semiring

semiring

Semiring = type['SemiringImpl'] module-attribute

ComplexLSESumSemiring

Bases: SemiringImpl

The complex log space computation.

Source code in cirkit/backend/torch/semiring.py

@SemiringImpl.register("complex-lse-sum")
class ComplexLSESumSemiring(SemiringImpl):
    """The complex log space computation."""

    @classmethod
    def cast(cls, x: Tensor) -> Tensor:
        if x.is_complex():
            return x
        if x.is_floating_point():
            return x.to(x.dtype.to_complex())
        default_float_dtype = torch.get_default_dtype()
        return x.to(default_float_dtype.to_complex())

    @classmethod
    def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        return x.logsumexp(dim=dim, keepdim=keepdim)

    @classmethod
    def add(cls, *xs: Tensor) -> Tensor:
        return functools.reduce(torch.logaddexp, xs)

    @classmethod
    def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        return x.sum(dim=dim, keepdim=keepdim)

    @classmethod
    def mul(cls, *xs: Tensor) -> Tensor:
        return functools.reduce(torch.add, xs)

    @classmethod
    def apply_reduce(
        cls,
        func: EinsumFunc,
        *xs: Tensor,
        dim: int,
        keepdim: bool,
    ) -> Tensor:
        # NOTE: Due to usage of intermediate results, they need to be instantiated in lists but not
        #       generators, because generators can't save much if we want to reuse.
        max_xs = [
            torch.clamp(
                torch.amax(xi.real, dim=dim, keepdim=True),
                min=torch.finfo(xi.real.dtype).min,
                max=torch.finfo(xi.real.dtype).max,
            )
            for xi in xs
        ]
        exp_xs = [torch.exp(xi - max_xi) for xi, max_xi in zip(xs, max_xs)]

        # NOTE: exp_x is not tuple, but list still can be unpacked with *.
        func_exp_xs = func(*cast(tuple[Tensor, ...], exp_xs))

        reduced_max_xs = functools.reduce(torch.add, max_xs)  # Do n-1 add instead of n.
        if not keepdim:
            reduced_max_xs = reduced_max_xs.squeeze(dim)  # To match shape of func_exp_x.

        # Compute log(x) and its gradients safely where x is a complex tensor.
        # The problem is that if x = 0 + 0j, then the complex gradient of log(x) yields NaNs.
        # Note that for real non-monotonic circuits this problem cannot be avoided by simply
        # clipping the parameters of e.g., dense layers. In fact, even if we clipped the
        # parameters to be sufficiently far from zero here, cancellations would still arise
        # from negations, which in turn might result in under-flows. This has been observed in
        # float32 for squared non-monotonic PCs with real parameters.
        # To solve this issue, here we use a 'safe' version of the complex logarithm whose gradients
        # are replaced with zero if NaN and to the largest/lowest representable values if +inf/-inf.
        return csafelog(func_exp_xs) + reduced_max_xs

add(*xs) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def add(cls, *xs: Tensor) -> Tensor:
    return functools.reduce(torch.logaddexp, xs)

apply_reduce(func, *xs, dim, keepdim) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def apply_reduce(
    cls,
    func: EinsumFunc,
    *xs: Tensor,
    dim: int,
    keepdim: bool,
) -> Tensor:
    # NOTE: Due to usage of intermediate results, they need to be instantiated in lists but not
    #       generators, because generators can't save much if we want to reuse.
    max_xs = [
        torch.clamp(
            torch.amax(xi.real, dim=dim, keepdim=True),
            min=torch.finfo(xi.real.dtype).min,
            max=torch.finfo(xi.real.dtype).max,
        )
        for xi in xs
    ]
    exp_xs = [torch.exp(xi - max_xi) for xi, max_xi in zip(xs, max_xs)]

    # NOTE: exp_x is not tuple, but list still can be unpacked with *.
    func_exp_xs = func(*cast(tuple[Tensor, ...], exp_xs))

    reduced_max_xs = functools.reduce(torch.add, max_xs)  # Do n-1 add instead of n.
    if not keepdim:
        reduced_max_xs = reduced_max_xs.squeeze(dim)  # To match shape of func_exp_x.

