On the Impossibility of Probabilistic Proofs in Relativized Worlds

Abstract

We initiate the systematic study of probabilistic proofs in relativized worlds. The goal is to understand, for a given oracle, if there exist non-trivial probabilistic proofs for checking deterministic or nondeterministic computations that make queries to the oracle. This question is intimately related to a recent line of work that builds cryptographic primitives (e.g., hash functions) via constructions that are friendly to known probabilistic proofs. This improves the efficiency of probabilistic proofs for computations calling these primitives. We prove that non-trivial probabilistic proofs relative to several natural oracles do not exist. Our results provide strong complexity-theoretic evidence that certain functionalities cannot be treated as black boxes, and thus investing effort to instantiate these functionalities via constructions tailored to known probabilistic proofs may be inherent