# Trains

 Z←  (f g) R is called a Monadic Hook — Z ≡ R f g R Z←L (f g) R is called a Dyadic Hook — Z ≡ L f g R Z←  (f g h) R is called a Monadic Fork — Z ≡ (f R) g h R Z←L (f g h) R is called a Dyadic Fork — Z ≡ (L f R) g L h R Z←  (f g h ...) RZ←L (f g h ...) R is also defined for longer Trains
L and R are arbitrary arrays, f, g, h, etc, are arbitrary functions of any type: primitive, user-defined, system, anonymous, and/or derived.

This clever idea from the designers of J is called Trains where a parenthesized sequence of functions (which normally would signal a SYNTAX ERROR) can be interpreted as per the above descriptions. Very nicely, they fit into and extend the spectrum of function definition syntax from user-defined to dynamic to trains to operator expressions. They are another and very interesting form of functional programming.

Note that the spacing between functions is for visual purposes only — it has no effect on the interpretation.

For example,

(,⍎)'2+3'
←→ '2+3',⍎'2+3'
←→ '2+3',5
2+3 5

avg←(+⌿ ÷ ≢) defines a function that computes the average of a numeric vector.
avg 1 2 3 4
←→ (+⌿ ÷ ≢) 1 2 3 4
←→ (+⌿1 2 3 4) ÷ ≢1 2 3 4
←→ 10 ÷ 4
2.5

Longer Trains are defined as follows:

(e f g h) ←→ (e (f g h))
(d e f g h) ←→ (d e (f g h))

and in general

 Even length: (a b c ...) ←→ (a (b c ...)) Odd length: (a b c ...) ←→ (a b (c ...))

For more applications of this concept, see the discussion in the Learning J manual.

There is also a series of tables of common function expressions and their corresponding Train.