    # Compute log(x) and its gradients safely where x is a complex tensor.
    # The problem is that if x = 0 + 0j, then the complex gradient of log(x) yields NaNs.
    # Note that for real non-monotonic circuits this problem cannot be avoided by simply
    # clipping the parameters of e.g., dense layers. In fact, even if we clipped the
    # parameters to be sufficiently far from zero here, cancellations would still arise
    # from negations, which in turn might result in under-flows. This has been observed in
    # float32 for squared non-monotonic PCs with real parameters.
    # To solve this issue, here we use a 'safe' version of the complex logarithm whose gradients
    # are replaced with zero if NaN and to the largest/lowest representable values if +inf/-inf.
    return csafelog(func_exp_xs) + reduced_max_xs

cast(x) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def cast(cls, x: Tensor) -> Tensor:
    if x.is_complex():
        return x
    if x.is_floating_point():
        return x.to(x.dtype.to_complex())
    default_float_dtype = torch.get_default_dtype()
    return x.to(default_float_dtype.to_complex())

mul(*xs) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def mul(cls, *xs: Tensor) -> Tensor:
    return functools.reduce(torch.add, xs)

prod(x, dim, *, keepdim=False) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    return x.sum(dim=dim, keepdim=keepdim)

sum(x, dim, *, keepdim=False) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    return x.logsumexp(dim=dim, keepdim=keepdim)

EinsumFunc

Bases: Protocol

Source code in cirkit/backend/torch/semiring.py

class EinsumFunc(Protocol):
    def __call__(self, *xs: Tensor) -> Tensor: ...

__call__(*xs)

Source code in cirkit/backend/torch/semiring.py

def __call__(self, *xs: Tensor) -> Tensor: ...

LSESumSemiring

Bases: SemiringImpl

The log space computation.

Source code in cirkit/backend/torch/semiring.py

@SemiringImpl.register("lse-sum")
class LSESumSemiring(SemiringImpl):
    """The log space computation."""

    @classmethod
    def cast(cls, x: Tensor) -> Tensor:
        if x.is_floating_point():
            return x
        if not x.is_complex():
            default_float_dtype = torch.get_default_dtype()
            return x.to(default_float_dtype)
        raise ValueError(f"Cannot cast a tensor of type '{x.dtype}' to the '{cls.__name__}'")

    @classmethod
    def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        return x.logsumexp(dim=dim, keepdim=keepdim)

    @classmethod
    def add(cls, *xs: Tensor) -> Tensor:
        return functools.reduce(torch.logaddexp, xs)

    @classmethod
    def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        return x.sum(dim=dim, keepdim=keepdim)

    @classmethod
    def mul(cls, *xs: Tensor) -> Tensor:
        return functools.reduce(torch.add, xs)

    @classmethod
    def apply_reduce(
        cls,
        func: EinsumFunc,
        *xs: Tensor,
        dim: int,
        keepdim: bool,
    ) -> Tensor:
        # NOTE: Due to usage of intermediate results, they need to be instantiated in lists but not
        #       generators, because generators can't save much if we want to reuse.
        max_xs = [
            torch.clamp(
                torch.amax(xi, dim=dim, keepdim=True),
                min=torch.finfo(xi.dtype).min,
                max=torch.finfo(xi.dtype).max,
            )
            for xi in xs
        ]
        exp_xs = [torch.exp(xi - max_xi) for xi, max_xi in zip(xs, max_xs)]

        # NOTE: exp_x is not tuple, but list still can be unpacked with *.
        func_exp_xs = func(*cast(tuple[Tensor, ...], exp_xs))

        reduced_max_xs = functools.reduce(torch.add, max_xs)  # Do n-1 add instead of n.
        if not keepdim:
            reduced_max_xs = reduced_max_xs.squeeze(dim)  # To match shape of func_exp_x.
        return torch.log(func_exp_xs) + reduced_max_xs

add(*xs) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def add(cls, *xs: Tensor) -> Tensor:
    return functools.reduce(torch.logaddexp, xs)

apply_reduce(func, *xs, dim, keepdim) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def apply_reduce(
    cls,
    func: EinsumFunc,
    *xs: Tensor,
    dim: int,
    keepdim: bool,
) -> Tensor:
    # NOTE: Due to usage of intermediate results, they need to be instantiated in lists but not
    #       generators, because generators can't save much if we want to reuse.
    max_xs = [
        torch.clamp(
            torch.amax(xi, dim=dim, keepdim=True),
            min=torch.finfo(xi.dtype).min,
            max=torch.finfo(xi.dtype).max,
        )
        for xi in xs
    ]
    exp_xs = [torch.exp(xi - max_xi) for xi, max_xi in zip(xs, max_xs)]

