[WIP] add a preprocessor

[jcmgray]

Description

This adds a potential preprocessor that takes an equation and converts it into a minimal form with the following simplifications:

• All indices which only appear on a single input (and not the output) are summed over.
• All indices which appear multiple times on the same term are traced.
• All scalars are multiplied into the smallest other term
• All terms with the same indices are multiplied (hamadard product / elementwise) into a single term ('de-duplication').

along with a function, that transforms input arrays into the inputs for the new simplified eq. It is a function so that it could be part of a contract 'expression'.

This would address (#114, #99, #112, #167, #189, ...).

Example from the docstring:

eq = ',ab,abee,,cd,cd,dd->ac'
arrays = helpers.build_views(eq)
new_eq, simplifier = make_simplifier(eq)
new_eq
# 'ab,cd,d->ac'
sarrays = simplifier(arrays)
oe.contract(new_eq, *sarrays)
# array([[0.65245661, 0.14684493, 0.42543411, 0.32350895],
#        [0.38357005, 0.08632807, 0.25010672, 0.19018636]])

Todos

Notable points that this PR has either accomplished or will accomplish.

• when and how should this be called?
• return a partial path rather than a 'simplifier'?
• should shape information (size_dict) be used and propagated for use with a contraction expression (and so that the smallest tensor to multiply scalars into can be most accurately chosen)?
• other simplifications are possible (e.g. hadamard deduplication up to transpose, performing all non-rank increasing contractions)...
• currently the parsing is a single sweep, should it iteratively sweep til no more changes? E.g. currently examples like ab,ab-> transform to ab-> but if processed again could transform to ->

Status

• Ready to go

[codecov]

Codecov Report

Merging #207 (d155c1f) into master (4c7eb61) will decrease coverage by 6.18%.
The diff coverage is 0.00%.

[dgasmith]

Makes sense, it would be good to start seeing some test here to find edge cases. A few notes:

• I would make a comment pass, see if we can highlight which operation we are reducing at each stage.
• The input/output pairing worries me a little bit. I wonder if we leave this as "internal" oe.contract(..., preprocess=True)
• It would be good to have a pathway for the propocess=True pathway.

Super cool addition!

[dgasmith]

Can we type these? I cannot quite tell what this is doing, somewhat looks like a set operation.

[dgasmith]

Suggested change

	def make_simplifier(*args, backend="numpy"):
	def expression_simplifier(*args, backend="numpy"):

[dgasmith]

We will need quite a few tests around this tech, too bad we cannot reproduce the original parsing tech here.