TY - GEN

T1 - Is it easier to prove theorems that are guaranteed to be true?

AU - Pass, Rafael

AU - Venkitasubramaniam, Muthuramakrishnan

N1 - Publisher Copyright: © 2020 IEEE.

PY - 2020/11

Y1 - 2020/11

N2 - 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- A nd 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.

AB - 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- A nd 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.

KW - TFNP, hard on the average, one way functions

UR - http://www.scopus.com/inward/record.url?scp=85100341380&partnerID=8YFLogxK

U2 - https://doi.org/10.1109/FOCS46700.2020.00119

DO - https://doi.org/10.1109/FOCS46700.2020.00119

M3 - منشور من مؤتمر

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 1255

EP - 1267

BT - Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020

PB - IEEE Computer Society

T2 - 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020

Y2 - 16 November 2020 through 19 November 2020

ER -