[shape_poly] Add documentation for workaround with dimension parameters.

docs/export/shape_poly.md

@@ -475,10 +475,82 @@ is unsound.
 
 ### Dimension variables must be solvable from the input shapes
 
-When an exported object is invoked on inputs with concrete shapes
-the dimension variables are derived from the input shapes.
-This works only if the symbolic dimensions in the input shapes are linear.
-See below an example of a failure:
+Currently, the only way to pass the values of dimension variables
+when an exported object is invoked is indirectly through the shapes
+of the array arguments. E.g., the value of `b` can be inferred at the
+call site from the shape of the first argument of type `f32[b]`.
+This works well for most use cases, and
+it mirrors the calling convention of JIT functions.
+
+Sometimes you may want to export a function parameterized
+by an integer values that determines some shapes in the program.
+For example, we may
+want to export the function `my_top_k` defined below,
+parameterized by the
+value of `k`, which determined the shape of the result.
+The following attempt will lead to an error since the dimension
+variable `k` cannot be derived from the shape of the input `x: i32[4, 10]`:
+
+```python
+>>> def my_top_k(k, x):  # x: i32[4, 10], k <= 10
+...   return lax.top_k(x, k)[0]  # : i32[4, 3]
+>>> x = np.arange(40, dtype=np.int32).reshape((4, 10))
+
+>>> # Export with static `k=3`. Since `k` appears in shapes it must be in `static_argnums`.
+>>> exp_static_k = export.export(jax.jit(my_top_k, static_argnums=0))(3, x)
+>>> exp_static_k.in_avals[0]
+ShapedArray(int32[4,10])
+
+>>> exp_static_k.out_avals[0]
+ShapedArray(int32[4,3])
+
+>>> # When calling the exported function we pass only the non-static arguments
+>>> exp_static_k.call(x)
+Array([[ 9,  8,  7],
+       [19, 18, 17],
+       [29, 28, 27],
+       [39, 38, 37]], dtype=int32)
+
+>>> # Now attempt to export with symbolic `k` so that we choose `k` after export.
+>>> k, = export.symbolic_shape("k", constraints=["k <= 10"])
+>>> export.export(jax.jit(my_top_k, static_argnums=0))(k, x)  # doctest: +IGNORE_EXCEPTION_DETAIL
+Traceback (most recent call last):
+KeyError: "Encountered dimension variable 'k' that is not appearing in the shapes of the function arguments
+
+```
+
+In the future, we may add an additional mechanism to pass the values of
+dimension variables, besides implicitly through the input shapes.
+Meanwhile, the workaround for the above use case is to replace the
+function parameter `k` with an array of shape `(0, k)`, so that
+`k` can be derived from the input shape of an array.
+The first dimension is 0 to ensure that the whole array is empty
+and there is no performance penalty when we call the exported function.
+
+```python
+>>> def my_top_k_with_dimensions(dimensions, x):  # dimensions: i32[0, k], x: i32[4, 10]
+...   return my_top_k(dimensions.shape[1], x)
+>>> exp = export.export(jax.jit(my_top_k_with_dimensions))(
+...     jax.ShapeDtypeStruct((0, k), dtype=np.int32),
+...     x)
+>>> exp.in_avals
+(ShapedArray(int32[0,k]), ShapedArray(int32[4,10]))
+
+>>> exp.out_avals[0]
+ShapedArray(int32[4,k])
+
+>>> # When we invoke `exp` we must construct and pass an array of shape (0, k)
+>>> exp.call(np.zeros((0, 3), dtype=np.int32), x)
+Array([[ 9,  8,  7],
+       [19, 18, 17],
+       [29, 28, 27],
+       [39, 38, 37]], dtype=int32)
+
+```
+
+Another situation when you may get an error is when some dimension
+variables do appear in the input shapes, but in a non-linear
+expression that JAX cannot currently solve:
 
 ```python
 >>> a, = export.symbolic_shape("a")
@@ -537,7 +609,6 @@ In particular, JAX will handle the cases when either (a) there
 is no remainder, or (b) the divisor is a constant
 in which case there may be a constant remainder.
 
-
 For example, the code below results in a division error when trying to
 compute the inferred dimension for a `reshape` operation:
 

jax/_src/export/shape_poly.py

@@ -215,7 +215,7 @@ class _DimFactor:
         return env[self.var]
       except KeyError:
         err_msg = (
-            f"Encountered dimension variable '{self.var}' that is not appearing in the shapes of the used function arguments.\n"
+            f"Encountered dimension variable '{self.var}' that is not appearing in the shapes of the function arguments.\n"
             "Please see https://jax.readthedocs.io/en/latest/export/shape_poly.html#dimension-variables-must-be-solvable-from-the-input-shapes for more details.")
         raise KeyError(err_msg)
     else: