Function axis: Difference between revisions
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{{Built-in|Function axis|[ax]}} is a special syntax which modifies the behavior of a function, for example <source lang=apl inline>⌽[2]</source> to [[Rotate]] along the second [[axis]]. Axis specification was a feature of [[Iverson notation]] and was ubiquitous in early APLs; many newer APLs which adhere to [[leading axis theory]] reject the use of axis specification in favor of the [[Rank operator]] because it is a fully general operator while the behavior of functions with axis must be defined for each function separately. | {{Built-in|Function axis|[ax]}} is a special syntax which modifies the behavior of a function, for example <source lang=apl inline>⌽[2]</source> to [[Rotate]] along the second [[axis]]. Axis specification was a feature of [[Iverson notation]] and was ubiquitous in early APLs; many newer APLs which adhere to [[leading axis theory]] reject the use of axis specification in favor of the [[Rank operator]] because it is a fully general operator while the behavior of functions with axis must be defined for each function separately. However, [[GNU APL]] embraces and extends the syntax so it applies to all user-defined functions. | ||
== Functions == | == Functions == | ||
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Axis specification always modifies the [[derived function]] obtained from an operator, and not the operator itself. The exact syntax may vary: in most cases a set of brackets is parsed like a monadic operator and applies directly to the derived function; sometimes brackets can be applied directly to an operator, in which case the derived function produced by this operator is modified. In [[Dyalog APL]], a slash with axis retains its [[function-operator overloading]]: it can be applied as an operator or as a dyadic function ([[Replicate]] or [[Expand]]). | Axis specification always modifies the [[derived function]] obtained from an operator, and not the operator itself. The exact syntax may vary: in most cases a set of brackets is parsed like a monadic operator and applies directly to the derived function; sometimes brackets can be applied directly to an operator, in which case the derived function produced by this operator is modified. In [[Dyalog APL]], a slash with axis retains its [[function-operator overloading]]: it can be applied as an operator or as a dyadic function ([[Replicate]] or [[Expand]]). | ||
== Generalisation == | |||
GNU APL generalises function axis to all functions, and beyond what could reasonably be called "axis", that is, referring to one or more argument axes. Instead, the syntax is used to provide an additional argument, which the function can use in whichever way. For example, <source lang=apl inline>⌹[ax] M</source> computes the [[wikipedia:QR factorization|QR factorization]] of <source lang=apl inline>M</source> with the [[comparison tolerance]] of <source lang=apl inline>ax</source>. Here, <source lang=apl inline>ax</source> is not an axis at all, and there's no clear connection between this monadic usage of Domino and its non-axis meaning of [[Matrix inverse]]. This further breaks a principle that, except for scalar functions and arrays, there always exists a value for <source lang=apl inline>ax</source> such that <source lang=apl inline>f[ax]</source> is the same as <source lang=apl inline>f</source>. Much like the left argument of [[Circular]] (<source lang=apl inline>○</source>), The GNU APL specific File Input Output [[system function]] (<source lang=apl inline>⎕FIO</source>) uses the bracket axis syntax as a function selector, as does the [[wikipedia:SQL|SQL]] interface function (<source lang=apl inline>⎕SQL</source>). However, some of the functionalities provided by this latter function take additional parameters as subsequent elements in the array inside the brackets. Similarly, the [[wikipedia:regular expression|regular expression]] interface (<source lang=apl inline>⎕RE</source>) uses [[character]] "axes" as flags for the regular expression engine. [[Dyalog APL]] instead uses a [[dyadic operator]], [[Variant]] (<source lang=apl inline>⍠</source>), to provide such auxiliary parameter, thus staying within the normal rules of APL syntax. | |||
GNU APL even gives user-defined functions, both [[dfn]]s and [[tradfn]]s, access to the bracket axis syntax, again while permitting its use for any purpose. The axis value is given For example, the [[root]] function could be implemented as a monadic function with axis specifying the degree, rather than as a dyadic function. In a dfn, the value given via axis notation is denoted <source lang=apl inline>χ</source>: | |||
<source lang=apl> | |||
Root←{⍵*÷χ} | |||
Root[3]8 | |||
2 | |||
</source> | |||
{{Works in|[[GNU APL]]}} | |||
Or as a tradfn, indicating axis notation in the header line: | |||
<source lang=apl> | |||
∇ r←Root[ax] y | |||
r←y*÷ax | |||
∇ | |||
</source> | |||
{{Works in|[[GNU APL]]}} | |||
== External links == | == External links == | ||
