How are APL's inner and outer product related?

Question

In APL, one can construct a generalized inner product using f.g. According to the manual, the result is an array in which each item is constructed from the vectors of the left and right operands as f/x g¨y (x and y being the vectors taken along a specific axis of said operands). On first glance, the outer product looks to be derived from this: it is ∘.g, which I obtain by setting f ← ∘. If I evaluate this using the definition of the inner product, though, I don't seem to get valid APL code (I get ∘/x g¨y, where ∘/ doesn't make a lot of sense in particular).

Are these two operators related with some magic underneath the hood, or is ∘. just interpreted as a different operator which has nothing to do with the inner product?

Answer 1

In all modern APLs, including Dyalog, ∘. is indeed just special-cased, and no left operand to . will make it behave like ∘. — indeed, ∘ is a dyadic operator, which cannot be an operand to another dyadic operator (.).

Historically, ∘.g was seen as a sort of deficient f.g since f.g can be seen as f/ over the diagonal values of ∘.g and thus, removing the reductions' part of the algorithm, an outer product remains. ∘ was used as a sort of "null function", but the symbol was later overloaded to become a regular dyadic (composition) operator.

It is however worth noting that Iverson later generalised the outer product in the form of the "tie" operator, where the left operand is a number indicating the number of "tied-up" dimensions, leaving all other dimensions to provide arguments that would be used in all combinations. Thus, 0 .g became equivalent of ∘.g and ∘ was defined as a boxed empty vector (essentially equivalent to today's ⊂⍬) and was treated as 0 when used with ..

Answer 2

According to the IBM APL2 language reference (page 165):

Z←L LO.RO R   ←→   Z←LO/¨(⊂[ρρL]L)∘.RO ⊂[1]R

(where ⎕IO←1 and LO and RO are your functions f and g respectively).