Is it Easier to Prove Theorems that are Guaranteed to be True?

Consider the following two fundamental open problems in complexity theory: •Does a hard-on-average language in NP imply the existence of one-way functions? •Does a hard-on-average language in NP imply a hard-on-average problem in TFNP (i.e., the class of total NP search problem)? Our main result is that the answer to (at least) one of these questions is yes. Both one-way functions and problems in TFNP can be interpreted as promise-true distributional NP search problems-namely, distributional search problems where the sampler only samples true statements. As a direct corollary of the above result, we thus get that the existence of a hard-on-average distributional NP search problem implies a hard-on-average promise-true distributional NP search problem. In other words, It is no easier to find witnesses (a.k.a. proofs) for efficiently-sampled statements (theorems) that are guaranteed to be true. This result follows from a more general study of interactive puzzles-a generalization of average-case hardness in NP- and in particular, a novel round-collapse theorem for computationally-sound protocols, analogous to Babai-Moran's celebrated round-collapse theorem for information-theoretically sound protocols. As another consequence of this treatment, we show that the existence of
$O(1)$
-round public-coin non-trivial arguments (i.e., argument systems that are not proofs) imply the existence of a hard-on-average problem in NP/poly.