Python Specific Functionality#

Alongside the support for builtin egglog functionality, egglog also provides functionality to more easily integrate with the Python ecosystem.

Retrieving Primitive Values#

If you have a egglog primitive, you can turn it into a Python object by using egraph.eval(...) method:

from __future__ import annotations

from egglog import *

egraph = EGraph()
assert egraph.eval(i64(1) + 20) == 21

Python Object Sort#

We define a custom “primitive sort” (i.e. a builtin type) for PyObjects. This allows us to store any Python object in the e-graph.

Saving Python Objects#

To create an expression of type PyObject, we call the call the constructor with any Python object. It will save a reference to the object:


We see that this as saved internally as a pointer to the Python object. For hashable objects like int we store two integers, a hash of the type and a has of the value.

We can also store unhashable objects in the e-graph like lists.

lst = PyObject([1, 2, 3])
PyObject([1, 2, 3])

We see that this is stored with one number, simply the id of the object.

Mutable Objects

While it is possible to store unhashable objects in the e-graph, you have to be careful defining any rules which create new unhashable objects. If each time a rule is run, it creates a new object, then the e-graph will never saturate.

Creating hashable objects is safer, since while the rule might create new Python objects each time it executes, they should have the same hash, i.e. be equal, so that the e-graph can saturate.

Retrieving Python Objects#

Like other primitives, we can retrieve the Python object from the e-graph by using the egraph.eval(...) method:

assert egraph.eval(lst) == [1, 2, 3]

Builtin methods#

Currently, we only support a few methods on PyObjects, but we plan to add more in the future.

Conversion to/from a string:


Conversion from an int:


We also support evaluating arbitrary Python code, given some locals and globals. This technically allows us to implement any Python method:

egraph.eval(py_eval("1 + 2"))

Executing Python code is also supported. In this case, the return value will be the updated globals dict, which will be copied first before using.

egraph.eval(py_exec("x = 1 + 2"))
{'x': 3}

Alongside this, we support a function dict_update method, which can allow you to combine some local egglog expressions alongside, say, the locals and globals of the Python code you are evaluating.

# Need this from our globals()
def my_add(a, b):
    return a + b

amended_globals = PyObject(globals()).dict_update("one", 1)
evalled = py_eval("my_add(one,  2)", locals(), amended_globals)
assert egraph.eval(evalled) == 3

Simpler Eval#

Instead of using the above low level primitive for evaluating, there is a higher level wrapper function, py_eval_fn.

It takes in a Python function and converts it to a function of PyObjects, by using py_eval under the hood.

The above code code be re-written like this:

def my_add(a, b):
    return a + b

evalled = py_eval_fn(lambda a: my_add(a, 2))(1)
assert egraph.eval(evalled) == 3


Type Promotion#

Similar to how an int can be automatically upcasted to an i64, we also support registering conversion to your custom types. For example:

class Math(Expr):
    def __init__(self, x: i64Like) -> None: ...

    def var(cls, name: StringLike) -> Math: ...

    def __add__(self, other: Math) -> Math: ...
    def __mul__(self, other: Math) -> Math: ...

converter(i64, Math, Math)
converter(String, Math, Math.var)

Math(2) + i64(30) + String("x")
# equal to
Math(2) + Math(i64(30)) + Math.var(String("x"))
(Math(2) + Math(30)) + Math.var("x")

You can also specify a “cost” for a conversion, which will be used to determine which conversion to use when multiple are possible. For example convert(i64, Math, 10).

Registering a conversion from A to B will also register all transitively reachable conversions from A to B, so you can also use:

Math(2) + 30 + "x"
(Math(2) + Math(30)) + Math.var("x")

If you want to have this work with the static type checker, you can define your own Union type, which MUST include the Expr class as the first item in the union. For example, in this case you could then define:

from typing import Union
MathLike = Union[Math, i64Like, StringLike]

def some_math_fn(x: MathLike) -> MathLike:


Keyword arguments#

All arguments for egg functions must be declared positional or keyword (the default argument type) currently. You can pass arguments variably or also as keyword arguments:

# egg: (function bar (i64 i64) i64)
def bar(a: i64Like, b: i64Like) -> i64:

# egg: (bar 1 2)
bar(1, 2)
bar(b=2, a=1)
bar(1, 2)

Default arguments#

Default argument values are also supported. They are not translated to egglog definition, which has no notion of optional values. Instead, they are added to args when the functions is called.

