I want to express the following two sentences in mathematical syntax (if possible):
The number of solutions to the problem should be finite and each solution should be of polynomial length
More difficult (maybe impossible to express in math syntax):
Given a Polynomial Length solution, we should be able to verify if an answer is true of false
Please Note: I just want to express these sentences as elegantly as possible, so please do not downvote me if this question does not meet the mathematical high standarts of yourself or this web site. If anything is unclear, please ask and I will edit my question to meet the high standarts of this community.