Log semiring

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. 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 b \to \infty (max-plus semiring) or to zero b \to 0 (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 ), log probability, or log-likelihoods. The operations on the log semiring can be defined extrinsically by mapping them to the non-negative real numbers, doing the operations there, and mapping them back.