In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base b for the exponent and logarithm (b is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base b < 1 is equivalent to using a negative sign and using the inverse 1/b > 1.[a] If not qualified, the base is conventionally taken to be e or 1/e, which corresponds to e with a negative.
The log semiring has the tropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity (max-plus semiring) or to zero (min-plus semiring), and thus can be viewed as a deformation ("quantization") of the tropical semiring. Notably, the addition operation, logadd (for multiple terms, LogSumExp) can be viewed as a deformation of maximum or minimum. The log semiring has applications in mathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see Decibel § Addition), log probability, or log-likelihoods.
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In mathematics, in the field of tropical analysis, the logsemiring is the semiring structure on the logarithmic scale, obtained by considering the extended...
and multiplicatively idempotent semiring, but sometimes also just idempotent semiring. The commutative, simple semirings with that property are exactly...
In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing...
Napier Level (logarithmic quantity) Log–log plot Logarithm Logarithmic mean Logsemiring Preferred number Semi-log plot Order of magnitude Entropy Entropy...
and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication)...
case of the Stolarsky mean. Logarithmic mean temperature difference Logsemiring Citations B. C. Carlson (1966). "Some inequalities for hypergeometric...
extended by an absorbing 0, forming the probability semiring, which is isomorphic to the logsemiring. Rational functions of the form f /g, where f and...
{\displaystyle \mathbb {R} _{\geq 0}} has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice...
arg min, corresponding to using the logsemiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg...
Extended complex plane Extended natural numbers Improper integral Infinity Logsemiring Series (mathematics) Projectively extended real line Computer representations...
instead. aProbLog generalizes ProbLog by allowing any commutative semiring instead of just probabilities. ProbFOIL: given a set of ProbLog facts as a probabilistic...
approach to these is to consider the two operations to be those of a semiring. Semiring multiplication is done along the path, and the addition is between...
on the tropical semiring. This is defined in two ways, depending on max or min convention. The min tropical semiring is the semiring ( R ∪ { + ∞ } , ⊕...
requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications...
Magnus ring over R. Given an alphabet Σ {\displaystyle \Sigma } and a semiring S {\displaystyle S} . The formal power series over S {\displaystyle S}...
generalization, this time in the context of a formal power series over a semiring. This approach gives rise to weighted rational expressions and weighted...
Strassen's algorithm works for any ring, such as plus/multiply, but not all semirings, such as min-plus or boolean algebra, where the naive algorithm still...
is valid more generally for two elements x and y in a ring, or even a semiring, provided that xy = yx. For example, it holds for two n × n matrices, provided...
automaton A is unambiguous, then the set of weight does not need to be a semiring, instead it suffices to consider a monoid. Indeed, there is at most one...
R r 1 + r 2 L ( x ) = ( log | x | v ) v {\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log |x|_{v})_{v}\end{cases}}}...
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