PEP 646 – Variadic Generics
- Author:
- Mark Mendoza <mendoza.mark.a at gmail.com>, Matthew Rahtz <mrahtz at google.com>, Pradeep Kumar Srinivasan <gohanpra at gmail.com>, Vincent Siles <vsiles at fb.com>
- Sponsor:
- Guido van Rossum <guido at python.org>
- Status:
- Final
- Type:
- Standards Track
- Topic:
- Typing
- Created:
- 16-Sep-2020
- Python-Version:
- 3.11
- Post-History:
- 07-Oct-2020, 23-Dec-2020, 29-Dec-2020
- Resolution:
- Python-Dev message
Table of Contents
- Abstract
- Acceptance
- Motivation
- Summary Examples
- Specification
- Type Variable Tuples
- Using Type Variable Tuples in Generic Classes
- Using Type Variable Tuples in Functions
- Type Variable Tuples Must Always be Unpacked
Unpack
for Backwards Compatibility- Variance, Type Constraints and Type Bounds: Not (Yet) Supported
- Type Variable Tuple Equality
- Multiple Type Variable Tuples: Not Allowed
- Type Concatenation
- Unpacking Tuple Types
*args
as a Type Variable Tuple- Type Variable Tuples with
Callable
- Behaviour when Type Parameters are not Specified
- Aliases
- Substitution in Aliases
- Overloads for Accessing Individual Types
- Type Variable Tuples
- Rationale and Rejected Ideas
- Alternatives
- Grammar Changes
- Backwards Compatibility
- Reference Implementation
- Appendix A: Shape Typing Use Cases
- Appendix B: Shaped Types vs Named Axes
- Footnotes
- Endorsements
- Acknowledgements
- Resources
- References
- Copyright
Abstract
PEP 484 introduced TypeVar
, enabling creation of generics parameterised
with a single type. In this PEP, we introduce TypeVarTuple
, enabling parameterisation
with an arbitrary number of types - that is, a variadic type variable,
enabling variadic generics. This enables a wide variety of use cases.
In particular, it allows the type of array-like structures
in numerical computing libraries such as NumPy and TensorFlow to be
parameterised with the array shape, enabling static type checkers
to catch shape-related bugs in code that uses these libraries.
Acceptance
This PEP was accepted for Python 3.11, with the caveat that details around multiple unpackings in a type expression aren’t specified precisely. This gives individual type checkers some leeway, but can be tightened in future PEPs.
Motivation
Variadic generics have long been a requested feature, for a myriad of use cases [4]. One particular use case - a use case with potentially large impact, and the main case this PEP targets - concerns typing in numerical libraries.
In the context of numerical computation with libraries such as NumPy and TensorFlow, the shape of variables is often just as important as the variable type. For example, consider the following function which converts a batch [1] of videos to grayscale:
def to_gray(videos: Array): ...
From the signature alone, it is not obvious what shape of array [2]
we should pass for the videos
argument. Possibilities include, for
example,
batch × time × height × width × channels
and
time × batch × channels × height × width. [3]
This is important for three reasons:
- Documentation. Without the required shape being clear in the signature, the user must hunt in the docstring or the code in question to determine what the input/output shape requirements are.
- Catching shape bugs before runtime. Ideally, use of incorrect shapes should be an error we can catch ahead of time using static analysis. (This is particularly important for machine learning code, where iteration times can be slow.)
- Preventing subtle shape bugs. In the worst case, use of the wrong shape will result in the program appearing to run fine, but with a subtle bug that can take days to track down. (See this exercise in a popular machine learning tutorial for a particularly pernicious example.)
Ideally, we should have some way of making shape requirements explicit in type signatures. Multiple proposals [6] [7] [9] have suggested the use of the standard generics syntax for this purpose. We would write:
def to_gray(videos: Array[Time, Batch, Height, Width, Channels]): ...
However, note that arrays can be of arbitrary rank - Array
as used above is
generic in an arbitrary number of axes. One way around this would be to use a different
Array
class for each rank…
Axis1 = TypeVar('Axis1')
Axis2 = TypeVar('Axis2')
class Array1(Generic[Axis1]): ...
class Array2(Generic[Axis1, Axis2]): ...
…but this would be cumbersome, both for users (who would have to sprinkle 1s and 2s and so on throughout their code) and for the authors of array libraries (who would have to duplicate implementations throughout multiple classes).
Variadic generics are necessary for an Array
that is generic in an arbitrary
number of axes to be cleanly defined as a single class.
Summary Examples
Cutting right to the chase, this PEP allows an Array
class that is generic
in its shape (and datatype) to be defined using a newly-introduced
arbitrary-length type variable, TypeVarTuple
, as follows:
from typing import TypeVar, TypeVarTuple
DType = TypeVar('DType')
Shape = TypeVarTuple('Shape')
class Array(Generic[DType, *Shape]):
def __abs__(self) -> Array[DType, *Shape]: ...
def __add__(self, other: Array[DType, *Shape]) -> Array[DType, *Shape]: ...
Such an Array
can be used to support a number of different kinds of
shape annotations. For example, we can add labels describing the
semantic meaning of each axis:
from typing import NewType
Height = NewType('Height', int)
Width = NewType('Width', int)
x: Array[float, Height, Width] = Array()
We could also add annotations describing the actual size of each axis:
from typing import Literal as L
x: Array[float, L[480], L[640]] = Array()
For consistency, we use semantic axis annotations as the basis of the examples
in this PEP, but this PEP is agnostic about which of these two (or possibly other)
ways of using Array
is preferable; that decision is left to library authors.
(Note also that for the rest of this PEP, for conciseness of example, we use
a simpler version of Array
which is generic only in the shape - not the
data type.)
Specification
In order to support the above use cases, we introduce
TypeVarTuple
. This serves as a placeholder not for a single type
but for a tuple of types.
In addition, we introduce a new use for the star operator: to ‘unpack’
TypeVarTuple
instances and tuple types such as Tuple[int,
str]
. Unpacking a TypeVarTuple
or tuple type is the typing
equivalent of unpacking a variable or a tuple of values.
Type Variable Tuples
In the same way that a normal type variable is a stand-in for a single
type such as int
, a type variable tuple is a stand-in for a tuple type such as
Tuple[int, str]
.
