Proofs of Proximity for Context-Free Languages and Read-Once Branching Programs

Abstract

Proofs of proximity are probabilistic proof systems in which the verifier only queries a sub-linear number of input bits, and soundness only means that, with high probability, the input is close to an accepting input. In their minimal form, called Merlin-Arthur proofs of proximity (${\mathcal {MAP}}$), the verifier receives, in addition to query access to the input, also free access to an explicitly given short (sub-linear) proof. A more general notion is that of an interactive proof of proximity (${\mathcal {IPP}}$), in which the verifier is allowed to interact with an all-powerful, yet untrusted, prover. ${\mathcal {MAP}}$s and ${\mathcal {IPP}}$s may be thought of as the ${\mathcal {NP}}$ and ${\mathcal {IP}}$ analogues of property testing, respectively.

In this work we construct proofs of proximity for two natural classes of properties: (1) context-free languages, and (2) languages accepted by small read-once branching programs. Our main results are:

• 1. ${\mathcal {MAP}}$s for these two classes, in which, for inputs of length $n$, both the verifier’s query complexity and the length of the ${\mathcal {MAP}}$ proof are $\widetilde{O}(\sqrt{n})$.

• 2. ${\mathcal {IPP}}$s for the same two classes with constant query complexity, poly-logarithmic communication complexity, and logarithmically many rounds of interaction.

Full version can be found in ECCC TR15-024 [GGR15].

This research was partially supported by the Israel Science Foundation (grant No. 671/13).