# egg: (function bar (i64 i64) i64)
def baz(a: i64Like, b: i64Like=i64(0)) -> i64:

# egg: (baz 1 0)


When defining a custom class, you are free to use any method names you like.

Builtin Methods#

Most of the Python special dunder (= “double under”) methods are supported as well:

  • __lt__

  • __le__

  • __eq__

  • __ne__

  • __ne__

  • __gt__

  • __ge__

  • __add__

  • __sub__

  • __mul__

  • __matmul__

  • __truediv__

  • __floordiv__

  • __mod__

  • __pow__

  • __lshift__

  • __rshift__

  • __and__

  • __xor__

  • __or__

  • __pos__

  • __neg__

  • __invert__

  • __getitem__

  • __call__

  • __setitem__

  • __delitem__

Currently __divmod__ is not supported, since it returns multiple results.

Also these methods are currently used in the runtime class and cannot be overridden currently, although we could change this if the need arises:

  • __getattr__

  • __repr__

  • __str__

  • _ipython_display_

  • __dir__

  • __getstate__

  • __setstate__

“Preserved” methods#

You can use the @method(preserve=True) decorator to mark a method as “preserved”, meaning that calling it will actually execute the body of the function and a corresponding egglog function will not be created,

Normally, all methods defined on a egglog Expr will ignore their bodies and simply build an expression object based on the arguments.

However, there are times in Python when you need the return type of a method to be an instance of a particular Python type, and some similar acting expression won’t cut it.

For example, let’s say you are implementing a Bool expression, but you want to be able to use it in if statements in Python. That means it needs to define a __bool__ methods which returns a Python bool, based on evaluating the expression.

class Boolean(Expr):
    def __bool__(self) -> bool:
        # Add this expression converted to a Python object to the e-graph
        # Run until the e-graph saturates
        # Extract the Python object from the e-graph
        return egraph.eval(self.to_py())

    def to_py(self) -> PyObject:

    def __or__(self, other: Boolean) -> Boolean:

TRUE = egraph.constant("TRUE", Boolean)
FALSE = egraph.constant("FALSE", Boolean)

def _bool(x: Boolean):
    return [
        rewrite(TRUE | x).to(TRUE),
        rewrite(FALSE | x).to(x),
/tmp/ipykernel_7132/ DeprecationWarning: Use top level constant function instead
  TRUE = egraph.constant("TRUE", Boolean)
/tmp/ipykernel_7132/ DeprecationWarning: Use top level constant function instead
  FALSE = egraph.constant("FALSE", Boolean)

Now whenever the __bool__ method is called, it will actually execute the body of the function, and return a Python bool based on the result.


Note that the following list of methods are only supported as “preserved” since they have to return a specific Python object type:

  • __bool__

  • __len__

  • __complex_

  • __int_

  • __float_

  • __hash_

  • __iter_

  • __index__

Reflected methods#

Note that reflected methods (i.e. __radd__) are handled as a special case. If defined, they won’t create their own egglog functions.

Instead, whenever a reflected method is called, we will try to find the corresponding non-reflected method and call that instead.

Also, if a normal method fails because the arguments cannot be converted to the right types, the reflected version of the second arg will be tried.

class Int(Expr):
    def __init__(self, i: i64Like) -> None:

    def var(cls, name: StringLike) -> Int:

    def __add__(self, other: Int) -> Int:

class Float(Expr):
    def __init__(self, i: f64Like) -> None:

    def var(cls, name: StringLike) -> Float:

    def from_int(cls, i: Int) -> Float:

    def __add__(self, other: Float) -> Float:

converter(f64, Float, Float)
converter(Int, Float, Float.from_int)

assert str(-1.0 + Int.var("x")) == "Float(-1.0) + Float.from_int(Int.var(\"x\"))"

For methods which allow returning NotImplemented, i.e. the comparison + binary math methods, we will also try upcasting both types to the type which is lowest cost to convert both to.

