1# The `Broadcastable` Trait 2 3[TOC] 4 5## Description 6 7The `Broadcastable` trait enforces the following properties on an operation: 8 9- The operation has at least one input operand. 10 11- The operation has exactly one result. 12 13- All input operands and result are of type `tensor` or `vector`. 14 15- A shape inference mechanism is able to compute the result shape solely based on input operand shapes. 16 17- Input operands have broadcast-compatible shapes, according to the verification rules presented below. 18 19- The operation's result shape is compatible with —though not necessarily identical to— the shape inferred from its input operands, according to the verification rules presented below. 20 21 22## Dimension inference 23 24Given an operation with two input operands, the size of dimension `i` of its result can be inferred from dimension `i` of the operands according to the table below. Here, `dim0` and `dim1` represent dimension `i` of the input operands in an interchangeable order, while `inferredDim` represents the inferred size for dimension `i` of the operation result. Dimensions are classified in three categories: dynamic ("?"), static equal to 1 ("1"), and static greater than 1 (">1"). 25 26 27| `dim0` | `dim1` | `inferredDim` | Notes | 28| -------- | -------- | ------------- | ----- | 29| ? | ? | ? | If `RuntimeSize(dim0)` is 1, dimension `dim0` is broadcast to `RuntimeSize(dim1)`. If `RuntimeSize(dim1)` is 1, dimension `dim1` is broadcast to `RuntimeSize(dim0)`. The operation produces undefined behavior if both runtime sizes are greater than 1 and not equal. | 30| ? | 1 | ? | Dimension `dim1` is broadcast to `RuntimeSize(dim0)`. | 31| ? | >1 | `dim1` | If `RuntimeSize(dim0)` is 1, `dim0` is broadcast to `dim1`. The operation produces undefined behavior if `RuntimeSize(dim0)` is greater than 1 and not equal to `dim1`. | 32| 1 | 1 | 1 | | 33| 1 | >1 | `dim1` | Dimension `dim0` is broadcast to `dim1`. | 34| >1 | >1 | `dim0` | The operation verifier produces a compile-time error if `dim0` != `dim1`. | 35 36 37The following pseudo-function is a formal representation of the dimension inference process: 38 39```python 40InferDim(dim0, dim1): 41 switch (dim0, dim1): 42 case (?, ?): 43 case (?, 1): 44 case (1, 1): 45 case (>1, ?): 46 case (>1, 1): 47 return dim0 48 case (?, >1): 49 case (1, ?): 50 case (1, >1): 51 return dim1 52 case (>1, >1): 53 ERROR_IF(dim0 != dim1) 54 return dim0 55``` 56 57## Shape inference 58 59The shape inference process begins by correcting rank differences in input operands. A shape is expanded by adding additional dimensions of size 1 on its left until the desired rank is reached, as shown here: 60 61```python 62ExpandRank(shape, rank): 63 while len(shape) < rank: 64 shape.prepend(1) 65``` 66 67Given the shapes of two ranked input operands, the result's shape is inferred by equalizing input ranks and inferring individual dimensions, as shown here: 68 69```python 70InferShape(shape0, shape1): 71 72 # Equalize ranks 73 rank = max(GetRank(shape0), GetRank(shape1)) 74 ExpandRank(shape0, rank) 75 ExpandRank(shape1, rank) 76 77 # Infer shape 78 inferredShape = [] 79 for (dim0, dim1) in zip(shape0, shape1): 80 inferredDim = InferDim(dim0, dim1) 81 inferredShape.append(inferredDim) 82 return inferredShape 83``` 84 85The result shape for an operation with an arbitrary number of input operands is then inferred by discarding unranked operands, applying shape inference on the first ranked operand pair, and updating the inferred shape with each additional ranked operand. If the operation has no ranked operands, the result shape cannot be inferred. If the operation has exactly one ranked operand, its shape is directly provided as the inferred result shape. Formally: 86 87```python 88InferResultShape(op): 89 90 # Filter ranked operands 91 rankedOperands = filter(op.operands, IsRanked) 92 if len(rankedOperands) == 0: 93 return None 94 95 # Infer result shape 96 inferredShape = GetShape(rankedOperands[0]) 97 for operand in rankedOperands[1:]: 98 inferredShape = InferShape(inferredShape, GetShape(operand)) 99 return inferredShape 100``` 101 102## Verification 103 104The legality of an operation with the `Broadcastable` trait is verified by first running the shape inference process. If a failure occurs during shape inference, it is concluded that input operands are not broadcast-compatible, and verification fails. If shape inference succeeds, verification continues. 105 106If either the result is unranked or all input operands are unranked, no further verification steps are needed, and the process ends here successfully. If, on the contrary, both the result and at least one input operand are ranked, verification continues by checking for a matching rank between the previously inferred shape and the result. 