Type variable tuples are created with:
from typing import TypeVarTuple
Ts = TypeVarTuple('Ts')
Using Type Variable Tuples in Generic Classes
Type variable tuples behave like a number of individual type variables packed in a
Tuple
. To understand this, consider the following example:
Shape = TypeVarTuple('Shape')
class Array(Generic[*Shape]): ...
Height = NewType('Height', int)
Width = NewType('Width', int)
x: Array[Height, Width] = Array()
The Shape
type variable tuple here behaves like Tuple[T1, T2]
,
where T1
and T2
are type variables. To use these type variables
as type parameters of Array
, we must unpack the type variable tuple using
the star operator: *Shape
. The signature of Array
then behaves
as if we had simply written class Array(Generic[T1, T2]): ...
.
In contrast to Generic[T1, T2]
, however, Generic[*Shape]
allows
us to parameterise the class with an arbitrary number of type parameters.
That is, in addition to being able to define rank-2 arrays such as
Array[Height, Width]
, we could also define rank-3 arrays, rank-4 arrays,
and so on:
Time = NewType('Time', int)
Batch = NewType('Batch', int)
y: Array[Batch, Height, Width] = Array()
z: Array[Time, Batch, Height, Width] = Array()
Using Type Variable Tuples in Functions
Type variable tuples can be used anywhere a normal TypeVar
can.
This includes class definitions, as shown above, as well as function
signatures and variable annotations:
class Array(Generic[*Shape]):
def __init__(self, shape: Tuple[*Shape]):
self._shape: Tuple[*Shape] = shape
def get_shape(self) -> Tuple[*Shape]:
return self._shape
shape = (Height(480), Width(640))
x: Array[Height, Width] = Array(shape)
y = abs(x) # Inferred type is Array[Height, Width]
z = x + x # ... is Array[Height, Width]
Type Variable Tuples Must Always be Unpacked
Note that in the previous example, the shape
argument to __init__
was annotated as Tuple[*Shape]
. Why is this necessary - if Shape
behaves like Tuple[T1, T2, ...]
, couldn’t we have annotated the shape
argument as Shape
directly?
This is, in fact, deliberately not possible: type variable tuples must always be used unpacked (that is, prefixed by the star operator). This is for two reasons:
- To avoid potential confusion about whether to use a type variable tuple
in a packed or unpacked form (“Hmm, should I write ‘
-> Shape
’, or ‘-> Tuple[Shape]
’, or ‘-> Tuple[*Shape]
’…?”) - To improve readability: the star also functions as an explicit visual indicator that the type variable tuple is not a normal type variable.
Unpack
for Backwards Compatibility
Note that the use of the star operator in this context requires a grammar change,
and is therefore available only in new versions of Python. To enable use of type
variable tuples in older versions of Python, we introduce the Unpack
type
operator that can be used in place of the star operator:
# Unpacking using the star operator in new versions of Python
class Array(Generic[*Shape]): ...
# Unpacking using ``Unpack`` in older versions of Python
class Array(Generic[Unpack[Shape]]): ...
Variance, Type Constraints and Type Bounds: Not (Yet) Supported
To keep this PEP minimal, TypeVarTuple
does not yet support specification of:
- Variance (e.g.
TypeVar('T', covariant=True)
) - Type constraints (
TypeVar('T', int, float)
) - Type bounds (
TypeVar('T', bound=ParentClass)
)
We leave the decision of how these arguments should behave to a future PEP, when variadic generics have been tested in the field. As of this PEP, type variable tuples are invariant.
Type Variable Tuple Equality
If the same TypeVarTuple
instance is used in multiple places in a signature
or class, a valid type inference might be to bind the TypeVarTuple
to
a Tuple
of a Union
of types:
def foo(arg1: Tuple[*Ts], arg2: Tuple[*Ts]): ...
a = (0,)
b = ('0',)
foo(a, b) # Can Ts be bound to Tuple[int | str]?
We do not allow this; type unions may not appear within the Tuple
.
If a type variable tuple appears in multiple places in a signature,
the types must match exactly (the list of type parameters must be the same
length, and the type parameters themselves must be identical):
def pointwise_multiply(
x: Array[*Shape],
y: Array[*Shape]
) -> Array[*Shape]: ...
x: Array[Height]
y: Array[Width]
z: Array[Height, Width]
pointwise_multiply(x, x) # Valid
pointwise_multiply(x, y) # Error
pointwise_multiply(x, z) # Error
Multiple Type Variable Tuples: Not Allowed
As of this PEP, only a single type variable tuple may appear in a type parameter list:
class Array(Generic[*Ts1, *Ts2]): ... # Error
The reason is that multiple type variable tuples make it ambiguous which parameters get bound to which type variable tuple:
x: Array[int, str, bool] # Ts1 = ???, Ts2 = ???
Type Concatenation
Type variable tuples don’t have to be alone; normal types can be prefixed and/or suffixed:
Shape = TypeVarTuple('Shape')
Batch = NewType('Batch', int)
Channels = NewType('Channels', int)
def add_batch_axis(x: Array[*Shape]) -> Array[Batch, *Shape]: ...
def del_batch_axis(x: Array[Batch, *Shape]) -> Array[*Shape]: ...
def add_batch_channels(
x: Array[*Shape]
) -> Array[Batch, *Shape, Channels]: ...
a: Array[Height, Width]
b = add_batch_axis(a) # Inferred type is Array[Batch, Height, Width]
c = del_batch_axis(b) # Array[Height, Width]
d = add_batch_channels(a) # Array[Batch, Height, Width, Channels]
Normal TypeVar
instances can also be prefixed and/or suffixed:
T = TypeVar('T')
Ts = TypeVarTuple('Ts')
def prefix_tuple(
x: T,
y: Tuple[*Ts]
) -> Tuple[T, *Ts]: ...
z = prefix_tuple(x=0, y=(True, 'a'))
# Inferred type of z is Tuple[int, bool, str]
Unpacking Tuple Types
We mentioned that a TypeVarTuple
stands for a tuple of types.
Since we can unpack a TypeVarTuple
, for consistency, we also
allow unpacking a tuple type. As we shall see, this also enables a
number of interesting features.
Unpacking Concrete Tuple Types
Unpacking a concrete tuple type is analogous to unpacking a tuple of
values at runtime. Tuple[int, *Tuple[bool, bool], str]
is
equivalent to Tuple[int, bool, bool, str]
.