    # NOTE: exp_x is not tuple, but list still can be unpacked with *.
    func_exp_xs = func(*cast(tuple[Tensor, ...], exp_xs))

    reduced_max_xs = functools.reduce(torch.add, max_xs)  # Do n-1 add instead of n.
    if not keepdim:
        reduced_max_xs = reduced_max_xs.squeeze(dim)  # To match shape of func_exp_x.
    return torch.log(func_exp_xs) + reduced_max_xs

cast(x) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def cast(cls, x: Tensor) -> Tensor:
    if x.is_floating_point():
        return x
    if not x.is_complex():
        default_float_dtype = torch.get_default_dtype()
        return x.to(default_float_dtype)
    raise ValueError(f"Cannot cast a tensor of type '{x.dtype}' to the '{cls.__name__}'")

mul(*xs) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def mul(cls, *xs: Tensor) -> Tensor:
    return functools.reduce(torch.add, xs)

prod(x, dim, *, keepdim=False) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    return x.sum(dim=dim, keepdim=keepdim)

sum(x, dim, *, keepdim=False) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    return x.logsumexp(dim=dim, keepdim=keepdim)

SemiringImpl

Bases: ABC

The abstract base class for semiring implementations.

Due to numerical precision, the actual units in computational graph may hold values in, e.g., log space, instead of linear space. And therefore, this provides a unified interface for the computations so that computation can be done in a space suitable to the implementation regardless of the global setting.

Source code in cirkit/backend/torch/semiring.py

class SemiringImpl(ABC):
    """The abstract base class for semiring implementations.

    Due to numerical precision, the actual units in computational graph may hold values in, e.g., \
    log space, instead of linear space. And therefore, this provides a unified interface for the \
    computations so that computation can be done in a space suitable to the implementation \
    regardless of the global setting.
    """

    # A registry from semiring string identifiers to semiring class implementations
    _registry: ClassVar[dict[str, Semiring]] = {}

    # A registry of morphisms between semiring class implementations
    _registry_morphisms: ClassVar[dict[tuple[Semiring, Semiring], Callable[[Tensor], Tensor]]] = {}

    @staticmethod
    def register(name: str) -> Callable[[Semiring], Semiring]:
        """Register a concrete semiring implementation by its name.

        Args:
            name: The name to register.

        Returns:
            Callable[[Semiring], Semiring]: The class decorator to register a subclass.
        """

        def _decorator(cls: Semiring) -> Semiring:
            """Register a concrete semiring implementation by its name.

            Args:
                cls: The semiring subclass to register.

            Returns:
                Semiring: The class passed in.
            """
            SemiringImpl._registry[name] = cls
            return cls

        return _decorator

    @classmethod
    def register_map_from(
        cls, other: Semiring
    ) -> Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]:
        """Register a concrete semiring morphism implementation.

        Args:
            other: The source semiring.

        Returns:
            Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]:
            The function decorator to register the morphism.
        """

        def _decorator(func: Callable[[Tensor], Tensor]) -> Callable[[Tensor], Tensor]:
            """Register a concrete semiring morphism implementation.

            Args:
                func: The morphism between semirings to register.

            Returns:
                Callable[[Tensor], Tensor]: The morphism passed in.
            """
            SemiringImpl._registry_morphisms[(other, cls)] = func
            return func

        return _decorator

    @staticmethod
    def list() -> Iterable[str]:
        """List all semiring names registered.

        Returns:
            Iterable[str]: An iterable over all names available.
        """
        return iter(SemiringImpl._registry)

    @staticmethod
    def from_name(name: str) -> Semiring:
        """Get a semiring by its registered name.

        Args:
            name (str): The name to probe.

