A note on the size of Craig Interpolants

Mundici considered the question of whether the interpolant of two propositional formulas of the form F → G can always have a short circuit description, and showed that if this is the case then every problem in NP ∩ co-NP would have polynomial size circuits. In this note we observe further consequences of the interpolant having short circuit descriptions, namely that UP ⊆ P/poly, and that every single valued NP function has a total extension in FP/poly. We also relate this question with other Complexity Theory assumptions.