prod(numeric(0)) surprise

On Mon, 9 Jan 2006, Martin Morgan wrote:

I guess I have to say yes, I'd exepct

x <- 1:10
sum(x[x>10]) ==> numeric(0)

this would be reinforced by recongnizing that numeric(0) is not zero,
but nothing. I guess the summation over an empty set is an empty set,
rather than a set containing the number 0. Certainly these

exp(x[x>10]) ==> numeric(0)
numeric(0) + 1 ==> numeric(0)

There are some fairly simple rules in how R does it.  You do need to
distinguish between functions (binary operators) that map two vectors of
length n to a vector of length n and functions such as prod and sum that
map a vector of length n to a vector of length 1.

The output of sum and prod is always of length 1, so sum(numeric(0)) and
prod(numeric(0)) should be of length 1 (or give an error).  It is
convenient that sum(c(x,y)) is equal to sum(x)+sum(y) and that
prod(c(x,y)) is equal to prod(x)*prod(y), which motivates making
sum(numeric(0)) give 0 and prod(numeric(0)) give 1.

Single argument functions such as exp(numeric(0)) seem fairly obvious: you
have no numbers and you exponentiate them so you still have no numbers.
You could also argue based on c() and exp() commuting.

The rules for binary operators are a little less tidy [my fault]. They
come from the idea that x+1 should always add 1 to each element of x.  If
you add 1 to each element of numeric(0) you get numeric(0).  The usual
recycling rule says that the shorter vector should be repeated to make it
the same length as the longer vector, so this is a wart.  On the other
hand, you can't recycle a vector of length 0 to have length 1, so the
usual recycling rule can't be applied here. This also makes matrix
operations work, at least in the sense of getting
matrices of the right dimension.