        Returns:
            Semiring: The retrieved concrete Semiring.
        """
        if name not in SemiringImpl._registry:
            raise IndexError(
                f"Unknown semiring '{name}'."
                f" Use @SemiringImpl.register(<name>) to register a new semiring"
            )
        return SemiringImpl._registry[name]

    @classmethod
    def map_from(cls, x: Tensor, semiring: Semiring) -> Tensor:
        """Map a tensor from the given semiring to `this` semiring.

        Args:
            x:
            semiring:

        Returns:

        """
        if cls == semiring:
            return x
        func: Callable[[Tensor], Tensor] | None = SemiringImpl._registry_morphisms.get(
            (semiring, cls), None
        )
        if func is None:
            raise NotImplementedError(
                f"Semiring map from '{semiring.__name__}' to '{cls.__name__}' is not implemented"
            )
        return func(x)

    def __new__(cls) -> "SemiringImpl":
        """Raise an error when this class is instantiated.

        Raises:
            TypeError: When this class is instantiated.

        Returns:
            SemiringImpl: This method never returns.
        """
        raise TypeError("This class cannot be instantiated")

    @classmethod
    def einsum(
        cls,
        equation: str | Sequence[Sequence[int]],
        *,
        inputs: tuple[Tensor, ...] | None = None,
        operands: tuple[Tensor, ...] | None = None,
        dim: int,
        keepdim: bool,
    ) -> Tensor:
        """Perform an einsum operation where sums and products are specified by the semiring.

        Args:
            equation: The einsum expression.
            inputs:  The inputs of the einsum.
            operands: Additional operands to pass to the einsum, after the inputs in the
                einsum expression.
            dim: The dimension of the inputs that get summed over in the einsum expression.
                This is useful to make the einsum computationally stable in some semirings,
                e.g., the log-sum-exp semiring.
            keepdim: Whether to keep the dimension that get summed over in the einsum
                expression.

        Returns:
            Tensor: the result of the einsum operation over the semiring.
        """
        # TODO: We need to remove this super general yet extremely complicated and hard
        #  to maintain einsum definition, which depends on the semiring. A future version of the
        #  compiler in cirkit will be able to emit pytorch code for every layer at compile time
        if inputs is None:
            inputs = ()
        if operands is None:
            operands = ()
        match equation:
            case str():

                def _einsum_str_func(*xs: Tensor) -> Tensor:
                    opds = tuple(cls.cast(opd) for opd in operands)
                    return torch.einsum(equation, *xs, *opds)

                einsum_func = _einsum_str_func
            case Sequence():

                def _einsum_seq_func(*xs: Tensor) -> Tensor:
                    opds = tuple(cls.cast(opd) for opd in operands)
                    einsum_args = tuple(
                        itertools.chain.from_iterable(zip(xs + opds, equation[:-1]))
                    )
                    return torch.einsum(*einsum_args, equation[-1])

                einsum_func = _einsum_seq_func
            case _:
                raise ValueError(
                    "The einsum expression must be either a string or a sequence of int sequences"
                )

        return cls.apply_reduce(einsum_func, *inputs, dim=dim, keepdim=keepdim)

    # NOTE: Subclasses should not touch any of the above final static methods but should implement
    #       all the following abstract class methods, and subclasses should be @final.

    @classmethod
    @abstractmethod
    def cast(cls, x: Tensor) -> Tensor:
        """Cast a tensor to the data type required by `this` semiring.

        Args:
            x: The tensor.

        Returns:
            Tensor: The tensor converted to the required data type.
        """

    @classmethod
    @abstractmethod
    def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        """

        Args:
            x:
            dim:
            keepdim:

        Returns:

        """

    @classmethod
    @abstractmethod
    def add(cls, *xs: Tensor) -> Tensor:
        """

        Args:
            *xs:

        Returns:

        """

    @classmethod
    @abstractmethod
    def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        """Do the product within a tensor on given dim(s).

        Args:
            x: The input tensor.
            dim: The dimension to reduce along.
            keepdim: Whether the dim is kept as a size-1 dim. Defaults to False.

        Returns:
            Tensor: The product result.
        """

    @classmethod
    @abstractmethod
    def mul(cls, *xs: Tensor) -> Tensor:
        """Multiply broadcastable tensors.

