More semi-ring algebras

[ArrogantGao]

I add more semi-ring algebras to the package, including the TropicalMaxMul, TropicalMinPlus and TropicalAndor, and I also give the original type Tropical an extra name TropicalMaxPlus, all tests passed for these algebras.

TropicalMinMul is not added, since $(\mathbb{R}^{-}, \cdot)$ do not have identity element.

I changed the rule about Bool type from

for T in [:Tropical, :TropicalMaxMul, :TropicalMinPlus, :CountingTropical]
    @eval begin
        Base.:(/)(a::$T, b::Bool) = b ? a : a / zero(a)
        Base.:(/)(b::Bool, a::$T) = b ? inv(a) : zero(a)
        Base.div(a::$T, b::Bool) = b ? a : a / zero(a)
        Base.div(b::Bool, a::$T) = b ? inv(a) : zero(a)
    end
end

to

for T in [:Tropical, :TropicalMaxMul, :TropicalMinPlus, :CountingTropical]
    @eval begin
        Base.:(/)(a::$T, b::Bool) = b ? a : a / zero(a)
        Base.:(/)(b::Bool, a::$T) = b ? inv(a) : zero(a)
        Base.div(a::$T, b::Bool) = b ? a : a ÷ zero(a)
        Base.div(b::Bool, a::$T) = b ? one(a) ÷ a : zero(a)
    end
end

The original one does not work well with TropicalMaxMul, since

julia> a = TropicalMaxMul(2.0)
2.0ₜ

julia> zero(TropicalMaxMul)
0ₜ

julia> a / zero(TropicalMaxMul)
Infₜ

julia> a ÷ zero(TropicalMaxMul)
NaNₜ

This change doesn't affect the original tests.

[ArrogantGao]

Another problem is that now all these different Tropical type are shown as xₜ, do we need to assign different indices to different types (indices like maxplus maybe to long)?

[GiggleLiu]

I think this is a very solid PR, I just have a few minor suggestions. I wish we could show the definitions of these algebras in the README, e.g. a table with (+, *, 1, 0, subscript) as columns and different algebra types as rows.

[GiggleLiu]

Good catch! Please remove the lines commented out.

[GiggleLiu]

I wish they could be printed with a different subscript.

[GiggleLiu]

Good job! This is exactly what I meant.

[GiggleLiu]

Another comment: we'd better add a new abstract type AbstractSemiring as their supertype.

[GiggleLiu]

About the subscripts, you may find a complete list of unicode here:
https://docs.julialang.org/en/v1/manual/unicode-input/

Among which I would suggest

• using ₜ for TropicalMaxPlus and TropicalAndOr
• using ₛ for TropicalMinPlus, because s and t have strong correlation when used in notation.
• using ₓ for TropicalMaxMul, because x can be related to multiplication.

I do not have strong opinion on how they look like. It is more important to explain these convensions in the documentation.

[ArrogantGao]

The following changes have been made:

1. An abstract type AbstractSemiring <: Number is added as the super type of all the tropical algebras.
2. Indices are changed, test sets are separated, and the test about TropicalMaxPlus are replaced by

@test Tropical == TropicalMaxPlus
@test TropicalF16 == TropicalMaxPlusF16
@test TropicalF32 == TropicalMaxPlusF32
@test TropicalF64 == TropicalMaxPlusF64

The latest test result is shown below (please see the indices):

     Testing Running tests...
trueₜ
Test Summary:   | Pass  Total  Time
tropical and or |   12     12  0.1s
3.0ₓ
Test Summary:    | Pass  Total  Time
tropical max mul |   44     44  0.2s
3.0ₛ
Test Summary:     | Pass  Total  Time
tropical min plus |   46     46  0.1s
3.0ₜ
Test Summary:     | Pass  Total  Time
tropical max plus |   50     50  0.1s
(3.0, 1.0)ₜ
Test Summary:     | Pass  Total  Time
counting_tropical |   25     25  0.1s
[ Info: SetupBuildDirectory: setting up build directory.
[ Info: Doctest: running doctests.
[ Info: Skipped ExpandTemplates step (doctest only).
[ Info: Skipped CrossReferences step (doctest only).
[ Info: Skipped CheckDocument step (doctest only).
[ Info: Skipped Populate step (doctest only).
[ Info: Skipped RenderDocument step (doctest only).
Test Summary:             | Pass  Total  Time
Doctests: TropicalNumbers |    1      1  5.4s
     Testing TropicalNumbers tests passed 

3. Documents and the readme file are revised, I added the description about semiring, tropical geometry and each of the algebras.

[ArrogantGao]

4. All the export are moved to the TropicalNumber.jl file to avoid repeat exportation.

[GiggleLiu]

Great work, I think all the changes make sense to me. I forgot to mention not using error handling in the consctructor.

We should remove all error handling in the default constructor of TropicalMaxMul, otherwise the code can not run on GPU.
A better practise would be using unsafe_xxx as the constructor without error handling. However, here to make life easier, we can also assume users know the negative numbers are forbidden in this algebra.

Either approach is good to me.

[ArrogantGao]

All the error handling are removed, here we may just assume all users know how to use it properly.

[codecov]

Codecov Report

Merging #9 (32272ca) into master (6200c3b) will decrease coverage by 1.36%.
The diff coverage is 83.78%.

@@            Coverage Diff             @@
##           master       #9      +/-   ##
==========================================
- Coverage   86.41%   85.06%   -1.36%     
==========================================
  Files           3        6       +3     
  Lines          81      154      +73     
==========================================
+ Hits           70      131      +61     
- Misses         11       23      +12     

Files Changed	Coverage Δ
src/TropicalNumbers.jl	84.00% <70.00%> (-9.75%)	⬇️
src/tropical_andor.jl	72.72% <72.72%> (ø)
src/tropical_maxmul.jl	83.33% <83.33%> (ø)
src/tropical_minplus.jl	92.85% <92.85%> (ø)
src/tropical_maxplus.jl	82.85% <100.00%> (ø)

[GiggleLiu]

Good job. I just registered a new version v0.6.0. 🎉

[ArrogantGao]

Wow, thanks a lot.
Then I will work on the tensor element interface of CuTropicalGEMM.jl.