TensorDSL

Prototype for a symbolic Tensor DSL in Julia.

Currently turns tensor operations into an abstract syntax tree.

Example:

julia> A = Tensor([1 2 ; 3 4], RnSpace(2), RnSpace(2))
julia> B = VariableTensor(RnSpace(2), RnSpace(2))
julia> x = FreeIndex(RnSpace(2), "x")

julia> (A + B)[1, x]
IndexingOperation{0}
        AddOperation{2}
                Tensor{Int64,2}, [1 2; 3 4], shape: R^2⊗ R^2
                VariableTensor{2}, shape: R^2⊗ R^2
        (1, x)

Supported operations

Vector Spaces

Vector spaces with or without a known number of dimensions.

V4 = VectorSpace(4)
V = VectorSpace()

Real vector spaces

R2 = RnSpace(2)

Dual spaces

julia> typeof(V4')
DualVectorSpace

Tensors

Tensors created from multidimensional array

A = Tensor([1 2 ; 3 4], R2, R2)

Tensors with unknown value

B = VariableTensor(R2, R2)

Tensors where all entries have the same value

julia> C = ConstantTensor(3, V4, V, V4')
ConstantTensor{Int64,3}, 3, shape: V4_2⊗ V_3⊗ V4_2*

Zero and and identity tensors

Z = ZeroTensor(V4)
I = DeltaTensor(V4, R2)

Tensor operations

Addition

julia> A + B
AddOperation{2}
        Tensor{Int64,2}, [1 2; 3 4], shape: R^2⊗ R^2
        VariableTensor{2}, shape: R^2⊗ R^2

Outer product

julia> A ⊗ C
OuterProductOperation{5}
        Tensor{Int64,2}, [1 2; 3 4], shape: R^2⊗ R^2
        ConstantTensor{Int64,3}, 3, shape: V4_2⊗ V_3⊗ V4_2*

Indexing

julia> x = FreeIndex(V4, "x")
julia> y = FreeIndex(V, "y")
julia> C[1, y, x']
IndexingOperation{0}
        ConstantTensor{Int64,3}, 3, shape: V4_2⊗ V_3⊗ V4_2*
        (1, y, x*)

Component tensor

julia>C[1, y]
ComponentTensorOperation{1}
        IndexingOperation{0}
                ConstantTensor{Int64,3}, 3, shape: V4_2⊗ V_3⊗ V4_2*
                (1, y, FreeIndex*)
        (FreeIndex*,)
julia> componenttensor(C[1, y, x'], x', y)
ComponentTensorOperation{2}
        IndexingOperation{0}
                ConstantTensor{Int64,3}, 3, shape: V4_2⊗ V_3⊗ V4_2*
                (1, y, x*)
        (x*, y)

Tensor contraction

julia> C[1, y, x']*Z[x]
IndexSumOperation{0}
        OuterProductOperation{0}
                IndexingOperation{0}
                        ConstantTensor{Int64,3}, 3, shape: V4_2⊗ V_3⊗ V4_2*
                        (1, y, x*)
                IndexingOperation{0}
                        ZeroTensor{1}, shape: V4_2
                        (x,)
        (x*,)