J Compositions

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 123

This is in response to Gordon Sutcliffe’s letter in which he asks how in J the results of combinations of & and @ can be "derived analytically". The answer is there is no problem — simply follow the rules stated explicitly in the J Introduction and Dictionary.

There appears to be a mistranscription of some sort in the second paragraph of the letter, however assuming that the problem is to predict the results of

1. x u & v & w y     2. x u & v @ w y
3. x u @ v & w y     4. x u @ v @ w y

start by observing the monadic and dyadic rules for & and @, namely

u & v   :  u(v y)            u @ v  :  u(v y)     Monadic
           (v x)u(v y)                 u(x v y)   Dyadic

Only conjunctions and verbs appear to the left of the colon, nouns are introduced to the right. (As an aside the monadic cases of @ and & are identical, a point made explicitly in the Dictionary.)

Now extend the above rules to pairs of conjunctions:

u & v & w  =  (u&v)&w  :  (w x)(u&v)(w y) = (v w x)u(v w y)
u & v @ w  =  (u&v)@w  :  (u&v)(x w y)    =  u v(x w y)
u @ v & w  =  (u@v)&w  :  u(w x)v(w y)    =  u(w x)v(w y)
u @ v @ w  =  (u@v)@w  :  (u@v)(x w y)    =  u v(x w y)

proving conclusively that cases 2 and 4 are identical. The derived expressions for 2 and 4 demonstrate well how @ should be thought of as "in sequence", a point elaborated at greater length (as well as much more on conjunctions) in J-ottings 3.