
Chapter 9: Trains of VerbsIn this chapter we continue the topic of trains of verbs begun in Chapter 03. Recall that a train is an isolated sequence of functions, written one after the other, such as (+ * ).9.1 Review: Monadic Hooks and ForksRecall from Chapter 03 the monadic hook, with the scheme:(f g) y means y f (g y)Here is an example, as a brief reminder: a whole number is equal to its floor:
Recall also the monadic fork, with the scheme: (f g h) y means (f y) g (h y)For example: the mean of a list of numbers is the sum divided by the numberofitems: sum =: +/ mean =: sum % #
Now we look at some further variations. 9.2 Dyadic Hooks3 hours and 15 minutes is 3.25 hours. A verb hr, such that (3 hr 15) is 3.25, can be written as a hook. We want x hr y to be x + (y%60) and so the hook is:hr =: + (%&60) 3 hr 15 3.25The scheme for dyadic hook is: x (f g) y means x f (g y)with the diagram: 9.3 Dyadic ForksSuppose we say that the expression "10 plus or minus 2" is to mean the list 12 8. A verb to compute x plusorminus y can be written as the fork (+,):
The scheme for a dyadic fork is: x (f g h) y means (x f y) g (x h y)Here is a diagram for this scheme:
9.4 ReviewThere are four basic schemes for trains of verbs.(f g h) y = (f y) g (h y) monadic fork x (f g h) y = (x f y) g (x h y) dyadic fork (f g) y = y f (g y) monadic hook x (f g) y = x f (g y) dyadic hook 9.5 Longer TrainsNow we begin to look at ways to broaden the class of functions which can be defined as trains.In general a train of any length can be analysed into hooks and forks. For a train of 4 verbs, e f g h, the scheme is that e f g h means e (f g h)that is, a 4train (e f g h) is a hook, where the first verb is e and the second is the fork (f g h). For example, Suppose that y is a list of numbers: y =: 2 3 4Then the "norm" of y is defined as (y  mean y), where mean is defined above as (sum % #). We see that the following expressions for the norm of y are all equivalent: y  mean y _1 0 1 ( mean) y NB. as a hook _1 0 1 ( (sum % #)) y NB. by definition of mean _1 0 1 ( sum % #) y NB. as 4train _1 0 1A certain amount of artistic judgement is called for with long trains. This last formulation as the 4train ( sum % #) does not bring out as clearly as it might that the key idea is subtracting the mean. The formulation (  mean) is clearer. For a train of 5 verbs d e f g h the scheme is: d e f g h means d e (f g h)That is, a 5train (d e f g h) is a fork with first verb d, second verb e and third verb the fork (f g h) For example, if we write a calendar date in the form day month year: date =: 28 2 1999and define verbs to extract the day month and year separately: Da =: 0 & { Mo =: 1 & { Yr =: 2 & {the date can be presented in different ways by 5trains:
The general scheme for a train of verbs (a b c ...) depends upon whether the number of verbs is even or odd: even: (a b c ...) means hook (a (b c ...)) odd : (a b c ...) means fork (a b (c ...)) 9.6 Identity FunctionsThere is a builtin verb, monadic [ (left bracket, called "Same"). It gives a result identical to its argument.
There is a dyadic case, and also a similar verb ] . Altogether we have these schemes: [ y means y x [ y means x ] y means y x ] y means y
Monadic [ and monadic ] are both called "Same". Dyadic [ is called "Left". Dyadic ] is "Right". The expression (+ % ]) is a fork; for arguments x and y it computes: (x+y) % (x ] y)that is, (x+y) % y
Another use for the identity function [ is to cause the result of an assignment to be displayed. The expression foo =: 42 is an assignment while the expression [ foo =: 42 is not: it merely contains an assignment. foo =: 42 NB. nothing displayed [ foo =: 42 42Yet another use for the [ verb is to allow several assignments to be combined on one line.