        Args:
            *xs (Tensor): The input tensors, should have broadcastable shapes.

        Returns:
            Tensor: The multiply result.
        """

    @classmethod
    @abstractmethod
    def apply_reduce(
        cls,
        func: EinsumFunc,
        *xs: Tensor,
        dim: int,
        keepdim: bool,
    ) -> Tensor:
        """Apply a sum-like functions to the tensor(s).

        The sum units may perform not just plain sum, but also weighted sum or even einsum. In \
        fact, it can possibly be any function that is linear to the each input. All that kind of \
        func can be used here.

        It is expected that func always does computation in the linear space, as with numerical \
        tricks, only relatively significant numbers contribute the final answer, and underflow \
        will not affect much. However, the input/output values may still be in another space, and \
        needs to be projected here.

        Args:
            func (Callable[[Unpack[Ts]], Tensor]): The sum-like function to be applied.
            *xs (Unpack[Ts]): The input tensors. Type expected to be Tensor.
            dim (int): The dimension along which the values are \
                correlated and must be scaled together, i.e., the dim to sum along. This should \
                match the actual operation done by func. The same dim is shared among all inputs.
            keepdim (bool): Whether the dim is kept as a size-1 dim, should match the actual \
                operation done by func.

        Returns:
            Tensor: The sum result.
        """

__new__()

Raise an error when this class is instantiated.

Raises:

Type	Description
TypeError	When this class is instantiated.

Returns:

Name	Type	Description
SemiringImpl	SemiringImpl	This method never returns.

Source code in cirkit/backend/torch/semiring.py

def __new__(cls) -> "SemiringImpl":
    """Raise an error when this class is instantiated.

    Raises:
        TypeError: When this class is instantiated.

    Returns:
        SemiringImpl: This method never returns.
    """
    raise TypeError("This class cannot be instantiated")

add(*xs) abstractmethod classmethod

Parameters:

Name	Type	Description	Default
*xs	Tensor		()

Returns:

Source code in cirkit/backend/torch/semiring.py

@classmethod
@abstractmethod
def add(cls, *xs: Tensor) -> Tensor:
    """

    Args:
        *xs:

    Returns:

    """

apply_reduce(func, *xs, dim, keepdim) abstractmethod classmethod

Apply a sum-like functions to the tensor(s).

The sum units may perform not just plain sum, but also weighted sum or even einsum. In fact, it can possibly be any function that is linear to the each input. All that kind of func can be used here.

It is expected that func always does computation in the linear space, as with numerical tricks, only relatively significant numbers contribute the final answer, and underflow will not affect much. However, the input/output values may still be in another space, and needs to be projected here.

Parameters:

Name	Type	Description	Default
func	Callable[[Unpack[Ts]], Tensor]	The sum-like function to be applied.	required
*xs	Unpack[Ts]	The input tensors. Type expected to be Tensor.	()
dim	int	The dimension along which the values are correlated and must be scaled together, i.e., the dim to sum along. This should match the actual operation done by func. The same dim is shared among all inputs.	required
keepdim	bool	Whether the dim is kept as a size-1 dim, should match the actual operation done by func.	required

Returns:

Name	Type	Description
Tensor	Tensor	The sum result.

Source code in cirkit/backend/torch/semiring.py

@classmethod
@abstractmethod
def apply_reduce(
    cls,
    func: EinsumFunc,
    *xs: Tensor,
    dim: int,
    keepdim: bool,
) -> Tensor:
    """Apply a sum-like functions to the tensor(s).

    The sum units may perform not just plain sum, but also weighted sum or even einsum. In \
    fact, it can possibly be any function that is linear to the each input. All that kind of \
    func can be used here.

    It is expected that func always does computation in the linear space, as with numerical \
    tricks, only relatively significant numbers contribute the final answer, and underflow \
    will not affect much. However, the input/output values may still be in another space, and \
    needs to be projected here.

    Args:
        func (Callable[[Unpack[Ts]], Tensor]): The sum-like function to be applied.
        *xs (Unpack[Ts]): The input tensors. Type expected to be Tensor.
        dim (int): The dimension along which the values are \
            correlated and must be scaled together, i.e., the dim to sum along. This should \
            match the actual operation done by func. The same dim is shared among all inputs.
        keepdim (bool): Whether the dim is kept as a size-1 dim, should match the actual \
            operation done by func.

