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LoopNest, Shape, ndim inference
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LoopNest and Shape Dialects and rules.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "6ac152c9",
"metadata": {},
"outputs": [],
"source": [
"from __future__ import annotations"
]
},
{
"cell_type": "markdown",
"id": "f766d37c",
"metadata": {},
"source": [
"## Define LoopNestAPI and ShapeAPI as Dialects"
]
},
{
"cell_type": "markdown",
"id": "68080f3b",
"metadata": {},
"source": [
"Pretend that Numba can define Pythonic API that acts like _dialects_/_DSL_. \n",
"We define `LoopNestAPI` and `ShapeAPI` _dialects_. These dialects are just like a normal Python class but will allow Numba to capture their high-level semantic in the compiler for analysis and transformation."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "f661a8bb",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.\n",
"Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.\n"
]
}
],
"source": [
"from __future__ import annotations\n",
"\n",
"import numpy as np\n",
"import itertools\n",
"\n",
"\n",
"class LoopNestAPI:\n",
" dim: tuple[int, ...]\n",
" inner: LoopNestAPI | None\n",
"\n",
" def __init__(self, dim, inner=None):\n",
" self.dim = dim\n",
" self.inner = inner\n",
"\n",
" @staticmethod\n",
" def from_tuple(args):\n",
" def expand(*args):\n",
" if not args:\n",
" return None\n",
" [head, *tail] = args\n",
" return LoopNestAPI(head, inner=expand(*tail))\n",
" return expand(*args)\n",
"\n",
" def __iter__(self):\n",
" dims = self.get_dims()\n",
" return iter(itertools.product(*map(lambda x: list(range(x)), dims)))\n",
"\n",
" def get_dims(self):\n",
" if self.inner is None:\n",
" return [self.dim]\n",
" else:\n",
" return [self.dim] + self.inner.get_dims()\n",
"\n",
" def reduce(self, fn, init):\n",
" res = init\n",
" for idx in self:\n",
" res = fn(res, idx)\n",
" return res\n",
"\n",
"\n",
"class ShapeAPI:\n",
" dims: tuple[int,...]\n",
"\n",
" def __init__(self, dims):\n",
" self.dims = tuple(dims)\n",
"\n",
" def deselect(self, axis):\n",
" dims = self.dims\n",
" return ShapeAPI([dims[i] for i in range(len(dims)) if i not in axis])\n",
"\n",
" def select(self, axis):\n",
" dims = self.dims\n",
" return ShapeAPI([dims[i] for i in range(len(dims)) if i in axis])\n",
"\n",
" def to_tuple(self):\n",
" return tuple(self.dims)"
]
},
{
"cell_type": "markdown",
"id": "a5cd52a8",
"metadata": {},
"source": [
"## Define the `np.linalg.norm` impl with the Dialects\n",
"\n",
"We will use a `np.linalge.norm` implementation as demo."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "cc6ba21f",
"metadata": {},
"outputs": [],
"source": [
"def linalg_norm_loopnest(X, axis):\n",
" # # If X ndim>=3 and axis is a 2-tuple\n",
" assert X.ndim >= 3\n",
" assert len(axis) == 2\n",
"\n",
" # peel off the outer shape for result array\n",
" outshape = ShapeAPI(X.shape).deselect(axis).to_tuple()\n",
" # get only the inner shape for reduction\n",
" reduce_axis = ShapeAPI(X.shape).select(axis).to_tuple()\n",
"\n",
" # alloc result array\n",
" res = np.zeros(outshape, dtype=X.dtype)\n",
"\n",
" # loop over the result array indices\n",
" for k in LoopNestAPI.from_tuple(outshape):\n",
" # reduction\n",
" def inner(carry, i):\n",
" idx = (*i, *k)\n",
" x = X[idx]\n",
" return carry + (x.conj() * x).real\n",
"\n",
" tmp = LoopNestAPI.from_tuple(reduce_axis).reduce(inner, init=0.0)\n",
" res[k] = np.sqrt(tmp)\n",
"\n",
" return res"
]
},
{
"cell_type": "markdown",
"id": "60a82062",
"metadata": {},
"source": [
"## Define a low-level impl\n",
"\n",
"The following is a way to implement it in the low-level that current Numba requires."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "670ad3fb",
"metadata": {},
"outputs": [],
"source": [
"def linalg_norm_low_level(X, axis):\n",
" # # If X ndim>=3 and axis is a 2-tuple\n",
" assert X.ndim >= 3\n",
" assert len(axis) == 2\n",
" outdim = [dim for dim in range(X.ndim) if dim not in axis]\n",
" outshape = tuple(np.asarray(X.shape)[outdim])\n",
"\n",
" res = np.zeros(outshape, dtype=X.dtype)\n",
" row_axis, col_axis = axis\n",
" for k in np.ndindex(outshape):\n",
" tmp = 0.0\n",
" for row in range(X.shape[row_axis]):\n",
" for col in range(X.shape[col_axis]):\n",
" idx = (row, col, *k)\n",
" x = X[idx]\n",
" tmp += (x.conj() * x).real\n",
" res[k] = np.sqrt(tmp)\n",
" return res"
]
},
{
"cell_type": "markdown",
"id": "f54bbd08",
"metadata": {},
"source": [
"## Test\n",
"\n",
"Just making sure the two implementation are working"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "d97edf57",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"=========================checking linalg_norm_low_level=========================\n",
"(3, 4) (3, 4)\n",