Since [ is a verb, its arguments must be nouns, (that is, not functions). Hence the assignments combined with [ must all evaluate to nouns. 9.6.1 Example: Hook as AbbreviationThe monadic hook (g h) is an abbreviation for the monadic fork ([ g h). To demonstrate, suppose we have:g =: , h =: *: y =: 3Then each of the following expressions is equivalent. ([ g h) y NB. a fork 3 9 ([ y) g (h y) NB. by defn of fork 3 9 y g (h y) NB. by defn of [ 3 9 (g h) y NB. by defn of hook 3 9 9.6.2 Example: Left HookRecall that the monadic hook has the general scheme(f g) y = y f (g y)How can we write, as a train, a function with the scheme ( ? ) y = (f y) g yThere are two possibilities. One is the fork (f g ]): f =: *: g =: , (f g ]) y NB. a fork 9 3 (f y) g (] y) NB. by meaning of fork 9 3 (f y) g y NB. by meaning of ] 9 3For another possibility, recall the ~ adverb with its scheme: (x f~ y) means y f xOur train can be written as the hook (g~ f). (g~ f) y NB. a hook 9 3 y (g~) (f y) NB. by meaning of hook 9 3 (f y) g y NB. by meaning of ~ 9 3 9.6.3 Example: DyadThere is a sense in which [ and ] can be regarded as standing for left and right arguments.f =: 'f' & , g =: 'g' & ,
9.7 The Capped ForkThe class of functions which can be written as unbroken trains can be widened with the aid of the "Cap" verb [: (leftbracket colon)The scheme is: for verbs f and g, the fork: [: f g means f @: gFor example, with f and g as above, we have
Notice how the sequence of three verbs ([: f g) looks like a fork, but with this "capped fork" it is the MONADIC case of the middle verb f which is applied. The [: verb is valid ONLY as the lefthand verb of a fork. It has no other purpose: as a verb it has an empty domain, that is, it cannot be applied to any argument. Its usefulness lies in building long trains. Suppose for example that: h =: 'h'&,then the expression (f , [: g h) is a 5train which denotes a verb: (f , [: g h) y NB. a 5train fyghy (f y) , (([: g h) y) NB. by meaning of 5train fyghy (f y) , (g @: h y) NB. by meaning of [: fyghy (f y) , (g h y) NB. by meaning of @: fyghy 'fy' , 'ghy' NB. by meaning of f g h fyghy 9.8 Constant FunctionsHere we continue looking at ways of broadening the class of functions that we can write as trains of verbs. There is a builtin verb 0: (zero colon) which delivers a value of zero regardless of its argument. There is a monadic and a dyadic case:
As well as 0: there are similar functions 1: 2: 3: and so on up to 9: and also the negative values: _9: to _1:
0: is said to be a constant function, because its result is constant. Constant functions are useful because they can occur in trains at places where we want a constant but must write a verb, (because trains of verbs, naturally, contain only verbs). For example, a verb to test whether its argument is negative (less than zero) can be written as (< & 0) but alternatively it can be written as a hook: negative =: < 0:
9.9 Constant Functions with the Rank ConjunctionThe constant functions _9: to 9: offer more choices for ways of defining trains. Neverthless they are limited to singledigit scalar constants. We look now at at a more general way of writing constant functions. Suppose that k is the constant in question:k =: 'hello'An explicit verb written as (3 : 'k') will give a constant result of k:
Since the verb (3 : 'k') is explicit, its rank is infinite. To apply it separately to scalars then (as we saw in Chapter 07) we need to specify a rank R of 0, with the aid of the Rank conjunction " :
The expression ((3 : 'k') " R) can be abbreviated as (k " R), because " can take, as its left argument, a verb, as above, or a noun:
Note that if k is a noun, then the verb (k"R) means: the constant value k produced for each rankR cell of the argument. By contrast, if v is a verb, then the verb (v"R) means: the verb v applied to each rankR cell of the argument. The general scheme for constant functions with " is: k " R means (3 : 'k') " R 9.9.1 A Special CaseGiven a temperature in degrees Fahrenheit, the equivalent in Celsius is computed by subtracting 32 and multiplying by fiveninths.Celsius =: ((5%9) & *) @: ( &32) Celsius 32 212 0 100Another way to define Celsius is as a fork  a train of three verbs. Celsius =: (5%9 "_ ) * (&32) Celsius 32 212 0 100Notice that the fork in Celsius above has its left verb as a constant function. Here we have a special case of a fork which can be abbreviated in the form (noun verb verb). Celsius =: (5%9) * (&32) Celsius 32 212 0 100The general scheme (new in J6) for this abbreviation for a fork is: if n is a noun, u and v are verbs, then n u v means the fork (n"_) u vWe have come to the end of of Chapter 9. 
The examples in this chapter
were executed using J version j701/beta/20101124/22:45.
This chapter last updated 22 Dec 2010
Copyright © Roger Stokes 2010.
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