J语言
引用

↑  Roger K.W. Hui, Kenneth E. Iverson, E. E. McDonnell（英语：）, Arthur T. Whitney. . 1990  [2022-06-12]. （原始内容存档于2022-06-14）. This paper describes a version of APL based upon the dictionary, but significantly simplified and enhanced, and directly usable on any machine that provides ASCII characters. It also describes salient features of a C implementation that has been tested on several machines, and is available as freeware.
↑  https://code.jsoftware.com/wiki/System/ReleaseNotes/J9.5.
↑  .   [2020-04-18]. （原始内容存档于2020-12-11）.
↑  K. E. Iverson. . 1991  [2022-06-12]. （原始内容存档于2022-06-12）. Roger and I then began a collaboration on the design and implementation of a dialect of APL (later named J by Roger), first deciding to roughly follow “A Dictionary of APL” and to impose no requirement of compatibility with any existing dialect. We were assisted by suggestions from many sources, particularly in the design of the spelling scheme (E.B. Iverson and A.T. Whitney) and in the treatment of cells, items, and formatting (A.T. Whitney, based on his work on SHARP/HP and on the dialect A reported at the APL89 conference in New York).
↑  Roger K. W. Hui, Morten J. Kromberg. . Proceedings of the ACM on Programming Languages, Volume 4, Issue HOPL. 2020  [2022-06-20]. （原始内容存档于2022-07-10）. In 1989, Iverson, together with Roger Hui and with input from Arthur Whitney, produced J, with a goal of providing a “shareware” APL implementation for use in teaching. The special APL characters were abandoned because it was felt that they require technical solutions which at that time were still prohibitively expensive in an educational environment. ……
J was clearly a “rationalization” of SHARP APL. ……
ACM SIGAPL, the ACM Special Interest Group on APL, reinterpreted the “APL” in its name in early 2009 as Array Programming Languages, so that J, k, Nial, etc. would be included in its purview.
↑  Kenneth E. Iverson. . 1987  [2022-06-08]. （原始内容存档于2022-06-12）. A dictionary should not be read as an introduction to a language, but should rather be consulted in conjunction with other material that uses the language in some context of interest to the reader. Even the general section on grammar, which may be intelligible even to the beginner, should perhaps be studied only after a certain amount of other exposure to the language.
On the other hand, a dictionary should not be used only to find the meanings of individual words, but should also be studied to gain an overall view of the language. In particular, the grammar may be profitably reviewed again and again in the light of increased knowledge of the language, and the study of groups of related verbs and adverbs can reveal important relationships otherwise easily overlooked.
↑  Kenneth E. Iverson. . 1978  [2022-06-21]. （原始内容存档于2022-06-24）. Nuclear Axis Operators － The nuax operator (denoted by ⍤) applies to a function left argument and a variable right argument to specify the axes which define the nuclei to which the function is to apply. ……The coax operator ⍥ is also provided; its argument specifies the axes complementary to the nuclear axes. 

Kenneth E. Iverson. . 1983  [2022-06-19]. （原始内容存档于2022-07-22）. In conventional APL, the scalar functions (which apply to scalar elements and produce scalar results) extend to higher rank arrays according to simple general rules; no corresponding general rules exist for the remaining so-called mixed functions. ……
Function rank is the most important notion needed to provide a simple and systematic basis for the uniform treatment of all “mixed” or non-scalar functions. ……
If f has rank r, then f⍵ is determined by applying f to each of the “cells” of shape (-r)↑⍴⍵, producing a common shape s for each, and assembling the whole into a result of shape ((-r)↓⍴⍵),s. ……
If the function g←f⍤r is to be applied dyadically as well as monadically (the only cases addressed in the preceding sections), then it is necessary that r specify three independent ranks, the monadic, the left, and the right. The general argument r is therefore a three-element vector that specifies the ranks in the order just indicated. Moreover, r is extended by reshape if necessary, so that f⍤r ←→ f⍤(⌽3⍴⌽r).