    Returns:
        Tensor: The sum result.
    """

cast(x) abstractmethod classmethod

Cast a tensor to the data type required by this semiring.

Parameters:

Name	Type	Description	Default
x	Tensor	The tensor.	required

Returns:

Name	Type	Description
Tensor	Tensor	The tensor converted to the required data type.

Source code in cirkit/backend/torch/semiring.py

@classmethod
@abstractmethod
def cast(cls, x: Tensor) -> Tensor:
    """Cast a tensor to the data type required by `this` semiring.

    Args:
        x: The tensor.

    Returns:
        Tensor: The tensor converted to the required data type.
    """

einsum(equation, *, inputs=None, operands=None, dim, keepdim) classmethod

Perform an einsum operation where sums and products are specified by the semiring.

Parameters:

Name	Type	Description	Default
equation	str | Sequence[Sequence[int]]	The einsum expression.	required
inputs	tuple[Tensor, ...] | None	The inputs of the einsum.	None
operands	tuple[Tensor, ...] | None	Additional operands to pass to the einsum, after the inputs in the einsum expression.	None
dim	int	The dimension of the inputs that get summed over in the einsum expression. This is useful to make the einsum computationally stable in some semirings, e.g., the log-sum-exp semiring.	required
keepdim	bool	Whether to keep the dimension that get summed over in the einsum expression.	required

Returns:

Name	Type	Description
Tensor	Tensor	the result of the einsum operation over the semiring.

Source code in cirkit/backend/torch/semiring.py

@classmethod
def einsum(
    cls,
    equation: str | Sequence[Sequence[int]],
    *,
    inputs: tuple[Tensor, ...] | None = None,
    operands: tuple[Tensor, ...] | None = None,
    dim: int,
    keepdim: bool,
) -> Tensor:
    """Perform an einsum operation where sums and products are specified by the semiring.

    Args:
        equation: The einsum expression.
        inputs:  The inputs of the einsum.
        operands: Additional operands to pass to the einsum, after the inputs in the
            einsum expression.
        dim: The dimension of the inputs that get summed over in the einsum expression.
            This is useful to make the einsum computationally stable in some semirings,
            e.g., the log-sum-exp semiring.
        keepdim: Whether to keep the dimension that get summed over in the einsum
            expression.

    Returns:
        Tensor: the result of the einsum operation over the semiring.
    """
    # TODO: We need to remove this super general yet extremely complicated and hard
    #  to maintain einsum definition, which depends on the semiring. A future version of the
    #  compiler in cirkit will be able to emit pytorch code for every layer at compile time
    if inputs is None:
        inputs = ()
    if operands is None:
        operands = ()
    match equation:
        case str():

            def _einsum_str_func(*xs: Tensor) -> Tensor:
                opds = tuple(cls.cast(opd) for opd in operands)
                return torch.einsum(equation, *xs, *opds)

            einsum_func = _einsum_str_func
        case Sequence():

            def _einsum_seq_func(*xs: Tensor) -> Tensor:
                opds = tuple(cls.cast(opd) for opd in operands)
                einsum_args = tuple(
                    itertools.chain.from_iterable(zip(xs + opds, equation[:-1]))
                )
                return torch.einsum(*einsum_args, equation[-1])

            einsum_func = _einsum_seq_func
        case _:
            raise ValueError(
                "The einsum expression must be either a string or a sequence of int sequences"
            )

    return cls.apply_reduce(einsum_func, *inputs, dim=dim, keepdim=keepdim)

from_name(name) staticmethod

Get a semiring by its registered name.

Parameters:

Name	Type	Description	Default
name	str	The name to probe.	required

Returns:

Name	Type	Description
Semiring	Semiring	The retrieved concrete Semiring.

Source code in cirkit/backend/torch/semiring.py

@staticmethod
def from_name(name: str) -> Semiring:
    """Get a semiring by its registered name.

    Args:
        name (str): The name to probe.