"=========================checking linalg_norm_loopnest==========================\n",
"(3, 4) (3, 4)\n"
]
}
],
"source": [
"np.random.seed(23)\n",
"X = np.random.random((3, 2, 3, 4))\n",
"axis = (0, 1)\n",
"expect = np.linalg.norm(X, axis=axis)\n",
"impl_list = [\n",
" linalg_norm_low_level,\n",
" linalg_norm_loopnest,\n",
"]\n",
"for impl in impl_list:\n",
" print(f'checking {impl.__name__}'.center(80, '='))\n",
" res = impl(X, axis=axis)\n",
" print(expect.shape, res.shape)\n",
" np.testing.assert_almost_equal(res, expect)"
]
},
{
"cell_type": "markdown",
"id": "7e4fe1e5",
"metadata": {},
"source": [
"# Propagate ndim using EGG"
]
},
{
"cell_type": "markdown",
"id": "756d00b2",
"metadata": {},
"source": [
"This is modelling the `ShapeAPI` dialect. `LoopNestAPI` needs to know the size of the input tuple to work correctly.\n",
"\n",
"In practice, we want users to define these rules so that any dialect can easily define custom inference logic."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "c01c60c7",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The result array will have nd = 1\n",
"The inner reduction loop will have nd = 2\n"
]
}
],
"source": [
"\n",
"from egglog import EGraph, Expr, String, i64, i64Like, var, rewrite, set_, rule, union, eq, StringLike, vars_, run\n",
"from typing import Union\n",
"\n",
"\n",
"egraph = EGraph()\n",
"\n",
"\n",
"# The following EGraph.class_ are like IR nodes but they are available for pattern matching\n",
"\n",
"@egraph.class_\n",
"class Tuple(Expr):\n",
" def __init__(self, nd: i64Like):\n",
" ...\n",
" \n",
" @classmethod\n",
" def symbolic(cls, name: StringLike) -> Tuple:\n",
" ...\n",
" \n",
"\n",
"@egraph.class_\n",
"class Shape(Expr):\n",
" def __init__(self, tup: Tuple):\n",
" ...\n",
" \n",
" def deselect(self, axis: Tuple) -> Shape:\n",
" ...\n",
" \n",
" def select(self, axis: Tuple) -> Shape:\n",
" ...\n",
"\n",
" def to_tuple(self) -> Tuple:\n",
" ...\n",
"\n",
" \n",
"# The following EGraph.functions are for inferring the ndim.\n",
" \n",
"@egraph.function\n",
"def ndim(x: Shape) -> i64:\n",
" ...\n",
" \n",
"@egraph.function\n",
"def length(x: Tuple) -> i64:\n",
" ...\n",
"\n",
"\n",
"# Below, we define the logic rules for the inference\n",
"[shape, shape1] = vars_(\"shape, shape1\", Shape)\n",
"[tup] = vars_(\"tup\", Tuple)\n",
"[n, p, q] = vars_(\"n, p, q\", i64)\n",
"egraph.register(\n",
" \n",
" # -- rule for Tuple length\n",
" rule( Tuple(n) ).then( set_( length(Tuple(n)) ).to( n ) ),\n",
" \n",
" # -- rule for Shape ctor\n",
" # Given p == length(tup)\n",
" # shape == Shape(tup)\n",
" # Then ndim(shape) = p\n",
" rule( \n",
" eq(p).to(length(tup)),\n",
" eq(shape).to(Shape(tup))\n",
" ).then( set_( ndim(shape) ).to( p ) ),\n",
" \n",
" # -- rule for Shape.deselect\n",
" # Given p == ndim(shape)\n",
" # q == length(tup)\n",
" # shape1 == shape.deselect(tup)\n",
" # Then ndim(shape1) = (p - q)\n",
" rule( \n",
" eq(p).to(ndim(shape)),\n",
" eq(q).to(length(tup)),\n",
" eq(shape1).to(shape.deselect(tup))\n",
" ).then( set_( ndim(shape1) ).to( p - q ) ),\n",
" \n",
" # -- rule for Shape.select\n",
" # Given q == length(tup)\n",
" # shape1 == shape.select(tup)\n",
" # Then ndim(shape1) = q\n",
" rule(\n",
" eq(q).to(length(tup)),\n",
" eq(shape1).to(shape.select(tup))\n",
" ).then( set_( ndim(shape1) ).to( q ) ),\n",
" \n",
" # -- rule for Shape.to_tuple\n",
" # Given p == ndim(shape)\n",
" # tup == shape.to_tuple()\n",
" # Then length(tup) = p\n",
" rule(\n",
" eq(p).to(ndim(shape)),\n",
" eq(tup).to(shape.to_tuple())\n",
" ).then( set_( length(tup) ).to( p ) ),\n",
" \n",
")\n",
"\n",
"# Here, we setup facts for the linalg.norm function call\n",
"\n",
"# Facts about the arguments\n",
"X_shape = egraph.let(\"X_shape\", Tuple(3))\n",
"axis = egraph.let(\"axis\", Tuple(2))\n",
"\n",
"# Expressions to evaluate\n",
"deselect_expr = Shape(X_shape).deselect(axis).to_tuple()\n",
"select_expr = Shape(X_shape).select(Tuple(2)).to_tuple()\n",
"egraph.let(\"y\", deselect_expr )\n",
"egraph.let(\"z\", select_expr )\n",
"\n",
"# Run\n",
"egraph.run(5)\n",
"\n",
"# Extract\n",
"print(\"The result array will have nd =\", egraph.extract(length(deselect_expr)))\n",
"print(\"The inner reduction loop will have nd =\", egraph.extract(length(select_expr)))"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "f722cd96",
"metadata": {},
"outputs": [
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@saulshanabrook
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saulshanabrook commented Feb 8, 2024 •
edited
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I am coming back to this after finishing some refactoring work on egglog and wanted to check to see if I am understanding things properly before trying to implement the features needed for this.

