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Tacit functions apply to implicit arguments | Tacit functions apply to implicit arguments. This is in contrast to the explicit use of arguments in [[dfn]]s (<source inline lang=apl>⍺ ⍵</source>) and [[tradfn]]s (which have named arguments). Some APL dialects allow to combine functions into '''trains''' following a small set of rules. This allows creating complex derived functions without specifying any arguments explicitly. | ||
Known dialects which implement trains are [[Dyalog APL]], [[dzaima/APL]], [[ngn/apl]] and [[NARS2000]]. | |||
== Primitives == | == Primitives == | ||
| Line 34: | Line 36: | ||
A train is a series of functions in isolation. An isolated function is either surrounded by parentheses or named. Below, <source lang=apl inline>⍺</source> and <source lang=apl inline>⍵</source> refer to the arguments of the train. <source lang=apl inline>f</source>, <source lang=apl inline>g</source>, and <source lang=apl inline>h</source> are functions (which themselves can be tacit or not), and <source lang=apl inline>A</source> is an array. The arguments are processed by the following rules: | A train is a series of functions in isolation. An isolated function is either surrounded by parentheses or named. Below, <source lang=apl inline>⍺</source> and <source lang=apl inline>⍵</source> refer to the arguments of the train. <source lang=apl inline>f</source>, <source lang=apl inline>g</source>, and <source lang=apl inline>h</source> are functions (which themselves can be tacit or not), and <source lang=apl inline>A</source> is an array. The arguments are processed by the following rules: | ||
=== Forks === | |||
A 3-train is a ''fork'': | A 3-train is a ''fork'': | ||
{| | {| | ||
| Line 53: | Line 50: | ||
|} | |} | ||
Only [[dzaima/APL]] allows <source lang=apl inline>(A h)</source>, which it treats as <source lang=apl inline>A∘h</source>.<ref>dzaima/APL: [https://github.com/dzaima/APL/blob/ceea05e25687988ed0980a4abf4b9249b736543f/docs/differences.txt#L19 Differences from Dyalog APL]. Retrieved 09 Jan 2020.</ref> | === Atops === | ||
A 2-train is an ''atop'': | |||
{| | |||
|<source lang=apl> (g h) ⍵</source>|| {{←→}} ||<source lang=apl>g ( h ⍵)</source> | |||
|- | |||
|<source lang=apl>⍺ (g h) ⍵</source>|| {{←→}} ||<source lang=apl>g (⍺ h ⍵)</source> | |||
|} | |||
Only [[dzaima/APL]] allows <source lang=apl inline>(A h)</source>, which it treats as <source lang=apl inline>A∘h</source>.<ref>dzaima/APL: [https://github.com/dzaima/APL/blob/ceea05e25687988ed0980a4abf4b9249b736543f/docs/differences.txt#L19 Differences from Dyalog APL]. Retrieved 09 Jan 2020.</ref> See [[Bind]]. | |||
== Debugging == | |||
In [[Dyalog APL]], analysis of trains is assisted by a [[user command]] <source lang=apl inline>]Boxing on</source>. This is achieved by executing the command <source lang=apl inline>]Boxing on</source> and then entering a train without any parameters. A structure of the train will be displayed. | |||
For example, the "accursed train" from the section below can be analysed like this: | |||
<source lang=apl> | |||
]Boxing on | |||
Was OFF | |||
((+.×⍨⊢~∘.×⍨)1↓⍳) ⍝ the train to be analysed | |||
┌───────────────────────────────┬───────┐ | |||
│┌───────────┬─────────────────┐│┌─┬─┬─┐│ | |||
││┌───────┬─┐│┌─┬─┬───────────┐│││1│↓│⍳││ | |||
│││┌─┬─┬─┐│⍨│││⊢│~│┌───────┬─┐│││└─┴─┴─┘│ | |||
││││+│.│×││ │││ │ ││┌─┬─┬─┐│⍨││││ │ | |||
│││└─┴─┴─┘│ │││ │ │││∘│.│×││ ││││ │ | |||
││└───────┴─┘││ │ ││└─┴─┴─┘│ ││││ │ | |||
││ ││ │ │└───────┴─┘│││ │ | |||
││ │└─┴─┴───────────┘││ │ | |||
│└───────────┴─────────────────┘│ │ | |||
└───────────────────────────────┴───────┘ | |||
</source> | |||
Alternatively, a train can be represented in form of a tree: | |||
<source lang=apl> | |||
]Boxing on -trains=tree | |||
Was ON -trains=box | |||
((+.×⍨⊢~∘.×⍨)1↓⍳) ⍝ the train to be analysed | |||
┌───┴───┐ | |||
┌─┴─┐ ┌─┼─┐ | |||
⍨ ┌─┼─┐ 1 ↓ ⍳ | |||
┌─┘ ⊢ ~ ⍨ | |||
. ┌─┘ | |||
┌┴┐ . | |||
+ × ┌┴┐ | |||
∘ × | |||
</source> | |||
Or fully parenthesised: | |||
<source lang=apl> | |||
]Boxing on -trains=parens | |||
Was OFF -trains=box | |||
((+.×⍨⊢~∘.×⍨)1↓⍳) ⍝ the train to be analysed | |||
(((+.×)⍨)(⊢~((∘.×)⍨)))(1↓⍳) | |||
</source> | |||
== Examples == | == Examples == | ||
One of the major benefits of tacit programming is the ability to convey a short, well-defined idea as an isolated expression. This aids both human readability ([[semantic density]]) and the computer's ability to interpret code, potentially executing special code for particular [[ | One of the major benefits of tacit programming is the ability to convey a short, well-defined idea as an isolated expression. This aids both human readability ([[semantic density]]) and the computer's ability to interpret code, potentially executing special code for particular [[idiom]]s. | ||
=== Plus and minus === | === Plus and minus === | ||
| Line 75: | Line 123: | ||
=== Fractions === | === Fractions === | ||
We can convert decimal numbers to fractions. For example, we can convert <math>2.625</math> to the improper fraction <math>21 | We can convert decimal numbers to fractions. For example, we can convert <math>2.625</math> to the improper fraction <math>\tfrac{21}{8}</math> with | ||
<source lang=apl> | <source lang=apl> | ||
(1∧⊢,÷)2.625 | (1∧⊢,÷)2.625 | ||
21 8 | 21 8 | ||
</source> | </source> | ||
Alternatively, we can convert it to the mixed fraction <math>2{5 | Alternatively, we can convert it to the mixed fraction <math>2\tfrac{5}{8}</math> with a mixed fraction: | ||
<source lang=apl> | <source lang=apl> | ||
(1∧0 1∘⊤,÷)2.625 | (1∧0 1∘⊤,÷)2.625 | ||
| Line 118: | Line 165: | ||
0 0 2 | 0 0 2 | ||
</source> | </source> | ||
For a more parallel comparison of the notations, see the [[Comparison_with_traditional_mathematics#Practical_example|comparison with traditional mathematics]]. | For a more parallel comparison of the notations, see the [[Comparison_with_traditional_mathematics#Practical_example|comparison with traditional mathematics]]. | ||
===The Number of the Beast=== | |||
The following expression for computing the [[wikipedia:666 (number)|number of the Beast]] (and of [[I.P. Sharp]]'s APL-based email system, [[666 BOX]]) nicely illustrates how to read a train. | |||
<source lang=apl> | |||
((+.×⍨⊢~∘.×⍨)1↓⍳)17 ⍝ Accursed train | |||
666 | |||
</source> | |||
First, <source lang=apl inline>((+.×⍨⊢~∘.×)1↓⍳)</source> is supplied with only one argument <source lang=apl inline>17</source> and is thus interpreted monadically. | |||
