Add grouping brackets

[fferflo]

Currently, there are no brackets that allow grouping subexpressions without also including additional semantics.

• Axis composition (a b) groups axes a and b, but also says that they are to be read as a single composed axis.
• Brackets [a b] group axes a and b, but also say that the operation should be applied to these axes.
• A new kind of grouping bracket, e.g. {a b}, would group the axes a and b without including additional meaning, i.e. it would be equivalent to just writing a b.

Some examples where this would be useful:

Example 1: Apply an ellipsis to a group

In the following expression, a and b are grouped in (a b), and the ellipsis therefore applies to the entire group:

"(a b)... -> a... b..."
# expands to (assuming the ellipsis is expanded to 2 repetitions)
"(a.0 b.0) (a.1 b.1) -> a.0 a.1 b.0 b.1"

If I don't want to compose a and b into a single new axis though, I could use grouping brackets instead:

"{a b}... -> a... b..."
# expands to (assuming the ellipsis is expanded to 2 repetitions)
"{a.0 b.0} {a.1 b.1} -> a.0 a.1 b.0 b.1"
# same as (grouping brackets add no semantics)
"a.0 b.0 a.1 b.1 -> a.0 a.1 b.0 b.1"

Example 2: Composability of -> and ,

The composability of expressions in einx also allows composing grouping brackets with -> and ,. This is useful for example to specify batch dimensions for all input and output tensors in an operation jointly:

einx.dot("... {a [b], [b] c -> a c}", ...) # specifiy batch dimensions jointly for all arguments
# is resolved to
einx.dot("... {a [b]}, ... {[b] c} -> ... {a c}", ...)
# same as
einx.dot("... a [b], ... [b] c -> ... a c", ...)

Using e.g. axis composition here changes the semantics of the expression:

einx.dot("... (a [b], [b] c -> a c)", ...)
# is resolved to
einx.dot("... (a [b]), ... ([b] c) -> ... (a c)", ...) # this is not what we wanted to express

Using brackets also changes the semantics:

einx.dot("... [a [b], [b] c -> a c]", ...)
# is resolved to
"einx.dot("... [a [b]], ... [[b] c] -> ... [a c]", ...)
# same as (brackets are idempotent)
"einx.dot("... [a b], ... [b c] -> ... [a c]", ...) # this is not what we wanted to express

---

Good:

• Allows writing some expressions more concisely.
• Probably easy to implement.

Bad:

• Adds another type of delimiter (e.g. braces {} as in the examples above) in addition to axis composition () and operation axes [], which adds to the complexity of einx expressions. This makes the notation more difficult to read/ learn and less intuitive.