Thank you again for posting this example! I am excited about getting this sort of thing to work and it's essential to have real use cases like this to drive the development...

Let's say we implement an API like you proposed on top of the existing array api bindings:

from typing import Callable, ClassVar, Iterator

import numpy as np

import egglog.exp.array_api as enp
from egglog import Expr, convert, rewrite, ruleset


class ShapeAPI(Expr):
    def __init__(self, dims: enp.TupleInt) -> None:
        ...

    def deselect(self, axis: enp.TupleInt) -> ShapeAPI:
        ...

    def select(self, axis: enp.TupleInt) -> ShapeAPI:
        ...

    def to_tuple(self) -> enp.TupleInt:
        ...

class OptionalLoopNestAPI(Expr):
    def __init__(self, value: LoopNestAPI) -> None:
        ...

    NONE: ClassVar[OptionalLoopNestAPI]

    def unwrap(self) -> LoopNestAPI:
        ...


class LoopNestAPI(Expr):
    def __init__(self, dim: enp.Int, inner: OptionalLoopNestAPI = OptionalLoopNestAPI.NONE) -> None:
        ...

    @classmethod
    def from_tuple(cls, args: enp.TupleInt) -> OptionalLoopNestAPI:
        ...

    def __iter__(self) -> Iterator[enp.TupleInt]:
        return iter(self.indices)

    @property
    def indices(self) -> enp.TupleTupleInt:
        ...

    def get_dims(self) -> enp.TupleInt:
        ...

    def reduce(self, fn: Callable[[enp.NDArray, enp.TupleInt], enp.NDArray], init: enp.NDArray) -> enp.NDArray:
        ...