Second, <source lang=apl inline>(+.×⍨⊢~∘.×⍨)1↓⍳</source> is a 4-train: reading right-to-left, the last 3 components are interpreted as the fork <source lang=apl inline>1↓⍳</source> and the 4-train is interpreted as the atop <source lang=apl inline>(+.×⍨⊢~∘.×⍨)(1↓⍳)</source>. | |||
Similarly, <source lang=apl inline>(+.×⍨⊢~∘.×⍨)</source> is also a 4-train and interpreted as the atop <source lang=apl inline>+.×⍨(⊢~∘.×⍨)</source>. | |||
Thus the accursed train is interpreted as <source lang=apl inline>((+.×⍨(⊢~∘.×⍨))(1↓⍳))17</source>. Having read the train, we now evaluate it monadically. | |||
<source lang=apl> | |||
((+.×⍨(⊢~∘.×⍨))(1↓⍳))17 ⍝ Accursed train as an atop over a fork atop a fork | |||
+.×⍨(⊢~∘.×⍨)1↓⍳17 ⍝ Atop evalution | |||
+.×⍨(⊢1↓⍳17)~∘.×⍨1↓⍳17 ⍝ Fork evalution | |||
+.×⍨(1↓⍳17)~∘.×⍨1↓⍳17 ⍝ ⊢ evaluation | |||
+.×⍨2 3 5 7 11 13 15 17 ⍝ numbers 2 through 17 without those appearing in their multiplication table are primes | |||
666 ⍝ the sum of the squares of the primes up to 17 | |||
</source> | |||
Note that <source lang=apl inline>((⊢⍨∘.×⍨)1↓⍳)</source> is a train computing primes up to the given input. | |||
A more satisfying variation of the accursed train is the following. | |||
<source lang=apl> | |||
(⍎⊢,⍕∘≢)'((+.×⍨⊢~∘.×⍨)1↓⍳)' ⍝ Accursed train 2.0 | |||
⍎(⊢,⍕∘≢)'((+.×⍨⊢~∘.×⍨)1↓⍳)' ⍝ 4-train intepreted as an atop | |||
⍎(⊢'((+.×⍨⊢~∘.×⍨)1↓⍳)'),⍕∘≢'((+.×⍨⊢~∘.×⍨)1↓⍳)' ⍝ fork evaluation | |||
⍎'((+.×⍨⊢~∘.×⍨)1↓⍳)','17' ⍝ ⊢ evaluation and ⍕∘≢ evaluation | |||
⍎'((+.×⍨⊢~∘.×⍨)1↓⍳)17' ⍝ , evaluation | |||
666 ⍝ ⍎ executes original Accursed train | |||
</source> | |||
== See also == | |||
* [[Function composition]] | |||
== External links == | == External links == | ||
== Tutorials == | === Tutorials === | ||
* Dyalog: [ | * Dyalog: [https://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm version 14.0 release notes] | ||
* APL Cultivation: [https://chat.stackexchange.com/rooms/52405/conversation/lesson-23-transcribing-to-and-reading-trains Transcribing to and reading trains] | * [[APL Cultivation]]: [https://chat.stackexchange.com/rooms/52405/conversation/lesson-23-transcribing-to-and-reading-trains Transcribing to and reading trains] | ||
* APLtrainer: [https://www.youtube.com/watch?v=kt4lMZbn-so How to read trains in Dyalog APL code] (video) | * [[APLtrainer]]: [https://www.youtube.com/watch?v=kt4lMZbn-so How to read trains in Dyalog APL code] (video) | ||
* APLtrainer: [https://www.youtube.com/watch?v=A2LqqBosvY0 Function trains in APL] (video) | * [[APLtrainer]]: [https://www.youtube.com/watch?v=A2LqqBosvY0 Function trains in APL] (video) | ||
* Dyalog | * [[Dyalog webinar]]: [https://www.youtube.com/watch?v=Enlh5qwwDuY?t=440 Train Spotting in Dyalog APL] (video) | ||
* Dyalog '13: [https://www.youtube.com/watch?v=7-93GzDqC08 Train Spotting in Version 14.0] (video) | * [[Dyalog '13]]: [https://www.youtube.com/watch?v=7-93GzDqC08 Train Spotting in Version 14.0] (video) | ||
== Documentation == | === Documentation === | ||
* [ | * [https://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm Announcement] | ||
* [ | * [https://help.dyalog.com/latest/Content/Language/Introduction/Trains.htm Dyalog] | ||
==References== | == References == | ||
<references/> | <references/> | ||
{{APL syntax}} | |||
{{APL syntax}}[[Category:Tacit programming| ]] | |||