They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff.
The numbers r, α, and β are collectively known as the format of the Pfaffian function, and give a useful measure of its complexity.
In the 1990s, Alex Wilkie showed that one has the same result if instead of adding every restricted analytic function, one just adds the unrestricted exponential function to R to get the ordered real field with exponentiation, Rexp, a result known as Wilkie's theorem.
[3] Wilkie also tackled the question of which finite sets of analytic functions could be added to R to get a model-completeness result.
It turned out that adding any Pfaffian chain restricted to the box [0, 1]m would give the same result.