For example, if you have Float and Int wrapper types and you have write the expr -1.0 + Int.var("x") you might want the result to be Float(-1.0) + Float.from_int(Int.var("x")):

Mutating arguments#

In order to support Python functions and methods which mutate their arguments, you can pass in the mutate_first_arg keyword argument to the @function decorator and the mutates_self argument to the @method decorator. This will cause the first argument to be mutated in place, instead of being copied.

from copy import copy

class Int(Expr):
    def __init__(self, i: i64Like) -> None:

    def __add__(self, other: Int) -> Int:  # type: ignore[empty-body]

def incr(x: Int) -> None:

i = var("i", Int)
incr_i = copy(i)

x = Int(10)
mutate_egraph = EGraph()
mutate_egraph.register(rewrite(incr_i).to(i + Int(1)), x)
mutate_egraph.check(eq(x).to(Int(10) + Int(1)))
%3 outer_cluster_0 cluster_0 outer_cluster_1 cluster_1 outer_cluster_2 cluster_2 outer_cluster_i64-3377577844511369682 cluster_i64-3377577844511369682 outer_cluster_i64-5871781006564002453 cluster_i64-5871781006564002453 Int___init__-3377577844511369682:s->i64-3377577844511369682 Int___add__-11743562013128004906:s->Int___init__-3377577844511369682 Int___add__-11743562013128004906:s->Int___init__-5871781006564002453 Int___init__-5871781006564002453:s->i64-5871781006564002453 incr-0:s->Int___init__-3377577844511369682 Int___init__-3377577844511369682 Int i64-3377577844511369682 10 Int___add__-11743562013128004906 · + · Int___init__-5871781006564002453 Int incr-0 incr i64-5871781006564002453 1

Any function which mutates its first argument must return None. In egglog, this is translated into a function which returns the type of its first argument.

Note that dunder methods such as __setitem__ will automatically be marked as mutating their first argument.

Functions as Values#

In Python, functions are first class objects, and can be passed around as values. You can use the builtin Callable type annotation to specify that a function is expected as an argument. You can then pass egglog functions directly and call them with rewrite rules. For example, here is how you could define a MathList class which supports mapping:

from import Callable
from typing import ClassVar

class MathList(Expr):
    EMPTY: ClassVar[MathList]

    def append(self, x: Math) -> MathList: ...

    def map(self, fn: Callable[[Math], Math]) -> MathList: ...

def math_list_ruleset(xs: MathList, x: Math, f: Callable[[Math], Math]):
    yield rewrite(
    yield rewrite(xs.append(x).map(f)).to(

To support partial application, you can use the builtin functools.partial function:

from functools import partial

x = MathList.EMPTY.append(Math(1))
added_two =, Math(2)))

check_eq(added_two, MathList.EMPTY.append(Math(2) + Math(1)), math_list_ruleset.saturate())
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
%3 outer_cluster_7 cluster_7 outer_cluster_2 cluster_2 outer_cluster_4 cluster_4 outer_cluster_9 cluster_9 outer_cluster_0 cluster_0 outer_cluster_3 cluster_3 outer_cluster_UnstableFn_Math_Math-0 cluster_UnstableFn_Math_Math-0 outer_cluster_i64-5871781006564002453 cluster_i64-5871781006564002453 outer_cluster_i64-11743562013128004906 cluster_i64-11743562013128004906 Math___add__-12139158412232607314:s->Math___init__-11743562013128004906 Math___add__-12139158412232607314:s->Math___init__-5871781006564002453 Math___init__-11743562013128004906:s->i64-11743562013128004906 Math___init__-5871781006564002453:s->i64-5871781006564002453 MathList_map-7619260772787498455:s->MathList_map-7619260772787498455 MathList_map-7619260772787498455:s->UnstableFn_Math_Math-0 UnstableFn_Math_Math-0:s->Math___init__-11743562013128004906 MathList_map-2868860904042873558:s->UnstableFn_Math_Math-0 MathList_map-2868860904042873558:s->MathList_append-14322442833369045165 MathList_append-14322442833369045165:s->Math___init__-5871781006564002453 MathList_append-14322442833369045165:s->MathList_map-7619260772787498455 MathList_append-13491041779351500908:s->Math___add__-12139158412232607314 MathList_append-13491041779351500908:s->MathList_EMPTY-0 Math___add__-12139158412232607314 · + · Math___init__-11743562013128004906 Math Math___init__-5871781006564002453 Math i64-5871781006564002453 1 i64-11743562013128004906 2 MathList_map-7619260772787498455 ·.map UnstableFn_Math_Math-0 · + · MathList_EMPTY-0 MathList.EMPTY MathList_map-2868860904042873558 ·.map MathList_append-14322442833369045165 ·.append MathList_append-13491041779351500908 ·.append

Note that this is all built on the unstable function support added as a sort to egglog. While this sort is exposed directly at the high level with the UnstableFn class, we don’t reccomend depending on it directly, and instead using the builtin Python type annotations. This will allow us to change the implementation in the future without breaking user code.

