J Essentials

At any time you may be given a pop quiz on anything in this document that we have gone over in class.

Definitions

1. Adverb  One of the primary parts of speech.  An adverb modifies the word or phrase to its left to produce a derived entity that can be any of the four primary parts of speech.
   • / is an adverb, because you combine it with a single verb on its left, as in +/ which means 'add items'.
2. Atom  Any single number, single character, or single box.  Also called a scalar.  An atom is a noun with rank 0 .
   • 5, 3.4 and 'a' are atoms. 3 4 and 'ab' are not atoms (they are 2-item lists)
3. Array  A noun comprising atoms arranged along 1 or more axes.  Each atom is identified by its index list.  J arrays are rectangular, meaning that all 1‑cells contain an identical number of 0‑cells, and all 2‑cells contain an identical number of 1‑cells, and so on.
   • Any noun that is not an atom is an array. 3 4 5 is an array; 'abcd' is an array.
     
     0 1 2
     4 5 6
     
     is an array of rank 2.
4. Axis  One of the dimensions along which the atoms of an array are arranged.  The atoms of every noun are arranged along zero or more axes where each axis has a corresponding length and each unique list of nonnegative integers in which each integer is less than the length of the corresponding axis designates a unique atom of the noun.
5. Boolean  Having a numeric value restricted to the values 0 and 1 .
6. Cell  A subarray of a noun consisting of all the atoms of the noun whose index lists have a given prefix.  For positive k, each k‑cell of a noun whose shape is s has the shape ((‑ k <. #s) {. s) and together they can be assembled to reconstruct the noun.  For negative k, the k‑cells are defined to be the (0 >. k + #s)‑cells.
   • 0-cells are atoms; 1-cells are lists; and so on.
     
   • 3 4 5 is an array of rank 1, in other words a list. It has (3) 0-cells, namely the atoms 3, 4, and 5. It has a single 1-cell, which is the entire list.
     
   • 0 1 2
     3 4 5
     is an array of rank 2. It has (6) 0-cells, namely the atoms 0 through 5. It has (2) 1-cells: the lists 0 1 2 and 3 4 5. It has a single 2-cell, which is the entire array.
   • In that same rank-2 array, the _1-cells are lists and the _2-cells are atoms.
7. Conjunction  One of the primary parts of speech.  A conjunction modifies the words or phrases to its left and right to produce a derived entity that can be any of the four primary parts of speech.
8. Copula  One of the parts of speech, signifying an assignment of a value to a name.  The copulas are =. and =: .
9. Derived Entity  The result of executing an adverb or conjunction.  If the part of speech of a derived entity is known, it may be called, for example, a derived verb.
10. Dyad (dyadic)  A verb with left and right operands (which must be nouns).  Any verb may be used as a monad or dyad, depending on whether it has a left noun operand when it is executed.
• To see whether a verb is a dyad, you have to look to see if it has a noun on its left. In
  
  3 <: 5
  
  the 3 is to the left of the <:, so the <: is used as a dyad (meaning, less-than-or-equal). The result is 1, meaning true. In
  
  <: 5
  
  there is no noun to the left of <:, so it is a monad (meaning subtract-one). The result is 4.
  
  
11. Fill  An atom appended to a noun to extend the length of the noun to a required length, especially when the noun must be extended because it is being made a cell in an array whose cells are bigger than the noun.
12. Frame  The k‑frame of a noun, or frame of a noun with respect to k‑cells, is the shape of the noun with the last r items removed, where r is the rank of a k-cell of the noun.  When a noun is viewed as an array of k‑cells, the frame with respect to k‑cells is the shape of the array of k‑cells.
13. Index  In an array, an integer indicating position along an axis.  The index of the first atom along an axis is 0 .
    • Consider the string 'Roger Hui'. The index of R is 0, and the index of u is 7.
    • Negative indexes count back from the end: the index of u could also be _2.
14. Index List  A list of integers with the same length as the shape of a noun, designating an atom of the noun by giving its position along each axis.
15. Inflection  Adding one or more of the characters . and : to a primitive to create a new primitive. Inflected primitives have entirely different meaning from uninflected ones. - means 'minus', but -: means 'take half of'.
16. Item  A _1‑cell of a noun.  An array is a vector of its items.  An atom has one item, itself.
    • Items are very important is J, because so many verbs operate on items. The items of any array are the cells that have rank 1 less than the rank of the array.
    • For example, the items of a list are atoms. The list is an array of rank 1, so the items have rank 0. Nouns with rank 0 are atoms.
    • The items of the list 'abcd' are the atoms 'a', 'b', 'c', and 'd'.
    • The items of a rank-2 array have rank 1. They are therefore lists. The items of
      
      0 1 2
      3 4 5
      
      are the two lists 0 1 2 and 3 4 5. These are the _1-cells.
    • Atoms are special, because they have no smaller parts that they can be broken into. An atom has one item, which is the atom itself.
17. List  An array of rank 1 .
• 'abc' is a list of 3 characters. 0 1 2 3 is a list of 4 numbers. 0 is not a list - it is an atom.
  
