Log semiring

Here’s a strange way to do arithmetic on the real numbers.

First, we’ll need to include +∞ and -∞ with the reals.

We define the new addition of two elements x and y to be -log (exp(-x) + exp(-y) ).

We define the new multiplication to be ordinary addition. (!)

In this new arithmetic +∞ is the additive identity and 0 is the multiplicative identity.

This new algebraic structure is called the log semiring. It’s called a semiring because it satisfies all the properties of a ring except that elements don’t necessarily have additive inverses. We’ll get into the details of the definition below.

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Let’s put a subscript S on everything associated with our semiring in order to distinguish them from their more familiar counterparts. Then we can summarize the definitions above as

• a +_S  b = -log (exp(-a) + exp(-b) )
• a *_S  b = a + b
• 0_S = +∞
• 1_S = 0

Note that if we define

f(a, b) = a +_S  b

then

f(a, b) = -g(-a, -b)

where g(a, b) is the soft maximum of a and b.

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Finally, we list the axioms of a semiring. Note that these equations all hold when we interpret +, *, 0, and 1 in the context of S, i.e. we imagine that each has a subscript S and is as defined above.

• (a + b) + c = a + (b + c)
• 0 + a = a + 0 = a
• a + b = b + a
• (a * b) * c = a * (b * c)
• 1 * a = a * 1 = a
• a * (b + c) = (a * b) + (a * c)
• (a + b) * c = (a * c) + (b * c)
• 0 * a = a * 0 = 0

Each of these follows immediately from writing out the definitions.