1 2  Kenneth E. Iverson, Eugene McDonnell（英语：）. . APL 89 Conference Proceedings. August 1989  [2022-06-09]. （原始内容存档于2022-06-12）. In combinatory logic one of the most useful primitive combinators is designated by S [Sch24]. Curry defines Sfgx in prefix notation to be fx(gx) [CuFeCr74]. In common mathematical infix notation this would be given by (x)f(g(x)), which one can write in APL as xfgx, and this is the hook form (fg)x. The combinatory logician appreciates this form because of its great expressiveness: it can be shown that S, along with K, the constancy combinator, suffice to define all other combinators of interest [Ro50]. (The constancy combinator K is defined in infix notation so that cKx has the value c for all x.) Users of APL will appreciate the hook for the same reasons.
1 2  Kenneth E. Iverson, Eugene McDonnell（英语：）. . APL 89 Conference Proceedings. August 1989  [2022-06-09]. （原始内容存档于2022-06-12）. Curry [Cu31] defines a formalizing combinator, Φ, in prefix notation, such that Φfghx means f(gx)(hx). In common mathematical infix notation this would be designated by (g(x))f(h(x)). An example of this form is Φ+sin2cos2θ, meaning sin2θ+cos2θ. The fork (f g h)⍵ has the same meaning, namely (f⍵)g(h⍵). Curry named this the formalizing combinator because of its role in defining formal implication in terms of ordinary implication.
Iverson and Whitney have made several earlier suggestions of ways to achieve what the fork form provides: the scalar operators of [Iv78], [Iv79a], [Iv 79b], the til operator of [Iv82], the union and intersection conjunctions of [Iv87], and the yoke adverb of [Iv88]. Benkard [Bk87] has also suggested a way to achieve the meaning of this form, in his proposal for ↑g/(f h)⍺ ⍵, using the notion of function pair (↑ is APL2’s first function). The present proposal has significant advantages over these earlier ones.
↑  Roger K.W. Hui, Kenneth E. Iverson, Eugene E. McDonnell（英语：）. . 1991  [2022-06-11]. （原始内容存档于2022-07-06）. To appreciate the more general use of tacit definition, it is necessary to understand three key notions of J: cells and rank, forks, and composition.……
The conjunction & is called with, and applies to nouns (variables) a and b as well as to verbs f and g as follows:
  a&g y is a g y
  f&b y is x f y
  f&g y is f g y
x f&g y is (g x) f (g y)
……
A number of other constructs in J similarly enhance the utility of tacit definitions. The more important are the under (or dual), atop (a second form of composition), the power conjunction ^:, and further forms of partitions. 

Jsoftware. . 1991  [2022-11-07]. （原始内容存档于2022-06-14）. 
@.    agenda
@:    at
&:    appose
↑  Robert Bernecky（英语：）, Roger K. W. Hui. . 1991  [2022-06-12]. （原始内容存档于2022-06-12）. Gerunds, verbal forms that can be used as nouns, are recognized as having utility in the realm of programming languages. We show that gerunds can be viewed as arrays of atomic repmentations of verbs (functions), in a way which is consistent with the syntax and semantics of APL, and which allows verbs to be first class objects in the language. We define derivations of verbs from gerunds in the J dialect of APL, and show how these derivations provide control structures for sequencing, selection (in the sense of generalized forms of CASE or SWITCH statements and IF/THEN/ELSE), iteration (DO UNTIL), recursion, and parallel computation (MIMD, or Multiple Instruction, Multiple Data). We conclude with alternative representations of verbs which are useful in other contexts. 

Jsoftware. . 1991  [2022-06-12]. （原始内容存档于2022-06-14）. 
`:1    replaced by u^:v
`:4    replaced by m~
`:5    replaced by @.