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* Dyalog: [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Operators%20Summarised.htm overview], [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Axis%20with%20Monadic%20Operand.htm monadic], [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Axis%20with%20Dyadic%20Operand.htm dyadic] | * Dyalog: [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Operators%20Summarised.htm overview], [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Axis%20with%20Monadic%20Operand.htm monadic], [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Operators/Axis%20with%20Dyadic%20Operand.htm dyadic] | ||
* [http://microapl.com/apl_help/ch_020_010_020.htm APLX] | * [http://microapl.com/apl_help/ch_020_010_020.htm APLX] | ||
* [https://www.gnu.org/software/apl/apl.html#Section-2_002e2 GNU APL] | |||
=== Lesson === | === Lesson === | ||
* [https://chat.stackexchange.com/rooms/52405/conversation/lesson-3-some-apl-operators-----#message-40898783 APL Cultivation] | * [https://chat.stackexchange.com/rooms/52405/conversation/lesson-3-some-apl-operators-----#message-40898783 APL Cultivation] | ||
{{APL syntax}}[[Category:Function characteristics]] | {{APL syntax}}[[Category:Function characteristics]] | ||
Revision as of 14:46, 21 July 2022
[ax]
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Function axis ([ax]) is a special syntax which modifies the behavior of a function, for example ⌽[2] to Rotate along the second axis. Axis specification was a feature of Iverson notation and was ubiquitous in early APLs; many newer APLs which adhere to leading axis theory reject the use of axis specification in favor of the Rank operator because it is a fully general operator while the behavior of functions with axis must be defined for each function separately. However, GNU APL embraces and extends the syntax so it applies to all user-defined functions.
Functions
Monadic functions
The following monads may allow an axis:
- Mix accepts a list of axes to specify where the axes of argument elements will be placed in the result.
- Ravel accepts a list of axes which are combined, or a single fractional number to add a length-1 axis.
- Enclose accepts a list of axes. Each subarray along these axes is enclosed.
- Split accepts a single axis, and encloses each vector along that axis.
- Reverse reverses along the specified axis.
Dyadic functions
The following dyads may allow one:
- Scalar dyadics accept a list of axes to override conformability rules: it specifies , for each axis in the lower-rank (or left, in case of a tie) argument, which axis in the other argument it is paired with.
- Catenate combines along the selected axis, adding a new axis if a non-integer axis is given.
- Rotate rotates the right argument along the selected axis.
- Replicate and Expand work on the specified right argument axis.
- Take and Drop modify the selected right argument axes.
- Squad indexing takes axes to specify which axis of the right argument corresponds to each left argument element.
- Partition and Partitioned Enclose have complicated and different behavior.
Operators
The following operators may admit axis specification:
- Reduction removes the specified right argument axis.
- Scan works on the specified right argument axis.
In SHARP APL, Replicate and Expand are also included in this category as they are operators and not functions.
Axis specification always modifies the derived function obtained from an operator, and not the operator itself. The exact syntax may vary: in most cases a set of brackets is parsed like a monadic operator and applies directly to the derived function; sometimes brackets can be applied directly to an operator, in which case the derived function produced by this operator is modified. In Dyalog APL, a slash with axis retains its function-operator overloading: it can be applied as an operator or as a dyadic function (Replicate or Expand).
Generalisation
GNU APL generalises function axis to all functions, and beyond what could reasonably be called "axis", that is, referring to one or more argument axes. Instead, the syntax is used to provide an additional argument, which the function can use in whichever way. For example, ⌹[ax] M computes the QR factorization of M with the comparison tolerance of ax. Here, ax is not an axis at all, and there's no clear connection between this monadic usage of Domino and its non-axis meaning of Matrix inverse. This further breaks a principle that, except for scalar functions and arrays, there always exists a value for ax such that f[ax] is the same as f. Much like the left argument of Circular (○), The GNU APL specific File Input Output system function (⎕FIO) uses the bracket axis syntax as a function selector, as does the SQL interface function (⎕SQL). However, some of the functionalities provided by this latter function take additional parameters as subsequent elements in the array inside the brackets. Similarly, the regular expression interface (⎕RE) uses character "axes" as flags for the regular expression engine. Dyalog APL instead uses a dyadic operator, Variant (⍠), to provide such auxiliary parameter, thus staying within the normal rules of APL syntax.
GNU APL even gives user-defined functions, both dfns and tradfns, access to the bracket axis syntax, again while permitting its use for any purpose. The axis value is given For example, the root function could be implemented as a monadic function with axis specifying the degree, rather than as a dyadic function. In a dfn, the value given via axis notation is denoted χ:
Root←{⍵*÷χ}
Root[3]8
2
Or as a tradfn, indicating axis notation in the header line:
∇ r←Root[ax] y r←y*÷ax ∇