In Finch, all tensors accessed by a particular index must have the same dimension along the corresponding mode. Finch determines the dimension of a loop index i from all of the tensors using i in an access, as well as the bounds in the loop itself.

For example, consider the following code

A = fsprand(3, 4, 0.5)
B = fsprand(4, 5, 0.5)
C = Tensor(Dense(SparseList(Element(0.0))))
@finch begin
C .= 0
for i = 1:3
for j = _
for k = _
C[i, j] += A[i, k] * B[k, j]
end
end
end

In the above code, the second dimension of A must match the first dimension of B. Also, the first dimension of A must match the i loop dimension, 1:3. Finch will also resize declared tensors to match indices used in writes, so C is resized to (1:3, 1:5). If no dimensions are specified elsewhere, then Finch will use the dimension of the declared tensor.

Dimensionalization occurs after wrapper arrays are de-sugared. You can therefore exempt a tensor from dimensionalization by wrapping the corresponding index in ~. For example,

@finch begin
y .= 0
for i = 1:3
y[~i] += x[i]
end

does not set the dimension of y, and y does not participate in dimensionalization.

In summary, the rules of index dimensionalization are as follows:

• Indices have dimensions
• Using an index in an access “hints” that the index should have the corresponding dimension
• Loop dimensions are equal to the “meet” of all hints in the loop body and the loop bounds
• The meet usually asserts that dimensions match, but may also e.g. propagate info about parallelization

The rules of declaration dimensionalization are as follows:

• Declarations have dimensions
• Left hand side (updating) tensor access “hint” the size of that tensor
• The dimensions of a declaration are the “meet” of all hints from the declaration to the first read
• The new dimensions of the declared tensor are used when the tensor is on the right hand side (reading) access.
Finch.FinchNotation.DimensionlessType
Dimensionless()

A singleton type representing the lack of a dimension. This is used in place of a dimension when we want to avoid dimensionality checks. In the @finch macro, you can write Dimensionless() with an underscore as for i = _, allowing finch to pick up the loop bounds from the tensors automatically.