1 2  Roger K. W. Hui, Morten J. Kromberg. . Proceedings of the ACM on Programming Languages, Volume 4, Issue HOPL. 2020  [2022-06-20]. （原始内容存档于2022-07-10）. For years, Iverson struggled to achieve in APL the effect of f+g and f×g as they are written in calculus. …… Finally, trains AKA forks were invented [Iverson and McDonnell 1989]. ……
Moreover, (f … h p q r) ↔ (f … h (p q r)), and an isolated sequence of two functions is also assigned a meaning (atop, described below), so that a train of any length, even or odd, is interpreted. ……
Subsequently, it was realized that trains greatly increase the possibilities for “tacit definition”, expressions consisting of compositions of functions which do not explicitly mention the arguments [Hui et al. 1991]. Trains are implemented in several dialects: J [Hui et al. 1991], NARS2000 [Smith 2020], NGN APL [Nickolov 2013], and Dyalog APL [Scholes 2013]. ……
The expressive completeness of trains depends on an atop composition of two functions …… Dyalog APL defines 2-trains as atop. That, together with the functions ⊣(left) and ⊢(right), allows many common compositions to be written as trains.
.   [2022-07-30]. （原始内容存档于2022-07-30）.
↑  .   [2022-07-30]. （原始内容存档于2022-07-30）.
↑  .   [2020-05-18]. （原始内容存档于2016-03-07）.
1 2  
Kenneth E. Iverson. . 1983. The enclose function as defined in [Operators and Enclosed Arrays] has made it possible to produce by straightforward APL functions the “index lists” required in indexing expressions of the form a[i;j], and therefore makes it possible to define a corresponding indexing function, which will be denoted by { and called from:
    i{a ←→ a[>i[0];>i[1]; ...]
Since the disclose function > is permissive, the selection of any single element of a can be written without enclosures as, for example, 1 2 3{a3. Moreover, the left rank of { is 1 and its right rank is infinite, so that …… a simple left argument i of rank greater than 1 produces an array of shape ¯1↓⍴i of elements chosen by the index vectors along its last axis, yielding what is sometimes called “scattered” indexing. For examp1e:
    (3 2⍴⍳6){a2 ←→ a2[0;1],a2[2;3],a2[4;5]
……
In forming the left arguments of the indexing function, it will often be convenient to use the link function ⊃ defined as follows:
     ⊃b ←→ <b if b is simple
            b if b is non-simple
    a⊃b ←→ (<a),⊃b
For example, (2 3⊃4⊃∘⊃5 6){a4 ←→ a[2 3;4;;5 6].
The indexing function { as defined thus far provides all of the facilities provided by conventional indexing, and “scattered” and “complementary” indexing as well. Its power is further enhanced by allowing negative indexing ……. 

Roger Hui. . 1987. Dyadic { encompasses all computations expressible by [;] indexing of APL\360, as well as the new negative indexing and complementary indexing.
1 2  IBM.  (PDF). June 1976  [2022-07-02]. （原始内容存档 (PDF)于2019-09-26）. For vectors R and X, the decode(or base-value) function R⊥X yields the value of the vector X evaluated in number system with radices R[1],R[2],...,R[⍴R]. ……
Scalar(or one-element vector) arguments are extended to conform, as required. ……
the decode function is extended to arrays in the manner of the inner product: each of the radix vectors along the last axis of the first argument is applied to each of the vectors along the first axis of the second argument. 

J语言的#.并未继承IBM对APL解码函数⊥的扩展规定，它可以实现为：
   decode=: {{x #."(1 _) (0|: y)}}
1 2  
Kenneth E. Iverson. . 1978  [2022-06-21]. （原始内容存档于2022-06-24）. We also introduce a form of indexing called from denoted by ⌷, ……. The basic definition is:
      i⌷a ↔ (,a)[⍉(⍴a)⊥⍉i]
The function ⌷ distributes over any scalar function; thus, i⌷a+b ↔ (i⌷a)+(i⌷b). …… For example:
      m←3 4⍴⍳12
      m
0 1  2  3
4 5  6  7
8 9 10 11
      2 2 ⌷ m
10
……
      (3 2⍴3|⍳6)⌷m
1 8 6 

J语言在定义上述的⌷（from）之时，解码前后不需要专门进行一元转置⍉运算：
   from=: {{(($ y) #. x) {::"(0 _) (, y)}}
   ]m=: 3 4$i.12
0 1  2  3
4 5  6  7
8 9 10 11
   2 2 from m
10
   3 2$3|i.6
0 1
2 0
1 2
   (3 2$3|i.6)from m
1 8 6
↑  IBM.  (PDF). June 1976  [2022-07-02]. （原始内容存档 (PDF)于2019-09-26）. APL functions apply to collections of individual items called arrays. Arrays range from scalars, which are dimensionless, to multi-dimensional arrays of arbitrary rank and size.