Unwrapped functions#

We also support using normal python functions, either named or anonymous, as values. These will automatically be wrapped as egglog functions when passed to a function which expects an egglog function.

x = MathList.EMPTY.append(Math(1))
added_two = x: x + Math(2))
check_eq(added_two, MathList.EMPTY.append(Math(1) + Math(2)), (math_list_ruleset + run()) * 10)
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
%3 outer_cluster_2 cluster_2 outer_cluster_6 cluster_6 outer_cluster_10 cluster_10 outer_cluster_9 cluster_9 outer_cluster_0 cluster_0 outer_cluster_3 cluster_3 outer_cluster_UnstableFn_Math_Math-0 cluster_UnstableFn_Math_Math-0 outer_cluster_i64-11743562013128004906 cluster_i64-11743562013128004906 outer_cluster_i64-5871781006564002453 cluster_i64-5871781006564002453 Math___init__-5871781006564002453:s->i64-5871781006564002453 Math___init__-11743562013128004906:s->i64-11743562013128004906 lambda x: x + Math(2)-11743562013128004906:s->Math___init__-5871781006564002453 Math___add__-8615755996610129082:s->Math___init__-5871781006564002453 Math___add__-8615755996610129082:s->Math___init__-11743562013128004906 MathList_map-7619260772787498455:s->MathList_map-7619260772787498455 MathList_map-7619260772787498455:s->UnstableFn_Math_Math-0 MathList_map-2868860904042873558:s->UnstableFn_Math_Math-0 MathList_map-2868860904042873558:s->MathList_append-14322442833369045165 MathList_append-14322442833369045165:s->Math___init__-5871781006564002453 MathList_append-14322442833369045165:s->MathList_map-7619260772787498455 MathList_append-4241682928276128773:s->Math___add__-8615755996610129082 MathList_append-4241682928276128773:s->MathList_EMPTY-0 Math___init__-5871781006564002453 Math i64-5871781006564002453 1 Math___init__-11743562013128004906 Math i64-11743562013128004906 2 lambda x: x + Math(2)-11743562013128004906 lambda x: x + Math(2) Math___add__-8615755996610129082 · + · MathList_map-7619260772787498455 ·.map UnstableFn_Math_Math-0 lambda x: x + Math(2) MathList_EMPTY-0 MathList.EMPTY MathList_map-2868860904042873558 ·.map MathList_append-14322442833369045165 ·.append MathList_append-4241682928276128773 ·.append

Their definition will be added to the default rulset, unless they are defined in the body of a function themselves or in a rule function:

def map_add_two(x: MathList) -> MathList:
    return x: x + Math(2))

check_eq(map_add_two(MathList.EMPTY.append(Math(1))), MathList.EMPTY.append(Math(1) + Math(2)), math_list_ruleset.saturate())
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
UnstableFn_Math_Math is a container sort
%3 outer_cluster_11 cluster_11 outer_cluster_2 cluster_2 outer_cluster_6 cluster_6 outer_cluster_3 cluster_3 outer_cluster_0 cluster_0 outer_cluster_10 cluster_10 outer_cluster_UnstableFn_Math_Math-0 cluster_UnstableFn_Math_Math-0 outer_cluster_i64-11743562013128004906 cluster_i64-11743562013128004906 outer_cluster_i64-5871781006564002453 cluster_i64-5871781006564002453 lambda x: x + Math(2)-11743562013128004906:s->Math___init__-5871781006564002453 Math___init__-5871781006564002453:s->i64-5871781006564002453 Math___add__-8615755996610129082:s->Math___init__-5871781006564002453 Math___add__-8615755996610129082:s->Math___init__-11743562013128004906 Math___init__-11743562013128004906:s->i64-11743562013128004906 MathList_append-16889558280977232211:s->Math___init__-5871781006564002453 MathList_append-16889558280977232211:s->MathList_EMPTY-0 MathList_map-5145996267849227305:s->MathList_map-5145996267849227305 MathList_map-5145996267849227305:s->UnstableFn_Math_Math-0 map_add_two-17615343019692007359:s->MathList_append-16889558280977232211 MathList_append-2651793105796594534:s->Math___add__-8615755996610129082 MathList_append-2651793105796594534:s->MathList_map-5145996267849227305 MathList_map-2868860904042873558:s->MathList_append-16889558280977232211 MathList_map-2868860904042873558:s->UnstableFn_Math_Math-0 lambda x: x + Math(2)-11743562013128004906 lambda x: x + Math(2) Math___init__-5871781006564002453 Math Math___add__-8615755996610129082 · + · Math___init__-11743562013128004906 Math i64-5871781006564002453 1 i64-11743562013128004906 2 MathList_append-16889558280977232211 ·.append MathList_EMPTY-0 MathList.EMPTY MathList_map-5145996267849227305 ·.map UnstableFn_Math_Math-0 lambda x: x + Math(2) map_add_two-17615343019692007359 map_add_two MathList_append-2651793105796594534 ·.append MathList_map-2868860904042873558 ·.map

