The [[Inner Product]] is written by placing one scalar function above another, e.g. <math>u\,^+_×v</math>, and the [[Outer Product]] by using <math>∘</math> in place of the top function with two vector arguments.

{{APL programming language}}