↑  
Bob Smith. . 1981  [2022-06-19]. （原始内容存档于2022-06-20）. The data structures of APL are rectangular, multi-dimensional, and flat -- the latter term meaning that all items of arrays are simple scalars. ……
In general, a system with nested arrays extends the power of APL by being able to represent data of non-zero depth. Also, in keeping with the past, the system includes a set of primitive functions and operators, tailor-made for manipulating these new data structures.
Continuing the above example, the names of the months of the year are best represented in a 12-item vector whose items are each character vectors. This structure has a depth of one. Note that because the individual names are in separate items, we need no longer resort to artifices like pad characters. Moreover, explicit delimiters are not needed as the separation between items is represented through structure rather than data. This particular representation is called a vector of vectors, and can be created as follows:
    MONTHS ← ('JANUARY') ('FEBRUARY') ...
The above line also illustrates strand notation, used to enter a nested vector in a simple and convenient manner. ……
Of the several new operators, the only one specific to nested arrays is the each operator(symbol ¨), which is monadic as an operator, and produces an ambivalent derived function. It is used to apply the f unction which is its argument to the items of an array to produce corresponding items in the result. For example, to determine the length of the names of the months in the above example, use
    ⍴¨MONTHS
(7) (8) (5) (5) (3) (4) (4) (6) (9) (7) (8) (8)
Since monadic rho returns a vector , each item of the above result is a vector (specifically in these cases a one-item vector). The parentheses in the display indicate that the item is not a simple scalar.
↑  James A. Brown（英语：）. . TR 03.247. IBM Santa Teresa Laboratory, San Jose, California. 1984  [2022-06-23]. （原始内容存档于2022-06-12）. APL2 is based on this writer' s PhD Thesis [Br1], the array theory of Trenchard More [Mo1] and most of all on APL1. ……
The arrays of APL2 are finite rectangular arrays which contain arrays as items. When the term array is used, it means this subset of all possible arrays.
The arrays of APL2 are the same as the arrays of Array Theory and in particular empty arrays have structure as defined by Array Theory [Mo1 etc.].
An array one of whose items is other than a single number or character (a simple scalar) is called a nested array. An array containing only numbers or containing only characters is called a homogeneous array. An array all of whose items are either single numbers or single characters is called a simple array. The arrays of APL1 are simple and homogeneous.
In some sense every array in APL2 is nested because it contains other arrays. The term is reserved for those which contain at least one item which is not a single number or character. Thus the universe of arrays is partitioned into two subsets: simple arrays and nested arrays. ……
A function is pervasive if pick distributes over it. ……
Since the pick function may select an item at an arbitrary depth in a nested array, it may select deep enough to access a simple scalar (because nested arrays have finite depth). Thus a pervasive function may be thought of as applying independently to each simple scalar in its argument(s). ……
In APL2 the scalar functions and only the scalar functions are pervasive.
1 2  Kenneth E. Iverson. . 1978  [2022-06-21]. （原始内容存档于2022-06-24）. The enclose function (denoted by <) produces a scalar representation of its argument in the sense that the result is of rank zero, and that there exists an inverse function (called disclose, and denoted by >) such that a ↔ ><a for all a. Any result producible by an expression which does not employ the enclose function is called a simple array, or is said to be simple.