Unpacking Unbounded Tuple Types
Unpacking an unbounded tuple preserves the unbounded tuple as it is.
That is, *Tuple[int, ...]
remains *Tuple[int, ...]
; there’s no
simpler form. This enables us to specify types such as Tuple[int,
*Tuple[str, ...], str]
- a tuple type where the first element is
guaranteed to be of type int
, the last element is guaranteed to be
of type str
, and the elements in the middle are zero or more
elements of type str
. Note that Tuple[*Tuple[int, ...]]
is
equivalent to Tuple[int, ...]
.
Unpacking unbounded tuples is also useful in function signatures where
we don’t care about the exact elements and don’t want to define an
unnecessary TypeVarTuple
:
def process_batch_channels(
x: Array[Batch, *Tuple[Any, ...], Channels]
) -> None:
...
x: Array[Batch, Height, Width, Channels]
process_batch_channels(x) # OK
y: Array[Batch, Channels]
process_batch_channels(y) # OK
z: Array[Batch]
process_batch_channels(z) # Error: Expected Channels.
We can also pass a *Tuple[int, ...]
wherever a *Ts
is
expected. This is useful when we have particularly dynamic code and
cannot state the precise number of dimensions or the precise types for
each of the dimensions. In those cases, we can smoothly fall back to
an unbounded tuple:
y: Array[*Tuple[Any, ...]] = read_from_file()
def expect_variadic_array(
x: Array[Batch, *Shape]
) -> None: ...
expect_variadic_array(y) # OK
def expect_precise_array(
x: Array[Batch, Height, Width, Channels]
) -> None: ...
expect_precise_array(y) # OK
Array[*Tuple[Any, ...]]
stands for an array with an arbitrary
number of dimensions of type Any
. This means that, in the call to
expect_variadic_array
, Batch
is bound to Any
and Shape
is bound to Tuple[Any, ...]
. In the call to
expect_precise_array
, the variables Batch
, Height
,
Width
, and Channels
are all bound to Any
.
This allows users to handle dynamic code gracefully while still
explicitly marking the code as unsafe (by using y: Array[*Tuple[Any,
...]]
). Otherwise, users would face noisy errors from the type
checker every time they tried to use the variable y
, which would
hinder them when migrating a legacy code base to use TypeVarTuple
.
Multiple Unpackings in a Tuple: Not Allowed
As with TypeVarTuples
, only one unpacking may appear in a tuple:
x: Tuple[int, *Ts, str, *Ts2] # Error
y: Tuple[int, *Tuple[int, ...], str, *Tuple[str, ...]] # Error
*args
as a Type Variable Tuple
PEP 484 states that when a type annotation is provided for *args
, every argument
must be of the type annotated. That is, if we specify *args
to be type int
,
then all arguments must be of type int
. This limits our ability to specify
the type signatures of functions that take heterogeneous argument types.
If *args
is annotated as a type variable tuple, however, the types of the
individual arguments become the types in the type variable tuple:
Ts = TypeVarTuple('Ts')
def args_to_tuple(*args: *Ts) -> Tuple[*Ts]: ...
args_to_tuple(1, 'a') # Inferred type is Tuple[int, str]
In the above example, Ts
is bound to Tuple[int, str]
. If no
arguments are passed, the type variable tuple behaves like an empty
tuple, Tuple[()]
.
As usual, we can unpack any tuple types. For example, by using a type variable tuple inside a tuple of other types, we can refer to prefixes or suffixes of the variadic argument list. For example:
# os.execle takes arguments 'path, arg0, arg1, ..., env'
def execle(path: str, *args: *Tuple[*Ts, Env]) -> None: ...
Note that this is different to
def execle(path: str, *args: *Ts, env: Env) -> None: ...
as this would make env
a keyword-only argument.
Using an unpacked unbounded tuple is equivalent to the
PEP 484
behavior of *args: int
, which accepts zero or
more values of type int
:
def foo(*args: *Tuple[int, ...]) -> None: ...
# equivalent to:
def foo(*args: int) -> None: ...
Unpacking tuple types also allows more precise types for heterogeneous
*args
. The following function expects an int
at the beginning,
zero or more str
values, and a str
at the end:
def foo(*args: *Tuple[int, *Tuple[str, ...], str]) -> None: ...
For completeness, we mention that unpacking a concrete tuple allows us
to specify *args
of a fixed number of heterogeneous types:
def foo(*args: *Tuple[int, str]) -> None: ...
foo(1, "hello") # OK
Note that, in keeping with the rule that type variable tuples must always
be used unpacked, annotating *args
as being a plain type variable tuple
instance is not allowed:
def foo(*args: Ts): ... # NOT valid
*args
is the only case where an argument can be annotated as *Ts
directly;
other arguments should use *Ts
to parameterise something else, e.g. Tuple[*Ts]
.
If *args
itself is annotated as Tuple[*Ts]
, the old behaviour still applies:
all arguments must be a Tuple
parameterised with the same types.
def foo(*args: Tuple[*Ts]): ...
foo((0,), (1,)) # Valid
foo((0,), (1, 2)) # Error
foo((0,), ('1',)) # Error
Finally, note that a type variable tuple may not be used as the type of
**kwargs
. (We do not yet know of a use case for this feature, so we prefer
to leave the ground fresh for a potential future PEP.)
# NOT valid
def foo(**kwargs: *Ts): ...
Type Variable Tuples with Callable
Type variable tuples can also be used in the arguments section of a
Callable
:
class Process:
def __init__(
self,
target: Callable[[*Ts], None],
args: Tuple[*Ts],
) -> None: ...
def func(arg1: int, arg2: str) -> None: ...
Process(target=func, args=(0, 'foo')) # Valid
Process(target=func, args=('foo', 0)) # Error
Other types and normal type variables can also be prefixed/suffixed to the type variable tuple:
T = TypeVar('T')
def foo(f: Callable[[int, *Ts, T], Tuple[T, *Ts]]): ...
The behavior of a Callable containing an unpacked item, whether the
item is a TypeVarTuple
or a tuple type, is to treat the elements
as if they were the type for *args
. So, Callable[[*Ts], None]
is treated as the type of the function:
def foo(*args: *Ts) -> None: ...
Callable[[int, *Ts, T], Tuple[T, *Ts]]
is treated as the type of
the function:
def foo(*args: *Tuple[int, *Ts, T]) -> Tuple[T, *Ts]: ...
Behaviour when Type Parameters are not Specified
When a generic class parameterised by a type variable tuple is used without
any type parameters, it behaves as if the type variable tuple was
substituted with Tuple[Any, ...]
:
def takes_any_array(arr: Array): ...
# equivalent to:
def takes_any_array(arr: Array[*Tuple[Any, ...]]): ...
x: Array[Height, Width]
takes_any_array(x) # Valid
y: Array[Time, Height, Width]
takes_any_array(y) # Also valid
This enables gradual typing: existing functions accepting, for example,
a plain TensorFlow Tensor
will still be valid even if Tensor
is made
generic and calling code passes a Tensor[Height, Width]
.
This also works in the opposite direction:
def takes_specific_array(arr: Array[Height, Width]): ...
z: Array
# equivalent to Array[*Tuple[Any, ...]]
takes_specific_array(z)
(For details, see the section on Unpacking Unbounded Tuple Types.)
This way, even if libraries are updated to use types like Array[Height, Width]
,
users of those libraries won’t be forced to also apply type annotations to
all of their code; users still have a choice about what parts of their code
to type and which parts to not.
Aliases
Generic aliases can be created using a type variable tuple in a similar way to regular type variables:
IntTuple = Tuple[int, *Ts]
NamedArray = Tuple[str, Array[*Ts]]
IntTuple[float, bool] # Equivalent to Tuple[int, float, bool]
NamedArray[Height] # Equivalent to Tuple[str, Array[Height]]
As this example shows, all type parameters passed to the alias are bound to the type variable tuple.
Importantly for our original Array
example (see Summary Examples), this
allows us to define convenience aliases for arrays of a fixed shape
or datatype:
Shape = TypeVarTuple('Shape')
DType = TypeVar('DType')
class Array(Generic[DType, *Shape]):
# E.g. Float32Array[Height, Width, Channels]
Float32Array = Array[np.float32, *Shape]
# E.g. Array1D[np.uint8]
Array1D = Array[DType, Any]
If an explicitly empty type parameter list is given, the type variable tuple in the alias is set empty:
IntTuple[()] # Equivalent to Tuple[int]
NamedArray[()] # Equivalent to Tuple[str, Array[()]]
If the type parameter list is omitted entirely, the unspecified type
variable tuples are treated as Tuple[Any, ...]
(similar to
Behaviour when Type Parameters are not Specified):
def takes_float_array_of_any_shape(x: Float32Array): ...
x: Float32Array[Height, Width] = Array()
takes_float_array_of_any_shape(x) # Valid
def takes_float_array_with_specific_shape(
y: Float32Array[Height, Width]
): ...
y: Float32Array = Array()
takes_float_array_with_specific_shape(y) # Valid
Normal TypeVar
instances can also be used in such aliases:
T = TypeVar('T')
Foo = Tuple[T, *Ts]
# T bound to str, Ts to Tuple[int]
Foo[str, int]
# T bound to float, Ts to Tuple[()]
Foo[float]
# T bound to Any, Ts to an Tuple[Any, ...]
Foo
Substitution in Aliases
In the previous section, we only discussed simple usage of generic aliases in which the type arguments were just simple types. However, a number of more exotic constructions are also possible.
Type Arguments can be Variadic
First, type arguments to generic aliases can be variadic. For example, a
TypeVarTuple
can be used as a type argument:
Ts1 = TypeVar('Ts1')
Ts2 = TypeVar('Ts2')
IntTuple = Tuple[int, *Ts1]
IntFloatTuple = IntTuple[float, *Ts2] # Valid
Here, *Ts1
in the IntTuple
alias is bound to Tuple[float, *Ts2]
,
resulting in an alias IntFloatTuple
equivalent to
Tuple[int, float, *Ts2]
.
Unpacked arbitrary-length tuples can also be used as type arguments, with similar effects:
IntFloatsTuple = IntTuple[*Tuple[float, ...]] # Valid
Here, *Ts1
is bound to *Tuple[float, ...]
, resulting in
IntFloatsTuple
being equivalent to Tuple[int, *Tuple[float, ...]]
: a tuple
consisting of an int
then zero or more float
s.
Variadic Arguments Require Variadic Aliases
Variadic type arguments can only be used with generic aliases that are themselves variadic. For example:
T = TypeVar('T')
IntTuple = Tuple[int, T]
IntTuple[str] # Valid
IntTuple[*Ts] # NOT valid
IntTuple[*Tuple[float, ...]] # NOT valid
Here, IntTuple
is a non-variadic generic alias that takes exactly one
type argument. Hence, it cannot accept *Ts
or *Tuple[float, ...]
as type
arguments, because they represent an arbitrary number of types.
Aliases with Both TypeVars and TypeVarTuples
In Aliases, we briefly mentioned that aliases can be generic in both
TypeVar
s and TypeVarTuple
s:
T = TypeVar('T')
Foo = Tuple[T, *Ts]
Foo[str, int] # T bound to str, Ts to Tuple[int]
Foo[str, int, float] # T bound to str, Ts to Tuple[int, float]
In accordance with Multiple Type Variable Tuples: Not Allowed, at most one
TypeVarTuple
may appear in the type parameters to an alias. However, a
TypeVarTuple
can be combined with an arbitrary number of TypeVar
s,
both before and after:
T1 = TypeVar('T1')
T2 = TypeVar('T2')
T3 = TypeVar('T3')
Tuple[*Ts, T1, T2] # Valid
Tuple[T1, T2, *Ts] # Valid
Tuple[T1, *Ts, T2, T3] # Valid
In order to substitute these type variables with supplied type arguments,
any type variables at the beginning or end of the type parameter list first
consume type arguments, and then any remaining type arguments are bound
to the TypeVarTuple
:
Shrubbery = Tuple[*Ts, T1, T2]
Shrubbery[str, bool] # T2=bool, T1=str, Ts=Tuple[()]
Shrubbery[str, bool, float] # T2=float, T1=bool, Ts=Tuple[str]
Shrubbery[str, bool, float, int] # T2=int, T1=float, Ts=Tuple[str, bool]
Ptang = Tuple[T1, *Ts, T2, T3]
Ptang[str, bool, float] # T1=str, T3=float, T2=bool, Ts=Tuple[()]
Ptang[str, bool, float, int] # T1=str, T3=int, T2=float, Ts=Tuple[bool]
Note that the minimum number of type arguments in such cases is set by
the number of TypeVar
s:
Shrubbery[int] # Not valid; Shrubbery needs at least two type arguments
Splitting Arbitrary-Length Tuples
A final complication occurs when an unpacked arbitrary-length tuple is used
as a type argument to an alias consisting of both TypeVar
s and a
TypeVarTuple
:
Elderberries = Tuple[*Ts, T1]
Hamster = Elderberries[*Tuple[int, ...]] # valid
In such cases, the arbitrary-length tuple is split between the TypeVar
s
and the TypeVarTuple
. We assume the arbitrary-length tuple contains
at least as many items as there are TypeVar
s, such that individual
instances of the inner type - here int
- are bound to any TypeVar
s
present. The ‘rest’ of the arbitrary-length tuple - here *Tuple[int, ...]
,
since a tuple of arbitrary length minus two items is still arbitrary-length -
is bound to the TypeVarTuple
.
Here, therefore, Hamster
is equivalent to Tuple[*Tuple[int, ...], int]
:
a tuple consisting of zero or more int
s, then a final int
.
Of course, such splitting only occurs if necessary. For example, if we instead did:
Elderberries[*Tuple[int, ...], str]
Then splitting would not occur; T1
would be bound to str
, and
Ts
to *Tuple[int, ...]
.
In particularly awkward cases, a TypeVarTuple
may consume both a type
and a part of an arbitrary-length tuple type:
Elderberries[str, *Tuple[int, ...]]
Here, T1
is bound to int
, and Ts
is bound to
Tuple[str, *Tuple[int, ...]]
. This expression is therefore equivalent to
Tuple[str, *Tuple[int, ...], int]
: a tuple consisting of a str
, then
zero or more int
s, ending with an int
.
TypeVarTuples Cannot be Split
Finally, although any arbitrary-length tuples in the type argument list can be
split between the type variables and the type variable tuple, the same is not
true of TypeVarTuple
s in the argument list:
Ts1 = TypeVarTuple('Ts1')
Ts2 = TypeVarTuple('Ts2')
Camelot = Tuple[T, *Ts1]
Camelot[*Ts2] # NOT valid
This is not possible because, unlike in the case of an unpacked arbitrary-length
tuple, there is no way to ‘peer inside’ the TypeVarTuple
to see what its
individual types are.
Overloads for Accessing Individual Types
For situations where we require access to each individual type in the type variable tuple,
overloads can be used with individual TypeVar
instances in place of the type variable tuple:
Shape = TypeVarTuple('Shape')
Axis1 = TypeVar('Axis1')
Axis2 = TypeVar('Axis2')
Axis3 = TypeVar('Axis3')
class Array(Generic[*Shape]):
@overload
def transpose(
self: Array[Axis1, Axis2]
) -> Array[Axis2, Axis1]: ...
@overload
def transpose(
self: Array[Axis1, Axis2, Axis3]
) -> Array[Axis3, Axis2, Axis1]: ...
(For array shape operations in particular, having to specify overloads for each possible rank is, of course, a rather cumbersome solution. However, it’s the best we can do without additional type manipulation mechanisms. We plan to introduce these in a future PEP.)
Rationale and Rejected Ideas
Shape Arithmetic
Considering the use case of array shapes in particular, note that as of
this PEP, it is not yet possible to describe arithmetic transformations
of array dimensions - for example,
def repeat_each_element(x: Array[N]) -> Array[2*N]
. We consider
this out-of-scope for the current PEP, but plan to propose additional
mechanisms that will enable this in a future PEP.
Supporting Variadicity Through Aliases
As noted in the introduction, it is possible to avoid variadic generics by simply defining aliases for each possible number of type parameters:
class Array1(Generic[Axis1]): ...
class Array2(Generic[Axis1, Axis2]): ...
However, this seems somewhat clumsy - it requires users to unnecessarily pepper their code with 1s, 2s, and so on for each rank necessary.
Construction of TypeVarTuple
TypeVarTuple
began as ListVariadic
, based on its naming in
an early implementation in Pyre.
We then changed this to TypeVar(list=True)
, on the basis that a)
it better emphasises the similarity to TypeVar
, and b) the meaning
of ‘list’ is more easily understood than the jargon of ‘variadic’.
Once we’d decided that a variadic type variable should behave like a Tuple
,
we also considered TypeVar(bound=Tuple)
, which is similarly intuitive
and accomplishes most what we wanted without requiring any new arguments to
TypeVar
. However, we realised this may constrain us in the future, if
for example we want type bounds or variance to function slightly differently
for variadic type variables than what the semantics of TypeVar
might
otherwise imply. Also, we may later wish to support arguments that should not be supported by regular type variables (such as arbitrary_len
[10]).
We therefore settled on TypeVarTuple
.
Unspecified Type Parameters: Tuple vs TypeVarTuple
In order to support gradual typing, this PEP states that both of the following examples should type-check correctly:
def takes_any_array(x: Array): ...
x: Array[Height, Width]
takes_any_array(x)
def takes_specific_array(y: Array[Height, Width]): ...
y: Array
takes_specific_array(y)
Note that this is in contrast to the behaviour of the only currently-existing
variadic type in Python, Tuple
:
def takes_any_tuple(x: Tuple): ...
x: Tuple[int, str]
takes_any_tuple(x) # Valid
def takes_specific_tuple(y: Tuple[int, str]): ...
y: Tuple
takes_specific_tuple(y) # Error
The rules for Tuple
were deliberately chosen such that the latter case
is an error: it was thought to be more likely that the programmer has made a
mistake than that the function expects a specific kind of Tuple
but the
specific kind of Tuple
passed is unknown to the type checker. Additionally,
Tuple
is something of a special case, in that it is used to represent
immutable sequences. That is, if an object’s type is inferred to be an
unparameterised Tuple
, it is not necessarily because of incomplete typing.
In contrast, if an object’s type is inferred to be an unparameterised Array
,
it is much more likely that the user has simply not yet fully annotated their
code, or that the signature of a shape-manipulating library function cannot yet
be expressed using the typing system and therefore returning a plain Array
is the only option. We rarely deal with arrays of truly arbitrary shape;
in certain cases, some parts of the shape will be arbitrary - for example,
when dealing with sequences, the first two parts of the shape are often
‘batch’ and ‘time’ - but we plan to support these cases explicitly in a
future PEP with a syntax such as Array[Batch, Time, ...]
.
We therefore made the decision to have variadic generics other than
Tuple
behave differently, in order to give the user more flexibility
in how much of their code they wish to annotate, and to enable compatibility
between old unannotated code and new versions of libraries which do use
these type annotations.
Alternatives
It should be noted that the approach outlined in this PEP to solve the issue of shape checking in numerical libraries is not the only approach possible. Examples of lighter-weight alternatives based on runtime checking include ShapeGuard [13], tsanley [11], and PyContracts [12].
While these existing approaches improve significantly on the default situation of shape checking only being possible through lengthy and verbose assert statements, none of them enable static analysis of shape correctness. As mentioned in Motivation, this is particularly desirable for machine learning applications where, due to library and infrastructure complexity, even relatively simple programs must suffer long startup times; iterating by running the program until it crashes, as is necessary with these existing runtime-based approaches, can be a tedious and frustrating experience.
Our hope with this PEP is to begin to codify generic type annotations as an official, language-supported way of dealing with shape correctness. With something of a standard in place, in the long run, this will hopefully enable a thriving ecosystem of tools for analysing and verifying shape properties of numerical computing programs.
Grammar Changes
This PEP requires two grammar changes.
Change 1: Star Expressions in Indexes
The first grammar change enables use of star expressions in index operations (that is, within square brackets), necessary to support star-unpacking of TypeVarTuples:
DType = TypeVar('DType')
Shape = TypeVarTuple('Shape')
class Array(Generic[DType, *Shape]):
...
Before:
slices:
| slice !','
| ','.slice+ [',']
After:
slices:
| slice !','
| ','.(slice | starred_expression)+ [',']
As with star-unpacking in other contexts, the star operator calls __iter__
on the callee, and adds the contents of the resulting iterator to the argument
passed to __getitem__
. For example, if we do foo[a, *b, c]
, and
b.__iter__
produces an iterator yielding d
and e
,
foo.__getitem__
would receive (a, d, e, c)
.
To put it another way, note that x[..., *a, ...]
produces the same result
as x[(..., *a, ...)]
(with any slices i:j
in ...
replaced with
slice(i, j)
, with the one edge case that x[*a]
becomes x[(*a,)]
).
TypeVarTuple Implementation
With this grammar change, TypeVarTuple
is implemented as follows.
Note that this implementation is useful only for the benefit of a) correct
repr()
and b) runtime analysers; static analysers would not use the
implementation.
class TypeVarTuple:
def __init__(self, name):
self._name = name
self._unpacked = UnpackedTypeVarTuple(name)
def __iter__(self):
yield self._unpacked
def __repr__(self):
return self._name
class UnpackedTypeVarTuple:
def __init__(self, name):
self._name = name
def __repr__(self):
return '*' + self._name
Implications
This grammar change implies a number of additional changes in behaviour not required by this PEP. We choose to allow these additional changes rather than disallowing them at a syntax level in order to keep the syntax change as small as possible.
First, the grammar change enables star-unpacking of other structures, such as lists, within indexing operations:
idxs = (1, 2)
array_slice = array[0, *idxs, -1] # Equivalent to [0, 1, 2, -1]
array[0, *idxs, -1] = array_slice # Also allowed
Second, more than one instance of a star-unpack can occur within an index:
array[*idxs_to_select, *idxs_to_select] # Equivalent to array[1, 2, 1, 2]
Note that this PEP disallows multiple unpacked TypeVarTuples within a single type parameter list. This requirement would therefore need to be implemented in type checking tools themselves rather than at the syntax level.
Third, slices may co-occur with starred expressions:
array[3:5, *idxs_to_select] # Equivalent to array[3:5, 1, 2]
However, note that slices involving starred expressions are still invalid:
# Syntax error
array[*idxs_start:*idxs_end]
Change 2: *args
as a TypeVarTuple
The second change enables use of *args: *Ts
in function definitions.
Before:
star_etc:
| '*' param_no_default param_maybe_default* [kwds]
| '*' ',' param_maybe_default+ [kwds]
| kwds
After:
star_etc:
| '*' param_no_default param_maybe_default* [kwds]
| '*' param_no_default_star_annotation param_maybe_default* [kwds] # New
| '*' ',' param_maybe_default+ [kwds]
| kwds
Where:
param_no_default_star_annotation:
| param_star_annotation ',' TYPE_COMMENT?
| param_star_annotation TYPE_COMMENT? &')'
param_star_annotation: NAME star_annotation
star_annotation: ':' star_expression
We also need to deal with the star_expression
that results from this
construction. Normally, a star_expression
occurs within the context
of e.g. a list, so a star_expression
is handled by essentially
calling iter()
on the starred object, and inserting the results
of the resulting iterator into the list at the appropriate place. For
*args: *Ts
, however, we must process the star_expression
in a
different way.
We do this by instead making a special case for the star_expression
resulting from *args: *Ts
, emitting code equivalent to
[annotation_value] = [*Ts]
. That is, we create an iterator from
Ts
by calling Ts.__iter__
, fetch a single value from the iterator,
verify that the iterator is exhausted, and set that value as the annotation
value. This results in the unpacked TypeVarTuple
being set directly
as the runtime annotation for *args
:
>>> Ts = TypeVarTuple('Ts')
>>> def foo(*args: *Ts): pass
>>> foo.__annotations__
{'args': *Ts}
# *Ts is the repr() of Ts._unpacked, an instance of UnpackedTypeVarTuple
This allows the runtime annotation to be consistent with an AST representation
that uses a Starred
node for the annotations of args
- in turn important
for tools that rely on the AST such as mypy to correctly recognise the construction:
>>> print(ast.dump(ast.parse('def foo(*args: *Ts): pass'), indent=2))
Module(
body=[
FunctionDef(
name='foo',
args=arguments(
posonlyargs=[],
args=[],
vararg=arg(
arg='args',
annotation=Starred(
value=Name(id='Ts', ctx=Load()),
ctx=Load())),
kwonlyargs=[],
kw_defaults=[],
defaults=[]),
body=[
Pass()],
decorator_list=[])],
type_ignores=[])
Note that the only scenario in which this grammar change allows *Ts
to be
used as a direct annotation (rather than being wrapped in e.g. Tuple[*Ts]
)
is *args
. Other uses are still invalid:
x: *Ts # Syntax error
def foo(x: *Ts): pass # Syntax error
Implications
As with the first grammar change, this change also has a number of side effects.
In particular, the annotation of *args
could be set to a starred object
other than a TypeVarTuple
- for example, the following nonsensical
annotations are possible:
>>> foo = [1]
>>> def bar(*args: *foo): pass
>>> bar.__annotations__
{'args': 1}
>>> foo = [1, 2]
>>> def bar(*args: *foo): pass
ValueError: too many values to unpack (expected 1)
Again, prevention of such annotations will need to be done by, say, static checkers, rather than at the level of syntax.
Alternatives (Why Not Just Use Unpack
?)
If these grammar changes are considered too burdensome, there are two alternatives.
The first would be to support change 1 but not change 2. Variadic generics
are more important to us than the ability to annotate *args
.
The second alternative would be to use ``Unpack`` instead, requiring no grammar changes. However, we regard this as a suboptimal solution for two reasons:
- Readability.
class Array(Generic[DType, Unpack[Shape]])
is a bit of a mouthful; the flow of reading is interrupted by length ofUnpack
and the extra set of square brackets.class Array(Generic[DType, *Shape])
is much easier to skim, while still markingShape
as special. - Intuitiveness. We think a user is more likely to intuitively understand
the meaning of
*Ts
- especially when they see thatTs
is a TypeVar**Tuple** - than the meaning ofUnpack[Ts]
. (This assumes the user is familiar with star-unpacking in other contexts; if the user is reading or writing code that uses variadic generics, this seems reasonable.)
If even change 1 is thought too significant a change, therefore, it might be better for us to reconsider our options before going ahead with this second alternative.
Backwards Compatibility
The Unpack
version of the PEP should be back-portable to previous
versions of Python.
Gradual typing is enabled by the fact that unparameterised variadic classes are compatible with an arbitrary number of type parameters. This means that if existing classes are made generic, a) all existing (unparameterised) uses of the class will still work, and b) parameterised and unparameterised versions of the class can be used together (relevant if, for example, library code is updated to use parameters while user code is not, or vice-versa).
Reference Implementation
Two reference implementations of type-checking functionality exist: one in Pyre, as of v0.9.0, and one in Pyright, as of v1.1.108.
A preliminary implementation of the Unpack
version of the PEP in CPython
is available in cpython/23527. A preliminary version of the version
using the star operator, based on an early implementation of PEP 637,
is also available at mrahtz/cpython/pep637+646.
Appendix A: Shape Typing Use Cases
To give this PEP additional context for those particularly interested in the array typing use case, in this appendix we expand on the different ways this PEP can be used for specifying shape-based subtypes.
Use Case 1: Specifying Shape Values
The simplest way to parameterise array types is using Literal
type parameters - e.g. Array[Literal[64], Literal[64]]
.
We can attach names to each parameter using normal type variables:
K = TypeVar('K')
N = TypeVar('N')
def matrix_vector_multiply(x: Array[K, N], y: Array[N]) -> Array[K]: ...
a: Array[Literal[64], Literal[32]]
b: Array[Literal[32]]
matrix_vector_multiply(a, b)
# Result is Array[Literal[64]]
Note that such names have a purely local scope. That is, the name
K
is bound to Literal[64]
only within matrix_vector_multiply
. To put it another
way, there’s no relationship between the value of K
in different
signatures. This is important: it would be inconvenient if every axis named K
were constrained to have the same value throughout the entire program.
The disadvantage of this approach is that we have no ability to enforce shape semantics across different calls. For example, we can’t address the problem mentioned in Motivation: if one function returns an array with leading dimensions ‘Time × Batch’, and another function takes the same array assuming leading dimensions ‘Batch × Time’, we have no way of detecting this.
The main advantage is that in some cases, axis sizes really are what we care about. This is true for both simple linear algebra operations such as the matrix manipulations above, but also in more complicated transformations such as convolutional layers in neural networks, where it would be of great utility to the programmer to be able to inspect the array size after each layer using static analysis. To aid this, in the future we would like to explore possibilities for additional type operators that enable arithmetic on array shapes - for example:
def repeat_each_element(x: Array[N]) -> Array[Mul[2, N]]: ...
Such arithmetic type operators would only make sense if names such as N
refer to axis size.
Use Case 2: Specifying Shape Semantics
A second approach (the one that most of the examples in this PEP are based around) is to forgo annotation with actual axis size, and instead annotate axis type.
This would enable us to solve the problem of enforcing shape properties across calls. For example:
# lib.py
class Batch: pass
class Time: pass
def make_array() -> Array[Batch, Time]: ...
# user.py
from lib import Batch, Time
# `Batch` and `Time` have the same identity as in `lib`,
# so must take array as produced by `lib.make_array`
def use_array(x: Array[Batch, Time]): ...
Note that in this case, names are global (to the extent that we use the
same Batch
type in different place). However, because names refer only
to axis types, this doesn’t constrain the value of certain axes to be
the same through (that is, this doesn’t constrain all axes named Height
to have a value of, say, 480 throughout).
The argument for this approach is that in many cases, axis type is the more important thing to verify; we care more about which axis is which than what the specific size of each axis is.
It also does not preclude cases where we wish to describe shape transformations without knowing the type ahead of time. For example, we can still write:
K = TypeVar('K')
N = TypeVar('N')
def matrix_vector_multiply(x: Array[K, N], y: Array[N]) -> Array[K]: ...
We can then use this with:
class Batch: pass
class Values: pass
batch_of_values: Array[Batch, Values]
value_weights: Array[Values]
matrix_vector_multiply(batch_of_values, value_weights)
# Result is Array[Batch]
The disadvantages are the inverse of the advantages from use case 1.
In particular, this approach does not lend itself well to arithmetic
on axis types: Mul[2, Batch]
would be as meaningless as 2 * int
.
Discussion
Note that use cases 1 and 2 are mutually exclusive in user code. Users can verify size or semantic type but not both.
As of this PEP, we are agnostic about which approach will provide most benefit. Since the features introduced in this PEP are compatible with both approaches, however, we leave the door open.
Why Not Both?
Consider the following ‘normal’ code:
def f(x: int): ...
