Changed Complex strict equality internally

[mahrud]

This changes how two complexes are compared to be identical internally, without incurring any computational cost. The point is that these basic things didn't work before: (I added these as a test)

R = QQ[x]
C = complex R
D = complex R
id_C |  id_D
id_C || id_D
id_C // id_D

But now complexes are compared like so:

isIdentical(Complex, Complex) := (C, D) -> C === D or C.module === D.module and C.dd.map === D.dd.map

isIdentical(ComplexMap, ComplexMap) := (f, g) -> f === g or (
    f.degree === g.degree and f.map === g.map
    and isIdentical(f.source, g.source)
    and isIdentical(f.target, g.target))

Fixes #3865 and #3868

[ggsmith]

This looks like a good idea to me. However, should the method name be areIdentical?

[mahrud]

Since it's an internal method I don't mind either way, but for the record I was going with matching isIsomorphic. Would you like me to change it?

[ggsmith]

Having investigated a bit further, I am still concerned that isIdentical is still too slow. In particular, it appears to take essentially the same amount of time as ==.

[mahrud]

What was the example you tried it on?

[ggsmith]

@mikestillman and I create some relatively large Koszul complexes to compare isIdentical and ==. We also looked at the source code for === focusing on its behaviour for matrices and modules.

[mahrud]

Ah, I see. Let me explain.

What you've observed about the difference between isIdentical and === is simply the fact that === for a HashTable (which like isIdentical checks === on all content) is slower than === for a MutableHashTable (which checks two fixed hash numbers)! In other words, even if you changed Complex into a HashTable, the timing would exactly match isIdentical, no better and no worse!

The reason in your examples == and isIdentical took the same amount is that internally == for modules and matrices first checks === before doing anything more complicated, and if you have large identical koszul complexes that === check succeeds immediately, which is exactly what isIdentical does, so they take the same amount. It's just that the matrices are large so checking entry by entry is taking a while.

In fact, in all cases simply constructing the koszul complex took an order of magnitude longer than isIdentical, so I'm not sure if the cost of using isIdentical is relevant at all. In other words, isIdentical is not going to be the bottleneck in a computation if constructing the objects in the first place takes an order of magnitude longer.

Finally, to respond to your initial concern, == and isIdentical are absolutely not the same when the modules are equal but not identical, which is the case you presumably want to avoid in liftMapAlongQuasiIsomorphism and other places. Here is an example:

debug needsPackage "FGLM"
I = cyclic(5)
C = complex module I
D = complex module ideal random I_*
elapsedTime isIdentical(C, D) -- false, .00001999s  elapsed
elapsedTime C === D           -- false, .000003078s elapsed
elapsedTime C == D            -- true,  .478898s    elapsed

If you raise that 5 to a 6 or 7, == should take up to half an hour!

[mahrud]

To recap the conversation during the meeting, the example above shows that this statement about isIdentical is incorrect:

In particular, it appears to take essentially the same amount of time as ==.

I spent quite a bit of time with liftMapAlongQuasiIsomorphism and I could not produce an example where switching from === to isIdentical causes a noticeable slowdown. I would like to see an example if you have one in mind.