Selection and reshaping functions apply without change to non-simple arrays. However, non-simple arrays are outside the domain of all other functions except for enclose, disclose, and equality (together with those functions such as ≠ and ∊ which are defined in terms of equality).
The equality function is extended to non-simple scalar arguments as follows:
    1. (<a)≠a for all a
    2. If a equals b(in rank, shape, and all elements), then (<a)=(<b) yields 1
……
The disclose function is scalar in the sense that it applies to each element of its argument, the new axes disclosed becoming the final axes of the result. ……
The disclose function applied to a simple array a produces a result identical to a. Thus (<a)=<>a is a test for whether a is simple.
↑  
Kenneth E. Iverson. . 1983. APL2 provides two significant facilities which apply at “depth” in the enclosure structure of an argument, the dyadic pick function, and the pervasive functions. RS provides no primitive corresponding to pick; it could be defined recursively by:
    pick←''∇('→2+0=⍴⍺'⊃'(>0{⍺){(1↓⍺)∆⍵'⊃'⍵')
……
Since pervasiveness is a property assigned to functions, it would, in the framework of RS, be provided by an operator. Such an operator could be applied to any function (defined or derived as well as primitive) and, if defined to be dyadic, could provide greater variety. ……
If each essential space in an expression is counted as a character, then the link function and strand notation used to form non-simple vectors from simple vectors require expressions of nearly identical length. ……
RS does not include the heterogeneous arrays of APL2, and the production of equivalent constructs requires greater use of enclosure. However, the structure of RS does not preclude their introduction. ……
The monadic enclose functions defined in RS (<) and in APL2 (⊂) differ in one respect: if s is a simple scalar, then s≡⊂s, but ~s≡<s. Although < can therefore produce some structures not producible by ⊂, the differences between them (in the contexts of the respective systems) cannot, in most cases, be discerned.
↑  
.   [2021-12-20]. （原始内容存档于2021-12-20）.
↑  Roger K. W. Hui, Morten J. Kromberg. . Proceedings of the ACM on Programming Languages, Volume 4, Issue HOPL. 2020  [2022-06-20]. （原始内容存档于2022-07-10）. In order to provide users with access to APIs and frameworks, APL language designers searched for ways to integrate into APL, where everything is an array, selected aspects of the OO paradigm, where everything is an object. ……
Some interpreters, like APL+Win, avoided incorporating objects into the APL heap (or “workspace”). ……
Other systems added features which allowed variables, functions and operators to be organized in dynamic objects in the workspace. The resulting containers are known as namespaces (Dyalog APL) and locales (J). k implements dictionaries, which are also containers for arrays, but are typically used to contain arrays representing the columns of a relational table. …… The same language features which supported namespaces or classes within APL were used to wrap the external objects used by object oriented APIs, or platforms like the Microsoft’s OLE and .NET frameworks.
The choice of syntax for referring to a member name of an object emp was not straightforward. ……
Most APL systems did adopt the notation from other OO languages — with a few exceptions.
  emp.name    in most APL systems
  name__emp   in J (retaining right-to-left evaluation)
  emp[`name]  in k (using a symbol within index brackets to select from a dictionary)
……
Although APL interpreters have had extensive support for object oriented programming for nearly two decades, most APL users still feel that object and array paradigms are an awkward fit. …… Many of the benefits of OO are related to taking advantage of types, while much of the strength of the APL family is that you can write code which is shape, rank, and type agnostic — achieving many of the same goals as OO through radically different mechanisms.
↑  .   [2022-09-29]. （原始内容存档于2022-09-29）.
1 2  Chris Burke. . 1998  [2022-11-08]. （原始内容存档于2022-11-08）. 
Roger Stokes. . 2015  [2020-05-18]. （原始内容存档于2020-06-25）.
↑  .   [2022-10-20]. （原始内容存档于2022-12-19）. 
标准库 （页面存档备份，存于）
↑  .   [2020-05-22]. （原始内容存档于2021-04-06）.