  0 1 2
  3 4 5
  
  is not a list - it is a rank-2 array.
18. List of  An array whose items are; as in 'list of 3-item lists'.
• Every array is a list of its items. The array 'abc' is a list of the items 'a', 'b', and 'c'. The array
  
  0 1 2
  3 4 5
  
  is a list of the two items 0 1 2 and 3 4 5.
19. Monad (monadic)  A verb with no left operand.  Any verb may be used as a monad or dyad, depending on whether it has a left noun operand when it is executed.
• To see whether a verb is a dyad, you have to look to see if it has a noun on its left. In
  
  3 <: 5
  
  the 3 is to the left of the <:, so the <: is used as a dyad (meaning, less-than-or-equal). The result is 1, meaning true. In
  
  <: 5
  
  there is no noun to the left of <:, so it is a monad (meaning subtract-one). The result is 4
  
20. Name  Either a simple name or a locative, to which an entity can be assigned.
    • We will not be dealing with locatives, so don't worry about that part.
    • Names include everything except primitives, punuctuation, and copulas. Names cannot end with . or :, and may not contain spaces.
    • a, total, my_word , and name123 are all names. a:, +, and if. are not names (a: and + are primitives; if. is a control word)
    • The part of speech a name is depends on what was assigned to it. If you assign a noun to a name, that name becomes a noun; if you assign a verb to a name, the name becomes a verb. If you assign one part of speech to a name that is already defined as a different part of speech, that's OK. Roger remembers only the last value and part of speech that is assigned.
21. Noun  One of the primary parts of speech.  An atom or array of atoms.  Nouns hold the data that verbs operate on.
    • Nouns are the things a J program works on.
    • Nouns have shape, type, and value.
    • Shape tells how many atoms there are, arranged in what way.
    • Type is numeric, literal, or boxed. Every atom in a noun has the same type
    • Value is the actual data in the noun. For numeric nouns, each atom has a number value. For literal nouns, each atom has a character value.
22. Part of Speech  One of the six categories into which words are classified; or, the word or entity so classified.  Every word has a part of speech: for primitives, the part of speech is defined by the language; for names, the part of speech is that of the entity assigned to the name.  The parts of speech are: noun, verb, conjunction, adverb, punctuation, and copula.  The primary parts of speech are noun, verb, conjunction, and adverb.
23. Primitive  A word with a built-in definition. Examples are verbs + and i. .
24. Punctuation  A part of speech.  Punctuation is not executed but it affects the execution of other words.  Punctuation in J comprises (, ), NB., and control words like if. and while. .
25. Rank  Of a noun, the number of axes along which the atoms of the noun are arranged; the number of items in the shape.  Of a verb, the highest rank of the noun operands that the verb can operate on.
    • The rank of a noun is the number of items in its shape.
    • Atoms have rank 0 . Example: 0   'a'    3.14159  all have rank 0.
    • Lists have rank 1.      1 2 3    'abcd'     (,1)    ''    all have rank 1.
    • A list can be empty, as in the case of '' . It still has rank 1. It is a list with no elements, which is not the same as nothing.
    • A list can have 1 atom, as in the case of (,1) . It still has rank 1. It is a list with one element, which is not the same as an atom.
    • Two-dimensional arrays have rank 2. An example is
      
      0 1 2
      3 4 5
26. Result  What a verb produces when it is executed.
• To be precise, adverbs and conjunctions also have results, called derived entities, but we won't worry about that.
27. Right-to-left  The order of execution in J. In the absence of parentheses, the rightmost verb in a sentence is executed first.  This may give a different result from standard mathematical usage. -3 - 4 is 1, not _7.
28. Scalar  See atom.
29. Sentence  A line that you give Roger to execute
30. Shape  The list of the lengths of the axes of a noun.
    • The shape is always a list.
    • To see the shape of a noun, use
      $ noun
    • An atom has no axes, so its shape is an empty list (a list with no items)
    • A list has 1 axis, so its shape is a list with 1 item. That one number is the number of items in the list.
    • A two-dimensional array has two axes, so its shape is a list of 2 items. The first item gives the number of rows in the array; the second number gives the number of columns.
    • The shape of the atom 5 is the list (0$0) - no numbers (an empty numeric list).
    • The shape of the list 1 2 3 is the list (,3) - one number.
    • The shape of the list 'abcd' is the list (,4) - one number.
    • The shape of an empty list like '' is the list (,0) - one number, 0, indicating that there are 0 items in the empty list ''.
    • The shape of
      
      0 1 2
      3 4 5
      
      is the list 2 3.
31. Type  An attribute of a noun: numeric, literal (also called string), boxed, or symbol.
32. Valence  Of a verb definition, an indication of the number of noun operands that the definition can accept: monadic if 1, dyadic if 2, dual-valence if either 1 or 2.
33. Vector  A sequence of cells arranged with a leading axis.  A non-scalar array can be construed as a vector of its items.  Synonymous with List or List of.
34. Verb  One of the primary parts of speech.  A verb operates on the noun to its right (and its left, if a noun is to its left) to produce a noun result.
    • Verbs are what make things happen in a J program.
    • Verbs can be either J primitives or verbs that you write using verb define.
    • Verbs operate on nouns, and produce a result that is a noun.
    • The verb and its noun operand(s) are replaced by the result of the verb. If there is another verb in the sentence, that verb operates on the results of earlier verbs.
35. Word  The fundamental unit of a sentence. Words can be names, numbers, primitives, punctuation, or copulas. To see the words in a sentence, use apply the ;: verb to the text of the sentence:
    
    ;: 'a =: b + 2 3'
    +-+--+-+-+---+
    |a|=:|b|+|2 3|
    +-+--+-+-+---+
    
    Note that 2 3 is a single word.
    

Summary of Terms

Verbs We Have Covered

• x + y (addition)
• x * y (multiplication)
• $ y (result is shape of y, always a list)
• # y (result is number of items in y, always an atom)
• i. y (result has shape y with values being sequential integers)
• x $ y (result has shape x with values coming from y)
• +/ y (result is the total of the items of y, with the shape of an item of y)
• >./ y (largest item of y)
• ? y (random value between 0 and y-1, or between 0 and 1 if y is 0)
• x { y (result is item number x of y. It has the shape of an item of y. The first item is item number 0.)