Add spec for Quantized Types and Constants

docs/spec.md

@@ -140,6 +140,7 @@ which may allow us to remove tuple types from StableHLO
 
 ```ebnf
 ElementType ::= BooleanType | IntegerType | FloatType | ComplexType
+                | QuantizedType
 BooleanType ::= 'i1'
 IntegerType ::= 'si4' | 'si8' | 'si16' | 'si32' | 'si64'
               | 'ui4' | 'ui8' | 'ui16' | 'ui32' | 'ui64'
@@ -173,10 +174,66 @@ values of type `tensor<T>`).
   and an **imaginary part** of the same **element type**. Supported complex
   types are `complex<f32>` (both parts are of type `f32`) and `complex<f64>`
   (both parts are of type `f64`).
-* In the future, we are also planning to introduce **quantized types** that
-  represent integer values obtained via uniform quantization of floating-point
-  values using given scales and zero points
-  ([#588](https://github.com/openxla/stablehlo/issues/588)).
+* **Uniform Quantized types** represent a mapping of real values to quantized
+  integers.
+
+```ebnf
+QuantizedType ::= PerTensorQuantizedType | PerAxisQuantizedType
+PerTensorQuantizedType ::= 'quantized.uniform' '<' 
+                              StorageType ':' ExpressedType ':' MinVal ':' 
+                              MaxVal ':' Scale ':' ZeroPoint '>'
+PerAxisQuantizedType ::= 'quantized.uniform' '<'
+                            StorageType ':' ExpressedType ':' MinVal ':'
+                            MaxVal ':' Scales ':' ZeroPoints ':'
+                            QuantizationDimension '>'
+StorageType ::= 'i4' | 'u4' | 'i8' | 'u8' | 'i16' | 'u16' | 'i32' | 'u32'
+ExpressedType ::= 'bf16' | 'f32'
+MinVal ::= IntegerConstant
+MaxVal ::= IntegerConstant
+Scales ::= '[' Scale {',' Scale} ']'
+ZeroPoints ::= '[' ZeroPoint {',' ZeroPoint} ']'
+Scale ::= '?' | FloatConstant
+ZeroPoint ::= '?' | IntegerConstant
+QuantizationDimension ::= IntegerConstant
+```
+
+  * `storage_type` is the integer type used for storage.
+  * `expressed_type` is the data type the quantized type represents.
+  * `min_val` is the minimum integer value that the quantized type can take.
+  * `min_val` is the maximum integer value that the quantized type can take.
+  * `scale` is the multiplicative factor used to dequantize back to the
+    `expressed_type`.
+  * `zero_point` is the offset used to dequantize back to the `expressed_type`.
+  * `per_tensor_quantizatied_types` are quantized per tensor.
+  * `per_axis_quantizatied_types` are quantized per dimension per tensor for the
+    given `quantization_dimension`.
+
+  For a given tensor or dimension the quantized value (`Quantization`) is
+  computed as follows -
+
+```
+quantized_value = clamp(min_val, 
+                        round(expressed_value / scale) + zero_point,
+                        max_val)
+```
+
+  For a given tensor or dimension the expressed value (`DeQuantization`) is
+  computed as follows -
+
+```
+dequantized_value = (quantized_value − zero_point) * scale
+```
+
+  Given `alpha` and `beta` are maximum and minimum values (per-tensor or
+  per-axis) needed to be represented in `expressed_type`, the values of `scale`
+  and `zero_point` can be computed as follows -
+
+```
+scale = (alpha -  beta) / (max_val - min_val)
+zero_point = min_val - round(beta * scale)
+```
+
+
 
 ```ebnf
 FunctionType ::= '(' [ValueType {',' ValueType}] ')' '->' '(' [ValueType {',' ValueType}] ')'
@@ -418,13 +475,25 @@ have the following constraints:
 
 * (C1) `is_wellformed(literal[:], element_type(type))`.
 
+```ebnf
+QuantizedConstant ::= QuantizedLiteral ':' QuantizedType
+QuantizedLiteral ::= IntegerLiteral
+```
+
+**Quantized constants** represent quantized values via integer literals.
+Quantized constants have the following constraints:
+
+* (C1) `is_wellformed(literal, type)`, i.e. `literal` can be parsed as
+  a value of type `type`.
+
 ```ebnf
 TensorConstant ::= TensorLiteral ':' TensorType
 TensorLiteral  ::= 'dense' '<' (DenseLiteral | ElementLiteral) '>'
 DenseLiteral   ::= DenseDimension | DenseElements
 DenseDimension ::= '[' [DenseLiteral {',' DenseLiteral}] ']'
 DenseElements  ::= [ElementLiteral {',' ElementLiteral}]
-ElementLiteral ::= BooleanLiteral | IntegerLiteral | FloatLiteral | ComplexLiteral
+ElementLiteral ::= BooleanLiteral | IntegerLiteral | FloatLiteral |
+                   ComplexLiteral | QuantizedLiteral
 ```
 
 **Tensor constants** represent tensor values using nested lists specified via