↑  插件 （页面存档备份，存于）
↑  .   [2022-10-30]. （原始内容存档于2022-10-30）. 
Dissect （页面存档备份，存于）
↑  .   [2022-09-25]. （原始内容存档于2022-12-06）.
↑  Bussell, Brian; Taylor, Stephen, , , Oulu, Finland: Springer: 21–31, 2006, ISBN 978-3-540-35094-1, In practice, competence in the business and in writing software coincide only where the business requires mathematical skills. In consequence, highly abstract executable notations such as APL, A+, J, K, Q, R and S flourish primarily among actuaries, financial traders and statisticians. ……
A key technique to the success of this has been programmers and users collaborating on writing the source code. ……
Programmers have made this possible by constructing local, domain-specific executable notations. The vocabulary of these notations is drawn from the users’ talk about the work. ……
Users and programmers can now converge quickly on and verify a common understanding. The notation enables them to avoid ambiguity; it is a “tool for thought” in the sense of Iverson’s Turing Award lecture [11]. Because the notation is executable (and interpreted), the running system animates the described behaviour in front of them.
↑  Holt, Alan, , Springer, 2007, ISBN 978-1-84628-822-7
↑  Eric Iverson. . J programming mailing list. 1 March 2011  [2020-05-18]. （原始内容存档于2016-09-23）.
↑  .   [2020-05-18]. （原始内容存档于2021-01-26）.
↑  NuVoc （页面存档备份，存于）
↑  词类（页面存档备份，存于）
↑  名词（页面存档备份，存于）
↑  动词（页面存档备份，存于）
↑  定语 （页面存档备份，存于）
1 2  IBM.  (PDF). June 1976  [2022-07-02]. （原始内容存档 (PDF)于2019-09-26）. For example, the grade function ⍋ is commonly used to produce indices needed to reorder a vector into ascending order (as in X[⍋X]), but may also be used in the treatment of permutations as the inverse function, that is, ⍋P yields the permutations as the inverse to P.
↑  .   [2022-06-14]. （原始内容存档于2022-07-07）. 
Addons/tables/csv （页面存档备份，存于）
↑  Roger K.W. Hui. . 1996  [2022-06-24]. （原始内容存档于2022-06-28）. Some verbs v signal domain error on some extended arguments because the result is not integral; however, <.@v and >.@v are closed on extended arguments.
↑  .   [2022-10-20]. （原始内容存档于2022-10-20）. 
内存映射文件 （页面存档备份，存于）
↑  K.E. Iverson. . 1982. The operator denoted by the dot has been extended to provide a monadic function (as in -.×m) as well as the established dyadic inner product function (as in n+.×m). ……
The determinant of a square matrix m is defined as the alternating sum (i.e. reduction by -) of the !n←1↑⍴m products over n elements chosen (in each of the !n possible ways) one from each row and column. Analogous calculations in which other function pairs are substituted for - and × lead to other useful functions; examples include the pairs ⌈⌊, ∧∨, and +×, the last (called the permanent) being useful in combinatorics.
1 2  IBM.  (PDF). June 1976  [2022-07-02]. （原始内容存档 (PDF)于2019-09-26）. If P and Q are vectors of the same shape, then the expression +/P×Q has a variety of useful interpretations. ……
The inner product produces functions equivalent to expressions in this form; it is denoted by a dot and applies the two function surround it. Thus P+.×Q is equivalent to +/P×Q, and P×.⋆Q is equivalent to ×/P⋆Q, and, in general, Pf.gQ is equivalent to f/PgQ, if P and Q are vectors.
The inner product is extended to arrays other than vectors along certain fixed axes, namely the last axis of the first argument and the first axis of the last argument. the lengths of these axes must agree. the shape of the result is obtained by deleting these axes and chaining the remaining shape vectors together. ……
The inner product M+.×N is commonly called the matrix product. ……
Either argument of the inner product may be a scalar or a one-element vectors; it is extended in the usual manner.