Note that we have symbols for both the value of the thing (x
) and the type of
the thing (int
). Why can’t we do the same with axes? For example, with an imaginary
syntax, we could write:
def f(array: Array[TimeValue: TimeType]): ...
This would allow us to access the axis size (say, 32) through the symbol TimeValue
and the type through the symbol TypeType
.
This might even be possible using existing syntax, through a second level of parameterisation:
def f(array: array[TimeValue[TimeType]]): ..
However, we leave exploration of this approach to the future.
Appendix B: Shaped Types vs Named Axes
An issue related to those addressed by this PEP concerns
axis selection. For example, if we have an image stored in an array of shape 64×64x3,
we might wish to convert to black-and-white by computing the mean over the third
axis, mean(image, axis=2)
. Unfortunately, the simple typo axis=1
is
difficult to spot and will produce a result that means something completely different
(all while likely allowing the program to keep on running, resulting in a bug
that is serious but silent).
In response, some libraries have implemented so-called ‘named tensors’ (in this context,
‘tensor’ is synonymous with ‘array’), in which axes are selected not by index but by
label - e.g. mean(image, axis='channels')
.
A question we are often asked about this PEP is: why not just use named tensors? The answer is that we consider the named tensors approach insufficient, for two main reasons:
- Static checking of shape correctness is not possible. As mentioned in Motivation, this is a highly desirable feature in machine learning code where iteration times are slow by default.
- Interface documentation is still not possible with this approach. If a function should only be willing to take array arguments that have image-like shapes, this cannot be stipulated with named tensors.
Additionally, there’s the issue of poor uptake. At the time of writing, named tensors have only been implemented in a small number of numerical computing libraries. Possible explanations for this include difficulty of implementation (the whole API must be modified to allow selection by axis name instead of index), and lack of usefulness due to the fact that axis ordering conventions are often strong enough that axis names provide little benefit (e.g. when working with images, 3D tensors are basically always height × width × channels). However, ultimately we are still uncertain why this is the case.
Can the named tensors approach be combined with the approach we advocate for in this PEP? We’re not sure. One area of overlap is that in some contexts, we could do, say:
Image: Array[Height, Width, Channels]
im: Image
mean(im, axis=Image.axes.index(Channels)
Ideally, we might write something like im: Array[Height=64, Width=64, Channels=3]
-
but this won’t be possible in the short term, due to the rejection of PEP 637.
In any case, our attitude towards this is mostly “Wait and see what happens before
taking any further steps”.
Footnotes
Endorsements
Variadic generics have a wide range of uses. For the fraction of that range involving numerical computing, how likely is it that relevant libraries will actually make use of the features proposed in this PEP?
We reached out to a number of people with this question, and received the following endorsements.
From Stephan Hoyer, member of the NumPy Steering Council: [14]
I just wanted to thank Matthew & Pradeep for writing this PEP and for clarifications to the broader context of PEP 646 for array typing in https://github.com/python/peps/pull/1904.As someone who is heavily involved in the Python numerical computing community (e.g., NumPy, JAX, Xarray), but who is not so familiar with the details of Python’s type system, it is reassuring to see that a broad range of use-cases related to type checking of named axes & shapes have been considered, and could build upon the infrastructure in this PEP.
Type checking for shapes is something the NumPy community is very interested in – there are more thumbs up on the relevant issue on NumPy’s GitHub than any others (https://github.com/numpy/numpy/issues/7370) and we recently added a “typing” module that is under active development.
It will certainly require experimentation to figure out the best ways to use type checking for ndarrays, but this PEP looks like an excellent foundation for such work.
From Bas van Beek, who has worked on preliminary support for shape-generics in NumPy:
I very much share Stephan’s opinion here and look forward to integrating the new PEP 646 variadics into numpy.In the context of numpy (and tensor typing general): the typing of array shapes is a fairly complicated subject and the introduction of variadics will likely play in big role in laying its foundation, as it allows for the expression of both dimensioability as well as basic shape manipulation.
All in all, I’m very interested in where both PEP 646 and future PEPs will take us and look forward to further developments.
From Dan Moldovan, a Senior Software Engineer on the TensorFlow Dev Team and author of the TensorFlow RFC, TensorFlow Canonical Type System: [15]
I’d be interested in using this the mechanisms defined in this PEP to define rank-generic Tensor types in TensorFlow, which are important in specifyingtf.function
signatures in a Pythonic way, using type annotations (rather than the custominput_signature
mechanism we have today - see this issue: https://github.com/tensorflow/tensorflow/issues/31579). Variadic generics are among the last few missing pieces to create an elegant set of type definitions for tensors and shapes.
(For the sake of transparency - we also reached out to folks from a third popular numerical computing library, PyTorch, but did not receive a statement of endorsement from them. Our understanding is that although they are interested in some of the same issues - e.g. static shape inference - they are currently focusing on enabling this through a DSL rather than the Python type system.)
Acknowledgements
Thank you to Alfonso Castaño, Antoine Pitrou, Bas v.B., David Foster, Dimitris Vardoulakis, Eric Traut, Guido van Rossum, Jia Chen, Lucio Fernandez-Arjona, Nikita Sobolev, Peilonrayz, Rebecca Chen, Sergei Lebedev, and Vladimir Mikulik for helpful feedback and suggestions on drafts of this PEP.
Thank you especially to Lucio for suggesting the star syntax (which has made multiple aspects of this proposal much more concise and intuitive), and to Stephan Hoyer and Dan Moldovan for their endorsements.
Resources
Discussions on variadic generics in Python started in 2016 with Issue 193 on the python/typing GitHub repository [4].
Inspired by this discussion, Ivan Levkivskyi made a concrete proposal at PyCon 2019, summarised in notes on ‘Type system improvements’ [5] and ‘Static typing of Python numeric stack’ [6].
Expanding on these ideas, Mark Mendoza and Vincent Siles gave a presentation on ‘Variadic Type Variables for Decorators and Tensors’ [8] at the 2019 Python Typing Summit.
Discussion over how type substitution in generic aliases should behave took place in cpython#91162.
References
Copyright
This document is placed in the public domain or under the CC0-1.0-Universal license, whichever is more permissive.
Source: https://github.com/python/peps/blob/main/peps/pep-0646.rst
Last modified: 2024-06-27 00:41:45 GMT