This is a beginner's introduction to tacit definition of verbs.
Although a motivation for APL and J is to have a precise language for mathematics, almost
nothing in J is exactly like its mathematical counterpart. This is one justification for J's
grammatical names noun, verb, adverb, conjunction for its objects.
As an example, although I said in the Language S thread
([: ^. ^) is a tacit verb equivalent to the mathematical composition (ln o
exp)
that assertion ignored the ambivalence of the verb ^ -- only in monadic usage does ^ correspond
to the exponential function of mathematics, and the train ([: ^. ^) permits
x ([: ^. ^) y <--> ^. (x ^ y)
as well as
([: ^. ^) y <--> ^. (^ y)
the latter corresponding to (ln o exp) y = ln(exp y) in mathematics.
This is a good thing. In the concept of verb J embraces two mathematical concepts, function
(monadic use) and operation (dyadic use), and J economically presents concepts of composition in
addition to the mathematical (f o g) y = f(g y) .
TACIT DEFINITION OF VERBS DEFINES VERBS IN TERMS OF OTHER VERBS. This occurs in mathematics
where you see h = f + g , h = f g, h = f o g, etc., but tacit definition is not much used in
math beyond differentiation formulas like (f + g)' = f' + g', (f g)' = f' g + f g', (f o g)' =
(f' o g) g' ; and you are more than likely to see the last expressed as f(g(x))' = f'(g(x))
g'(x) . Likewise, you are more likely to see (x^n)' = n x^(n-1) than (id^n)' = n id^(n-1) .
That is, math is more likely to use informal definitions with x's than tacit definitions without
x's.
Above, id is the identity function defined by
for every real number t, id(t) = t
In terms of id you can define many other functions, for example f = (1 + id)/(1
- id) which means
for every real number t except 1, f(t) = (1 + id(t))/(1 - id(t)) = (1 + t)/(1
- t) .
In J, the tacit definition of f would be expressed
f =: (1 + ]) % (1 - ])
where in monadic use ] is in fact an identity verb: x ] y is y , and ] y is y .
You may object that I have violated my definition, "TACIT DEFINITION OF VERBS DEFINES VERBS IN
TERMS OF OTHER VERBS" because there is a number, 1, in my definition of f . In fact, until
recently, you would have had to use
f =: (1"_ + ]) % (1"_ - ]) or f =: (1: + ]) % (1: - ])
where both 1"_ and 1: are constant verbs which return only the value 1. Now
the form
Noun Verb Verb (for examples 1 + ] and 1 - ]) is given special dispensation in tacit
definitions. It is understood to mean Noun"_ Verb Verb.
That is my beginner's introduction to tacit definition of verbs.
Kip Murray
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