Proof of non-knowledge
Yes. ​ Just prove that such information doesn't exist.

(For example, that can be used to prove that one

doesn't know a boolean value x such that ​ x and not x .)

For P/poly provers, if such proofs exist then UNSAT has a usually-correct promiseMA protocol.

However, if public-coin collision-resistant hash families exist, then

a common random string is enough for arguments of non-knowledge.

For example, to argue that one does not know a string

such that ​ Hcommon_random_string(0||x) = y , ​ argue that one

does know a string x such that ​ Hcommon_random_string(1||x) = y .