
Chapter 8: Composing VerbsThis chapter is concerned with operators which combine two verbs to produce new composite verbs. 8.1 Composition of Monad and MonadRecall the composition operator @: from Chapter 03. Given verbs sum and square we can define a composite verb, sumofthesquares. sum =: +/ square =: *:
The general scheme is that if f and g are monads then (f @: g) y means f (g y) Note in particular that f is applied to the whole result (g y). To illustrate, suppose g applies separately to each row of a table, so we have: g =: sum " 1 f =: <
We have just seen the most basic of kind of composition. Now we look at some variations. 8.2 Composition: Monad And DyadIf f is a monad and g is a dyad, then (f @: g) is a dyadic verb such that x (f @: g) y means f (x g y) For example, the sum of the product of two vectors x and y is called the "scalar product". sp =: +/ @: *
The last example showed that, in the expression (x (f @: g) y) the verb f is applied once to the whole of (x g y) 8.3 Composition: Dyad And MonadThe conjunction &: (ampersand colon, called "Appose") will compose dyad f and monad g. The scheme is: x (f &: g) y means (g x) f (g y) For example, we can test whether two lists are equal in length, with the verb (= &: #) eqlen =: = &: #
Here f is applied once to the whole of (g x) and (g y). 8.4 Ambivalent CompositionsTo review, we have seen three different schemes for composition. These are: (f @: g) y = f (g y) x (f @: g) y = f (x g y) x (f &: g) y = (g x) f (g y) There is a fourth scheme, (f &: g) y = f (g y) which is, evidently, the same as the first. This apparent duplication is useful for the following reason. Suppose verb g is ambivalent, that is, has both a monadic and dyadic case. It follows from the first two schemes that the composition (f @: g) is also ambivalent. Similarly, if verb f is ambivalent, it follows from the third and fourth schemes that (f &: g) is ambivalent. To illustrate, let g be the ambivalent builtin verb (.) with (. y) being the reverse of y and x . y being the rotation of y by x places.
For an example of ambivalent (f &: g), let f be the verb %  reciprocal or divide.
8.5 More on Composition: Monad Tracking MonadThe conjunction @ is a variation of the @: conjunction. Here is an example to show the difference between (f @: g) and (f @ g). y =: 2 2 $ 0 1 2 3
We see that with (f @: g) verb f is applied once. However, with (f@g), for each separate application of g there is a corresponding application of f. We could say that applications of f track the applications of g. Suppose that the monadic rank of g is G. Then (f @ g) means (f @: g) applied separately to each Gcell, that is, (f @: g)"G. RANKS =: 1 : 'x. b. 0'
and so the general scheme is: (f @ g) y means (f @: g) " G y There is also the & operator. For reasons of symmetry, as with the ambivalent functions mentioned above, (f&g) y means the same as (f@g) y. 8.6 Composition: Monad Tracking DyadNext we look at the composition (f @ g) for a dyadic g. Suppose f and g are defined by: f =: < g =: . " 0 1 NB. dyadic Here x g y means: rotate vectors in y by corresponding scalars in x. For example:
Here now is an example to show the difference between f @: g and f @ g
We see that with (f @: g) verb f is applied once. With (f@g), for each separate application of g there is a corresponding application of f. Suppose that the left and right ranks of dyad g are L and R. Then (f @ g) means (f @: g) applied separately to each pair of an Lcell from x and corresponding Rcell from y. That is, (f@g) means (f @: g)"G where G = L,R.
The scheme is: x (f@g) y = x (f@:g) " G y 8.7 Composition: Dyad Tracking Monadhere we look at the composition (f & g) for dyadic f. Suppose g is the "Square" function, and f is the "comma" function which joins two lists. f =: , g =: *:
Here now is an example to show the difference between (f &: g) and (f & g)
We see that in (f &: g) the verb f is applied just once, to join the two lists of squares. By contrast, in (f & g) each separate pair of squares is combined with a separate application of f The scheme is that x (f & g) y means (g x) (f " G,G) (g y) where G is the monadic rank of g. Here f is applied separately to each combination of a Gcell from x and a corresponding Gcell from y. To illustrate:
8.8 SummaryHere is a summary of the 8 cases we have looked at so far. @: (f @: g) y = f (g y) @: x (f @: g) y = f (x g y) &: (f &: g) y = f (g y) &: x (f &: g) y = (g x) f (g y) @ (f @ g) y = (f @: g) " G y @ x (f @ g) y = x (f @: g) " LR y & (f & g) y = (f @: g) " G y & x (f & g) y = (g x) (f " (G,G)) (g y) where G is the monadic rank of g and LR is the vector of left and right ranks of g. 8.9 InversesThe "Square" verb, (*:), is said to be the inverse of the "Squareroot" verb (%:). The reciprocal verb is its own inverse.
Many verbs in J have inverses. The adverb (^: _1) produces the inverse verb of its argument verb. Let us call this adverb INV. INV produces "Squareroot" from "Square":
INV can automatically find inverses, not only of builtin verbs, but of userdefined verbs such as compositions. For example, the inverse of (1 + the squareroot) of y is (the square of 1 minus)y.
8.10 Composition: Verb Under VerbWe now look at composition with the conjunction &. (ampersand dot, called "Under"). The idea is that the composition "f Under g" means: apply g, then f, then the inverse of g. For an example, suppose first that f is the verb which rounds a number to the nearest integer:
A number can be rounded to the nearest 10, say, by dividing by 10, rounding to nearest integer, then multiplying by 10 again. Let g be division by 10, and then (g INV) will be the inverse, multiplication by 10. g =: % & 10
The general scheme is that (f &. g) y means (g INV) f g y This is the end of Chapter 8. 
Copyright © Roger Stokes 2000. This material may be freely reproduced, provided that this copyright notice and provision is also reproduced.
last updated 10 March 00