    Returns:
        Semiring: The retrieved concrete Semiring.
    """
    if name not in SemiringImpl._registry:
        raise IndexError(
            f"Unknown semiring '{name}'."
            f" Use @SemiringImpl.register(<name>) to register a new semiring"
        )
    return SemiringImpl._registry[name]

list() staticmethod

List all semiring names registered.

Returns:

Type	Description
Iterable[str]	Iterable[str]: An iterable over all names available.

Source code in cirkit/backend/torch/semiring.py

@staticmethod
def list() -> Iterable[str]:
    """List all semiring names registered.

    Returns:
        Iterable[str]: An iterable over all names available.
    """
    return iter(SemiringImpl._registry)

map_from(x, semiring) classmethod

Map a tensor from the given semiring to this semiring.

Parameters:

Name	Type	Description	Default
x	Tensor		required
semiring	Semiring		required

Returns:

Source code in cirkit/backend/torch/semiring.py

@classmethod
def map_from(cls, x: Tensor, semiring: Semiring) -> Tensor:
    """Map a tensor from the given semiring to `this` semiring.

    Args:
        x:
        semiring:

    Returns:

    """
    if cls == semiring:
        return x
    func: Callable[[Tensor], Tensor] | None = SemiringImpl._registry_morphisms.get(
        (semiring, cls), None
    )
    if func is None:
        raise NotImplementedError(
            f"Semiring map from '{semiring.__name__}' to '{cls.__name__}' is not implemented"
        )
    return func(x)

mul(*xs) abstractmethod classmethod

Multiply broadcastable tensors.

Parameters:

Name	Type	Description	Default
*xs	Tensor	The input tensors, should have broadcastable shapes.	()

Returns:

Name	Type	Description
Tensor	Tensor	The multiply result.

Source code in cirkit/backend/torch/semiring.py

@classmethod
@abstractmethod
def mul(cls, *xs: Tensor) -> Tensor:
    """Multiply broadcastable tensors.

    Args:
        *xs (Tensor): The input tensors, should have broadcastable shapes.

    Returns:
        Tensor: The multiply result.
    """

prod(x, dim, *, keepdim=False) abstractmethod classmethod

Do the product within a tensor on given dim(s).

Parameters:

Name	Type	Description	Default
x	Tensor	The input tensor.	required
dim	int	The dimension to reduce along.	required
keepdim	bool	Whether the dim is kept as a size-1 dim. Defaults to False.	False

Returns:

Name	Type	Description
Tensor	Tensor	The product result.

Source code in cirkit/backend/torch/semiring.py

@classmethod
@abstractmethod
def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    """Do the product within a tensor on given dim(s).

    Args:
        x: The input tensor.
        dim: The dimension to reduce along.
        keepdim: Whether the dim is kept as a size-1 dim. Defaults to False.

    Returns:
        Tensor: The product result.
    """

register(name) staticmethod

Register a concrete semiring implementation by its name.

Parameters:

Name	Type	Description	Default
name	str	The name to register.	required

Returns:

Type	Description
Callable[[Semiring], Semiring]	Callable[[Semiring], Semiring]: The class decorator to register a subclass.

Source code in cirkit/backend/torch/semiring.py

@staticmethod
def register(name: str) -> Callable[[Semiring], Semiring]:
    """Register a concrete semiring implementation by its name.

    Args:
        name: The name to register.

    Returns:
        Callable[[Semiring], Semiring]: The class decorator to register a subclass.
    """

    def _decorator(cls: Semiring) -> Semiring:
        """Register a concrete semiring implementation by its name.

        Args:
            cls: The semiring subclass to register.

        Returns:
            Semiring: The class passed in.
        """
        SemiringImpl._registry[name] = cls
        return cls

    return _decorator

register_map_from(other) classmethod

Register a concrete semiring morphism implementation.

Parameters:

Name	Type	Description	Default
other	Semiring	The source semiring.	required

Returns:

Type	Description
Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]	Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]:
Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]	The function decorator to register the morphism.

Source code in cirkit/backend/torch/semiring.py

@classmethod
def register_map_from(
    cls, other: Semiring
) -> Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]:
    """Register a concrete semiring morphism implementation.

    Args:
        other: The source semiring.