With that, a user could write a loopnest function with it:

def linalg_norm_loopnest_egglog(X: enp.NDArray, axis: enp.TupleInt) -> enp.NDArray:
    # peel off the outer shape for result array
    outshape = ShapeAPI(X.shape).deselect(axis).to_tuple()
    # get only the inner shape for reduction
    reduce_axis = ShapeAPI(X.shape).select(axis).to_tuple()

    return enp.NDArray.from_fn(
        outshape,
        X.dtype,
        lambda k: enp.sqrt(
            LoopNestAPI.from_tuple(reduce_axis)
            .unwrap()
            .reduce(lambda carry, i: carry + enp.real(enp.conj(x := X[i + k]) * x), init=0.0)
        ).to_value(),
    )

By applying some rewrite rules (included below in a dropdown), we could translate away the custom APIs and end up with something like this expression:

def linalg_norm_array_api(X: enp.NDArray, axis: enp.TupleInt) -> enp.NDArray:
    outdim = enp.range_(X.ndim).filter(lambda x: ~axis.contains(x))
    outshape = convert(convert(X.shape, enp.NDArray)[outdim], enp.TupleInt)
    row_axis, col_axis = axis
    return enp.NDArray.from_fn(
        outshape,
        X.dtype,
        lambda k: enp.sqrt(
            enp.int_product(enp.range_(X.shape[row_axis]), enp.range_(X.shape[col_axis]))
            .map_to_ndarray(lambda rc: enp.real(enp.conj(x := X[rc + k]) * x))
            .sum()
        ).to_value(),
    )

Then, using the existing source generation, we could create this Python source from that function:

def linalg_norm_low_level(
    X: np.ndarray[tuple, np.dtype[np.float64]], axis: tuple[int, int]
) -> np.ndarray[tuple, np.dtype[np.float64]]:
    # # If X ndim>=3 and axis is a 2-tuple
    assert X.ndim >= 3
    assert len(axis) == 2
    #  Y - 2
    outdim = [dim for dim in range(X.ndim) if dim not in axis]

    outshape = tuple(np.asarray(X.shape)[outdim])

    res = np.zeros(outshape, dtype=X.dtype)
    row_axis, col_axis = axis
    for k in np.ndindex(outshape):
        tmp = 0.0
        for row in range(X.shape[row_axis]):
            for col in range(X.shape[col_axis]):
                idx = (row, col, *k)
                x = X[idx]
                tmp += (x.conj() * x).real
        res[k] = np.sqrt(tmp)
    return res

Does that seem about right?

Implementation

I haven't tried any of this code, and to implement it would require adding functions as values to egglog. I believe this could be done without any (or minimal) changes to the core, but before starting on it, I wanted to check in to see if this use case is accurate enough to give it a go.

Also implementing generic functions and types would go a long way in making the code a bit more fluid, but in the examples I put here, I assume we don't have that feature yet. It should all still be possible without it, just more repetition.

I believe I could add them as a Python pre-processor and this use case would also be a good model to show how they help with readability.

Rewrite rules: 
@ruleset
def shape_api_ruleset(dims: enp.TupleInt, axis: enp.TupleInt):  # noqa: ANN201
    yield rewrite(ShapeAPI(dims).deselect(axis)).to(
        ShapeAPI(enp.range_(dims.length()).filter(lambda i: ~axis.contains(i)).map(lambda i: dims[i]))
    )
    yield rewrite(ShapeAPI(dims).select(axis)).to(
        ShapeAPI(enp.range_(dims.length()).filter(lambda i: axis.contains(i)).map(lambda i: dims[i]))
    )
    yield rewrite(ShapeAPI(dims).to_tuple()).to(dims)


@ruleset
def loopnest_api_ruleset(
    head: enp.Int,
    tail: enp.TupleInt,
    lna: LoopNestAPI,
    fn: Callable[[enp.NDArray, enp.TupleInt], enp.NDArray],
    init: enp.NDArray,
    dim: enp.Int,
):
    # from_tuple
    yield rewrite(LoopNestAPI.from_tuple(enp.TupleInt.EMPTY)).to(OptionalLoopNestAPI.NONE)
    yield rewrite(
        LoopNestAPI.from_tuple(enp.TupleInt.some(head, tail)),
    ).to(
        OptionalLoopNestAPI(LoopNestAPI(head, LoopNestAPI.from_tuple(tail))),
    )
    # reduce
    yield rewrite(lna.reduce(fn, init)).to(lna.indices.reduce_ndarray(fn, init))
    # get_dims
    yield rewrite(LoopNestAPI(dim, OptionalLoopNestAPI.NONE).get_dims()).to(enp.TupleInt(dim))
    yield rewrite(LoopNestAPI(dim, OptionalLoopNestAPI(lna)).get_dims()).to(enp.TupleInt(dim) + lna.get_dims())
    # indices
    yield rewrite(lna.indices).to(lna.get_dims().map_tuple_int(enp.range_).product())

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