prod() over zero-length tuple generator should just return 1?

[davidavdav]

Hello,

This might be related to #28028, #36262 and similar issues.

Observing the following statements:

## These all work as expected
julia> collect(i for i in Int[]) == Int[]
true
julia> sum(i for i in Int[]) == 0
true
julia> prod(i for i in Int[]) == 1
true
julia> collect(i for (i, j) in zip(Int[], Int[])) == Int[]
true
## but here I would expect result `1`
julia> prod(i for (i, j) in zip(Int[], Int[]))
ERROR: ArgumentError: reducing over an empty collection is not allowed
...

I can't really say where this goes wrong, whether it is in the zip generator or in the reduction in prod().

The following "identity" function id(n) evaluates for all integers except 1:

import Primes
id(n::Integer) = prod(prime^power for (prime, power) in Primes.factor(n))

Of course such functions could check for cases where n==1 but that is somewhat clumsy, I think. My actual use is in a function computing the sum of the divisors of a positive natural number:

sigma(n::Integer) = prod((prime ^ (power + 1 ) - 1 ) ÷ (prime - 1) for (prime, power) in Primes.factor(n))
sigma(28) == 2 * 28
sigma(1)
ERROR: ArgumentError: reducing over an empty collection is not allowed

[JeffBezanson]

The difficulty with this is that without any elements, we don't know what the domain is, so we don't know if 1 is the correct identity element.

(We have prod(()) == 1 but it's a legacy thing that's not ideal.)

[davidavdav]

Is prod(()) what prod(i for i in Int[]) in the end evaluates to? I can imagine it is difficult to deduce the type of the argument from an empty generator expression, but julia is still able to do this for the non-tuple cases

prod(i for i in Int[]) === 1
prod(i for i in Float64[]) === 1.0

but tuples start to make a difference between collect and prod

collect(i for (i,j) in NTuple{2, Int}[]) == Int[]
collect(j for (i,j) in Tuple{Int, Float64}[]) == Float64[]
prod(i for (i,j) in NTuple{2, Int}[]) ## Error

Thanks, anyways, I can definitely live with the current behavior, I just wondered if this is intended or just hard to solve case.

[StefanKarpinski]

For products and sums, I feel like using true and false as "weak" identities would be ok, but it's a fairly big decision. And even if it works computationally, people will be surprised and complain about it, so 🤷🏻‍♂️

[JeffBezanson]

Is prod(()) what prod(i for i in Int[]) in the end evaluates to?

Not exactly, we just have some special cases for the identity function, allowing i for i in ... to work.

[vtjnash]

similar to #36262 we have init now for this. See also triage's rationale at #29919 (comment) for why this is not implemented