Their name will just be the body of the function, so that two anonymous functions with the same body will be considered equal.

MathList.EMPTY.append(Math(1)).map(lambda x: x + Math(2))

Default Replacements#

When defining a function or a constant, you can also provide a default replacement value. This is useful when you might want both the original value and the replaced value in the e-graph, so that later rules could reference either.

def math_float(f: f64Like) -> Math:

# Can add a default replacement value for a constants
pi = constant("pi", Math, math_float(3.14))

# or for a function by providing a body
def square(x: Math) -> Math:
    return x * x

# thse rewrites will be added to the e-graph under the default ruleset
egraph = EGraph()
egraph.check(eq(square(Math.var('x'))).to(Math.var('x') * Math.var('x')))
%3 outer_cluster_2 cluster_2 outer_cluster_1 cluster_1 outer_cluster_0 cluster_0 outer_cluster_String-17097879294545007654 cluster_String-17097879294545007654 outer_cluster_f64-3617621074547945739 cluster_f64-3617621074547945739 square-5871781006564002453:s->Math_var-17097879294545007654 Math_var-17097879294545007654:s->String-17097879294545007654 Math___mul__-6828067974578293639:s->Math_var-17097879294545007654 Math___mul__-6828067974578293639:s->Math_var-17097879294545007654 math_float-3617621074547945739:s->f64-3617621074547945739 square-5871781006564002453 square Math_var-17097879294545007654 Math.var Math___mul__-6828067974578293639 · * · String-17097879294545007654 "x" math_float-3617621074547945739 math_float f64-3617621074547945739 3.14 pi-0 pi

This is equivalent to adding the rewrite rules to the e-graph directly, like this, but just more succinct:

x  = var("x", Math)
egraph.register(rewrite(square(x)).to(x * x))

You can also specify a ruleset to add the rewrites to, by passing in the ruleset keyword argument:

math_ruleset = ruleset()

e_constant = constant("e", Math, math_float(2.71), ruleset=math_ruleset)

def cube(x: Math) -> Math:
    return x * x * x

egraph.check(eq(cube(Math.var('x'))).to(Math.var('x') * Math.var('x') * Math.var('x')))

Default Replacement for Classes#

In classes, you can also provide a default replacement value for constants and methods, and an optional ruleset on the class constructor:

other_math_ruleset = ruleset()

class WrappedMath(Expr, ruleset=other_math_ruleset):
    PI: ClassVar[Math] = math_float(3.14)

    def __init__(self, x: Math) -> None: ...

    def double(self) -> WrappedMath:
        return self + self

    def __add__(self, other: WrappedMath) -> WrappedMath: ...

x = WrappedMath(WrappedMath.PI).double()
egraph = EGraph()
egraph.register(x) * 2)
egraph.check(eq(x).to(WrappedMath(math_float(3.14)) + WrappedMath(math_float(3.14))))
%3 outer_cluster_0 cluster_0 outer_cluster_1 cluster_1 outer_cluster_2 cluster_2 outer_cluster_f64-3617621074547945739 cluster_f64-3617621074547945739 math_float-3617621074547945739:s->f64-3617621074547945739 WrappedMath___init__-0:s->math_float-3617621074547945739 WrappedMath_double-5871781006564002453:s->WrappedMath___init__-0 WrappedMath___add__-6828067974578293639:s->WrappedMath___init__-0 WrappedMath___add__-6828067974578293639:s->WrappedMath___init__-0 math_float-3617621074547945739 math_float f64-3617621074547945739 3.14 WrappedMath_PI-0 WrappedMath.PI WrappedMath___init__-0 WrappedMath WrappedMath_double-5871781006564002453 ·.double WrappedMath___add__-6828067974578293639 · + ·