    Returns:
        Callable[[Callable[[Tensor], Tensor]], Callable[[Tensor], Tensor]]:
        The function decorator to register the morphism.
    """

    def _decorator(func: Callable[[Tensor], Tensor]) -> Callable[[Tensor], Tensor]:
        """Register a concrete semiring morphism implementation.

        Args:
            func: The morphism between semirings to register.

        Returns:
            Callable[[Tensor], Tensor]: The morphism passed in.
        """
        SemiringImpl._registry_morphisms[(other, cls)] = func
        return func

    return _decorator

sum(x, dim, *, keepdim=False) abstractmethod classmethod

Parameters:

Name	Type	Description	Default
x	Tensor		required
dim	int		required
keepdim	bool		False

Returns:

Source code in cirkit/backend/torch/semiring.py

@classmethod
@abstractmethod
def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    """

    Args:
        x:
        dim:
        keepdim:

    Returns:

    """

SumProductSemiring

Bases: SemiringImpl

The linear space computation.

Source code in cirkit/backend/torch/semiring.py

@SemiringImpl.register("sum-product")
class SumProductSemiring(SemiringImpl):
    """The linear space computation."""

    @classmethod
    def cast(cls, x: Tensor) -> Tensor:
        """Cast a tensor to the data type required by the semiring.

        Args:
            x: The tensor.

        Returns:
            Tensor: The tensor converted to the required data type.
        """
        if x.is_floating_point():
            return x
        if not x.is_complex():
            default_float_dtype = torch.get_default_dtype()
            return x.to(default_float_dtype)
        raise ValueError(f"Cannot cast a tensor of type '{x.dtype}' to the '{cls.__name__}'")

    @classmethod
    def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        return x.sum(dim=dim, keepdim=keepdim)

    @classmethod
    def add(cls, *xs: Tensor) -> Tensor:
        return functools.reduce(torch.add, xs)

    @classmethod
    def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
        return torch.prod(x, dim=dim, keepdim=keepdim)

    @classmethod
    def mul(cls, *xs: Tensor) -> Tensor:
        return functools.reduce(torch.mul, xs)

    @classmethod
    def apply_reduce(
        cls,
        func: EinsumFunc,
        *xs: Tensor,
        dim: int,
        keepdim: bool,
    ) -> Tensor:
        return func(*xs)

add(*xs) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def add(cls, *xs: Tensor) -> Tensor:
    return functools.reduce(torch.add, xs)

apply_reduce(func, *xs, dim, keepdim) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def apply_reduce(
    cls,
    func: EinsumFunc,
    *xs: Tensor,
    dim: int,
    keepdim: bool,
) -> Tensor:
    return func(*xs)

cast(x) classmethod

Cast a tensor to the data type required by the semiring.

Parameters:

Name	Type	Description	Default
x	Tensor	The tensor.	required

Returns:

Name	Type	Description
Tensor	Tensor	The tensor converted to the required data type.

Source code in cirkit/backend/torch/semiring.py

@classmethod
def cast(cls, x: Tensor) -> Tensor:
    """Cast a tensor to the data type required by the semiring.

    Args:
        x: The tensor.

    Returns:
        Tensor: The tensor converted to the required data type.
    """
    if x.is_floating_point():
        return x
    if not x.is_complex():
        default_float_dtype = torch.get_default_dtype()
        return x.to(default_float_dtype)
    raise ValueError(f"Cannot cast a tensor of type '{x.dtype}' to the '{cls.__name__}'")

mul(*xs) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def mul(cls, *xs: Tensor) -> Tensor:
    return functools.reduce(torch.mul, xs)

prod(x, dim, *, keepdim=False) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def prod(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    return torch.prod(x, dim=dim, keepdim=keepdim)

sum(x, dim, *, keepdim=False) classmethod

Source code in cirkit/backend/torch/semiring.py

@classmethod
def sum(cls, x: Tensor, dim: int, *, keepdim: bool = False) -> Tensor:
    return x.sum(dim=dim, keepdim=keepdim)

_(x)

Source code in cirkit/backend/torch/semiring.py

@ComplexLSESumSemiring.register_map_from(LSESumSemiring)
def _(x: Tensor) -> Tensor:
    return ComplexLSESumSemiring.cast(x)