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

## TL;DR

This paper proves that if certain hard problems exist on average in NP, then related problems are also hard, implying that proving true theorems is not easier than finding witnesses for guaranteed true statements.

## Contribution

It establishes a novel connection between average-case hardness in NP and the complexity of proof and witness-finding, introducing a new round-collapse theorem for interactive puzzles.

## Key findings

- Existence of hard-on-average NP problems implies existence of one-way functions.
- Hard-on-average distributional NP search problems lead to hard promise-true problems.
- O(1)-round public-coin arguments imply NP/poly-hard problems.

## Abstract

Consider the following two fundamental open problems in complexity theory: (a) Does a hard-on-average language in NP imply the existence of one-way functions?, or (b) 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.