1 2  IBM.  (PDF). June 1976  [2022-07-02]. （原始内容存档 (PDF)于2019-09-26）. The outer product operator, denoted by by the symbols ∘. preceding the function symbol, applies to any dyadic primitive scalar function, so that the function is evaluated for each member of the left argument paired with each member of the right argument. ……
Such tables may be better understood if labelled in a way widely used in elementary arithmetic texts: values of the arguments are placed beside and above the table, and the function whose outer product is being computed is shown at the corner.
↑  Kenneth E. Iverson. . 1987. The case 3⍤v also has left rank 2 , and ⍺3⍤v⍵ applies v to each element produced by a tessellation of ⍵, using a size 1{⍺, and beginning points that are multiples of the “shift” 0{⍺.
↑  .   [2023-05-08]. （原始内容存档于2023-05-08）.
↑  秩 （页面存档备份，存于）
↑  Roger K. W. Hui, Morten J. Kromberg. . Proceedings of the ACM on Programming Languages, Volume 4, Issue HOPL. 2020  [2022-06-20]. （原始内容存档于2022-07-10）. In the dyadic case, two frames lf and rf are involved, from the left and right arguments. Several different treatments are possible:
• scalar agreement: (lf≡rf)∨(lf≡⍬)∨(rf≡⍬), the left and right frames match, or one is the empty vector (⍬). If the frames match, there are an equal number of left and right cells, and the operand function applies to corresponding cells. If they do not match, one frame must be ⍬, that is, there is one cell on one side, whence that one cell is applied against every cell on the other side. Scalar agreement is implemented in Dyalog APL [Dyalog 2015].
• prefix agreement: (p↑lf)≡(p↑rf) ⊣ p←(≢lf)⌊(≢rf), one frame must be a prefix of the other. Let ff be the longer frame (that is, ff←lf,p↓rf). In this case a cell of the argument with the shorter frame is applied against ×⌿p↓ff cells of the other argument. Prefix agreement is implemented in J [Hui and Iverson 2004], and is consistent with the emphasis on the leading axis (§2).
• suffix agreement: one frame must be a suffix of the other. J had suffix agreement before it switched to prefix agreement in 1992 on a suggestion by Whitney [Whitney 1992].
• strict agreement: the frames must match. No dialect has ever implemented this.
↑  .   [2022-06-04]. （原始内容存档于2022-07-06）.
1 2  .   [2022-06-07]. （原始内容存档于2022-07-06）.
↑  .   [2022-06-04]. （原始内容存档于2022-07-07）.
↑  Kenneth E. Iverson. . 1978  [2022-06-21]. （原始内容存档于2022-06-24）. The dual operator, denoted by ⍢, is a slight extension of the notion of dual functions implicit in deMorgan’s law (∨⍢~ ↔ ^ and ≠⍢~ ↔ =), the extension being to include a monadic left argument, as in ⌊⍢-x ↔ ⌈x. ……
Composition and the dual operator applied to a divalent left argument and a monadic (or divalent) right argument yield parallel definitions of divalent derived functions as follows:
……
Dual:  f⍢g y ↔ (g⍣¯1) f (g y)
    x f⍢g y ↔ (g⍣¯1) (g x) f (g y)
It should be noted that the extension of the dual to include the monadic definition makes the identities ⌈⍢- ↔ ⌊ and ⌊⍢- ↔ ⌈ hold for both the monadic case (floor and ceiling) and for the dyadic case (minimum and maximum). Moreover, for the dyadic case the exponential function yields the identities ×⍢* ↔ + and +⍢⍟ ↔ ×, the latter of which provides the basis for the use of natural logarithms in multiplication, just as the identity +⍢(10¨⍟) ↔ x forms the basis for the use of base ten logarithms.
↑  Kenneth E. Iverson. . 1978  [2022-06-21]. （原始内容存档于2022-06-24）. The expression f⍢> produces a derived function which applies the function f to its argument in an “item-wise” fashion, by disclosing each element of the argument, applying f, and enclosing the result to produce the corresponding element of the overall result.