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PEP 646 -- Variadic Generics

PEP:646
Title: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:Draft
Type:Standards Track
Created:16-Sep-2020
Python-Version:3.10
Post-History:07-Oct-2020, 23-Dec-2020, 29-Dec-2020

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.

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 [18] 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 an arbitrary number of types, and behaving like a number of TypeVar instances packed in a Tuple.

In addition, we introduce a new use for the star operator: to 'unpack' TypeVarTuple instances, in order to access the type variables contained in the tuple.

Type Variable Tuples

In the same way that a normal type variable is a stand-in for a single type, a type variable tuple is a stand-in for an arbitrary number of types (zero or more) in a flat ordered list.

Type variable tuples are created with:

from typing import TypeVarTuple

Ts = TypeVarTuple('Ts')

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()

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.

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 its type parameters are 'Any, ...' (an arbitrary number of Any):

def takes_any_array(arr: Array): ...

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
takes_specific_array(z)

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.

Type Variable Tuples Must Have Known Length

Type variables tuples may not be bound to a type with unknown length. That is:

def foo(x: Tuple[*Ts]): ...

x: Tuple[float, ...]
foo(x)  # NOT valid; Ts would be bound to ``Tuple[float, ...]``

If this is confusing - didn't we say that type variable tuples are a stand-in for an arbitrary number of types? - note the difference between the length of the type variable tuple itself, and the length of the type it is bound to. Type variable tuples themselves can be of arbitrary length - that is, they can be bound to Tuple[int], Tuple[int, int], and so on - but the types they are bound to must be of known length - that is, Tuple[int, int], but not Tuple[int, ...].

Note that, as a result of this rule, omitting the type parameter list is the only way of instantiating a generic type with an arbitrary number of type parameters. (We plan to introduce a more deliberate syntax for this case in a future PEP.) For example, an unparameterised Array may behave like Array[Any, ...], but it cannot be instantiated using Array[Any, ...], because this would bind its type variable tuple to Tuple[Any, ...]:

x: Array            # Valid
y: Array[int, ...]  # Error
z: Array[Any, ...]  # Error

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

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]

*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]

If no arguments are passed, the type variable tuple behaves like an empty tuple, Tuple[()].

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

Following Type Variable Tuples Must Have Known Length, note that the following should not type-check as valid (even though it is, of course, valid at runtime):

def foo(*args: *Ts): ...

def bar(x: Tuple[int, ...]):
  foo(*x)  # NOT valid

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], Any],
    args: Tuple[*Ts]
  ): ...

def func(arg1: int, arg2: str): ...

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]]): ...

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 alias is compatible with arbitrary type parameters:

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 arbitrary number of Any
Foo

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.

Behaviour when Type Parameters are not Specified

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. Full diffs of python.gram and simple tests to confirm correct behaviour are available at https://github.com/mrahtz/cpython/commits/pep646-grammar.

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_to_select = (1, 2)
array[0, *idxs_to_select, -1]  # Equivalent to [0, 1, 2, -1]

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

This accomplishes the desired outcome (making *args: *Ts not be a syntax error) while matching the behaviour of star-unpacking in other contexts: at runtime, __iter__ is called on the starred object, and a tuple containing the items of the resulting iterator is set as the type annotion for args. In other words, at runtime *args: *foo is equivalent to *args: tuple(foo).

>>> Ts = TypeVarTuple('Ts')
>>> def foo(*args: *Ts): pass  # Equivalent to `*args: tuple(Ts)`
>>> foo.__annotations__
{'args': (*Ts,)}
# *Ts is the repr() of Ts._unpacked, an instance of UnpackedTypeVarTuple

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 annotation is possible:

>>> foo = [1, 2, 3]
>>> def bar(*args: *foo): pass  # Equivalent to `*args: tuple(foo)`
>>> bar.__annotations__
{'args': (1, 2, 3)}

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 of Unpack and the extra set of square brackets. class Array(Generic[DType, *Shape]) is much easier to skim, while still marking Shape as special.
  • Intuitiveness. We think a user is more likely to intuitively understand the meaning of *Ts - especially when they see that Ts is a TypeVar**Tuple** - than the meaning of Unpack[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 [16]. 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 [17].

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], 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], 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

[1]'Batch' is machine learning parlance for 'a number of'.
[2]We use the term 'array' to refer to a matrix with an arbitrary number of dimensions. In NumPy, the corresponding class is the ndarray; in TensorFlow, the Tensor; and so on.
[3]If the shape begins with 'batch × time', then videos_batch[0][1] would select the second frame of the first video. If the shape begins with 'time × batch', then videos_batch[1][0] would select the same frame.

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 [19]: [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 specifying tf.function signatures in a Pythonic way, using type annotations (rather than the custom input_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.

References

[4](1, 2) Python typing issue #193: https://github.com/python/typing/issues/193
[5]Ivan Levkivskyi, 'Type system improvements', PyCon 2019: https://paper.dropbox.com/doc/Type-system-improvements-HHOkniMG9WcCgS0LzXZAe
[6](1, 2) Ivan Levkivskyi, 'Static typing of Python numeric stack', PyCon 2019: https://paper.dropbox.com/doc/Static-typing-of-Python-numeric-stack-summary-6ZQzTkgN6e0oXko8fEWwN
[7]Stephan Hoyer, 'Ideas for array shape typing in Python': https://docs.google.com/document/d/1vpMse4c6DrWH5rq2tQSx3qwP_m_0lyn-Ij4WHqQqRHY/edit
[8]Mark Mendoza, 'Variadic Type Variables for Decorators and Tensors', Python Typing Summit 2019: https://github.com/facebook/pyre-check/blob/ae85c0c6e99e3bbfc92ec55104bfdc5b9b3097b2/docs/Variadic_Type_Variables_for_Decorators_and_Tensors.pdf
[9]Matthew Rahtz et al., 'Shape annotation syntax proposal': https://docs.google.com/document/d/1But-hjet8-djv519HEKvBN6Ik2lW3yu0ojZo6pG9osY/edit
[10]Discussion on Python typing-sig mailing list: https://mail.python.org/archives/list/typing-sig@python.org/thread/SQVTQYWIOI4TIO7NNBTFFWFMSMS2TA4J/
[11]tsanley: https://github.com/ofnote/tsanley
[12]PyContracts: https://github.com/AndreaCensi/contracts
[13]ShapeGuard: https://github.com/Qwlouse/shapeguard
[14]https://mail.python.org/archives/list/python-dev@python.org/message/UDM7Y6HLHQBKXQEBIBD5ZLB5XNPDZDXV/
[15]https://mail.python.org/archives/list/python-dev@python.org/message/HTCARTYYCHETAMHB6OVRNR5EW5T2CP4J/
[16]https://github.com/python/cpython/pull/24527
[17]https://github.com/mrahtz/cpython/tree/pep637%2B646
[18]https://spinningup.openai.com/en/latest/spinningup/exercise2_2_soln.html
[19]https://github.com/tensorflow/community/pull/208
Source: https://github.com/python/peps/blob/master/pep-0646.rst