$\begingroup$

I was recently reading The Two Dualities of Computation: Negative and Fractional Types. The paper expands on sum-types and product-types, giving semantics to the types a - b and a/b .

Unlike addition and multiplication, there are not one but two inverses of exponentiation, logarithms and rooting. If function types (a → b) are type-theoretic exponentiation, given the type a → b (or b^a ) what does it mean to have the type logb(c) or the type a√c ?

Does it make sense to extend logarithms and roots to types at all?

If so, has there been any work in this area, and what are some good directions on how to comprehend the repercussions?

I tried looking up information on this via logic, hoping the Curry-Howard correspondence could help me, but to no avail.