107 108Once a rank match is guaranteed, each dimension of the inferred shape is compared with the corresponding dimension of the actual result shape according to the following table table: 109 110 111| `inferredDim` | `actualDim` | Verification outcome | 112| ------------- | ----------- | -------------------- | 113| ? | ? | **OK** | 114| ? | static | **OK** <br> A failure to guarantee that the runtime dimension size of the result is equal to `actualDim` causes undefined behavior. While unusual, this implicit dynamic-to-static cast is convenient in certain scenarios, such as an intermediate state of a shape inference pass. Ultimately, a static dimension in the result implies that all input dimension sizes are also known at compile time and may therefore become static as well, preferably. | 115| static | ? | **OK** <br> The actual result dimension may be dynamic even when a static size can be inferred at compile time. The programmer may choose to relax the specificity of the result dimension for forward compatibility of the result type. | 116| static | static | **OK if equal** <br> When both the inferred and actual dimensions are static, they must be set to the same size. | 117 118 119The full verification process can be formally specified as follows: 120 121```python 122Verify(op): 123 124 # Run shape inference 125 inferredShape = InferResultShape(op.operands) 126 127 # Done if result is unranked or all operands are unranked 128 if not IsRanked(op.result) or inferredShape is None: 129 return 130 131 # Rank must match 132 actualShape = GetShape(op.result): 133 ERROR_IF(len(inferredShape) != len(actualShape)) 134 135 # Verify 136 for (inferredDim, actualDim) in zip(inferredShape, actualShape): 137 ERROR_IF(IsStatic(actualDim) and inferredDim != actualDim) 138``` 139 140## Examples 141 142The following are correct uses of broadcastable ops: 143 144```mlir 145// Exact match of static sizes. 146%result = "test.broadcastable"(%arg0, %arg1) : (tensor<1x2xi32>, tensor<1x2xi32) -> tensor<1x2xi32> 147 148// Dynamic sizes match. The programmer must guarantee that the runtime sizes of 149// %arg0 and %arg1 are equal at runtime. 150%result = "test.broadcastable"(%arg0, %arg1) : (tensor<?xi32>, tensor<?xi32) -> tensor<?xi32> 151 152// The shape of %arg0 is broadcast from tensor<1xi32> to tensor<4xi32>. 153%result = "test.broadcastable"(%arg0, %arg1) : (tensor<1xi32>, tensor<4xi32) -> tensor<4xi32> 154 155// The shape of %result is inferred as tensor<4xi32>, while the actual result 156// type is tensor<?xi32>. The inferred shape is compatible with the actual shape. 157%result = "test.broadcastable"(%arg0) : (tensor<4xi32) -> tensor<?xi32> 158 159// The shape of %arg0 is first expanded to tensor<1x1x4xi32> and then broadcast 160// to tensor<2x3x4xi32>. 161%result = "test.broadcastable"(%arg0, %arg1) : (tensor<4xi32>, tensor<2x3x4xi32) -> tensor<2x3x4xi32> 162 163// Input and results tensors have different element types (i1, i32, i64). The 164// 'Broadcastable' trait has no restrictions on element types. 165%result = "test.broadcastable"(%arg0, %arg1) : (tensor<2xi1>, tensor<2xi32) -> tensor<2xi64> 166 167// No result shape verification is needed when the result is unranked. 168%result = "test.broadcastable"(%arg0) : (tensor<2xi32>) -> tensor<*xi32> 169 170// No result shape verification needed when all inputs are unranked. 171%result = "test.broadcastable"(%arg0, %arg1) : (tensor<*xi32>, tensor<*xi32>) -> tensor<2xi32> 172``` 173 174 175The following are incorrect uses of broadcastable ops: 176 177```mlir 178// Dimension 0 of input operands is static but not equal. 179%result = "test.broadcastable"(%arg0, %arg1) : (tensor<3xi32>, tensor<2xi32) -> tensor<?xi32> 180 181// The inferred result shape is tensor<3xi32>, but the actual result shape is 182// tensor<1x3xi32>. Inferred and actual shapes differ in rank. 183%result = "test.broadcastable"(%arg0, %arg1) : (tensor<3xi32>, tensor<3xi32) -> tensor<1x3xi32> 184 185// The inferred result shape is tensor<?xi32>, but the actual shape is 186// tensor<4xi32>. The inferred shape is not compatible with the actual shape. 187%result = "test.broadcastable"(%arg0, %arg1) : (tensor<?xi32>, tensor<?xi32) -> tensor<4xi32> 188 189// The inferred result shape is tensor<2xi32>, but the actual result shape is 190// tensor<4xi32>, which is not compatible. 191%result = "test.broadcastable"(%arg0, %arg1) : (tensor<2xi32>, tensor<2xi32) -> tensor<4xi32> 192 193// The inferred result shape is tensor<1xi32>, but the actual result shape is 194// tensor<4xi32>. Broadcast semantics are not applicable for results. 195%result = "test.broadcastable"(%arg0, %arg1) : (tensor<1xi32>, tensor<1xi32) -> tensor<4xi32> 196``` 197 198 199