Library Reference

Semiring{T}

Abstract Semiring type. T is the type of the value wrapped by the semiring structure (e.g. Float32, Bool, ...).

val(x::Semiring)

Return the value wrapped by the semiring type

valtype(x::Semiring{T})
valtype(::Type{<:Semiring{T}}) where T

Return the type T of the value wrapped by the semiring.

parse(S::Type{<:Semiring{T}}, str) where T

Return S(parse(T, str)).

zero(::Semiring)
zero(::Type{<:Semiring})

Return the additive neutral element of the semiring $\bar{0}$.

one(::Semiring)
one(::Type{<:Semiring})

Return the multiplicative neutral element of the semiring $\bar{1}$.

+(x::S, y::S) where S<:Semiring
x + y

Return $x \oplus y$ for the semiring S.

*(x::S, y::S) where S<:Semiring
x + y

Return $x \otimes y$ for the semiring S.

*(n::Integer, x::Semiring)
n * x
x * n

Return the n-times repeated addition $x \oplus x \oplus ...$.

inv(x::Semiring)

If defined, return the multiplicative inverse of x. Not implemented for semiring type that does not support multiplicative inverse.

SemiringTangent{T} <: ChainRulesCore.AbstractTangent

Tangent type of semirings type.

∂sum(u::S, x::S) where S<:Semiring

Compute $\frac{\partial u}{\partial x}$ where $u = x \oplus y \oplus z \oplus ...$.

∂rmul(x::S, a::S) where S<:Semiring

Compute $\frac{\partial u}{\partial x}$ where $u = x \otimes a$.

∂lmul(a::S, x::S) where S<:Semiring

Compute $\frac{\partial u}{\partial x}$ where $u = a \otimes x$.

ArcticSemiring{T<:AbstractFloat} <: Semiring{T}

Arctic semiring: $( \mathbb{R} \cup \{\infty \}, \max, +, -\infty, 0 )$.

See also LogSemiring and TropicalSemiring.

BoolSemiring <: Semiring{Bool}

Boolean semiring: $(\{\text{false}, \text{true} \}, \lor, \land, \text{false}, \text{true} )$ where $\lor$ and $\land$ are the logical "or" and logical "and operators.

ExpectationSemiring{T<:AbstractFloat,F} <: Semiring{T}

Expectation semiring: $( (\mathbb{R}_+, \mathbb{R}), +, \cdot, (0, 0), (1, 0) )$.

LogSemiring{T<:AbstractFloat,τ} <: Semiring{T}

Logarithmic semiring: $(\mathbb{R} \cup \{\text{sgn}(\tau) \cdot +\infty \}, \oplus_{\log}, +, -\infty, 0)$ where

\[x \oplus y = \frac{1}{\tau} \log ( e^{\tau x} + e^{\tau y} ).\]

TropicalSemiring{T<:AbstractFloat} <: Semiring{T}

Tropical semiring: $(\mathbb{R} \cup \{- \infty \}, \min, +, \infty, 0)$.

See also LogSemiring and ArcticSemiring.

exp(x::LogSemiring{T,1})::ProbSemiring{T}

Exponential morphism of the log-semiring.

See also Base.log.

log(x::ProbSemiring{T})::LogSemiring{T,1}

Logarithm morphism of the probaility semiring.

See also Base.exp.