We explore the power of interactive proofs with a distributed verifier. In this
setting, the verifier consists of $n$ nodes and a graph $G$ that defines their
communication pattern. The prover is a single entity that communicates with all
nodes by short messages.
The goal is to verify that the graph $G$ belongs to some language in a small
number of rounds, and with small communication bound, \ie the proof size.

This interactive model was introduced by Kol, Oshman and Saxena (PODC 2018) as a generalization of non-interactive distributed proofs. They demonstrated the power of interaction in this setting by constructing protocols for problems as Graph Symmetry and Graph Non-Isomorphism -- both of which require proofs of $\Omega(n^2)$-bits without interaction.

In this work, we provide a new general framework for distributed interactive proofs that allows one to
translate standard interactive protocols (i.e., with a centralized verifier) to ones where the verifier is distributed with a proof size that depends on the computational complexity of the verification algorithm run by the centralized verifier.
We show the following:
* Every (centralized) computation performed in time $O(n)$ on a RAM can
be translated into three-round distributed interactive protocol with
$O(\log n)$ proof size. This implies that many graph problems for sparse graphs
have succinct proofs (e.g., testing planarity).

* Every (centralized) computation implemented by either a small space or by uniform $NC$ circuit can
be translated into a
distributed protocol with $O(1)$ rounds and $O(\log n)$ bits proof size for the low space
case and $polylog(n)$ many rounds and proof size for $NC$.

* We show that for Graph Non-Isomorphism, one of the striking demonstrations of
the power of interaction, there is a 4-round protocol with $O(\log n)$ proof size, improving upon the $O(n \log n)$ proof size of Kol et al.

* For many problems, we show how to reduce proof
size
below the seemingly natural barrier of $\log n$. By employing our RAM compiler,
we get a 5-round
protocol with proof size $O(\log \log n)$ for a family of problems including
Fixed
Automorphism, Clique and Leader Election (for the latter two problems we
actually get $O(1)$ proof size).
* Finally, we discuss how to make these proofs non-interactive {\em
arguments} via random
oracles.

Our compilers capture many natural problems and demonstrate the difficulty in
showing lower bounds in these regimes.

We explore the power of interactive proofs with a distributed verifier. In this setting, the verifier consists of $n$ nodes and a graph $G$ that defines their communication pattern. The prover is a single entity that communicates with all nodes by short messages. The goal is to verify that the graph $G$ belongs to some language in a small number of rounds, and with small communication bound, i.e., the proof size.

This interactive model was introduced by Kol, Oshman and Saxena (PODC 2018) as a generalization of non-interactive distributed proofs. They demonstrated the power of interaction in this setting by constructing protocols for problems as Graph Symmetry and Graph Non-Isomorphism -- both of which require proofs of $\Omega(n^2)$-bits without interaction.

In this work, we provide a new general framework for distributed interactive proofs that allows one to translate standard interactive protocols (i.e., with a centralized verifier) to ones where the verifier is distributed with a proof size that depends on the computational complexity of the verification algorithm run by the centralized verifier.
We show the following:

* Every (centralized) computation that can be performed in time $O(n)$ can be translated into three-round distributed interactive protocol with $O(\log n)$ proof size. This implies that many graph problems for sparse graphs have succinct proofs (e.g., testing planarity).

* Every (centralized) computation implemented by either a small space or by uniform NC circuit can be translated into a distributed protocol with $O(1)$ rounds and $O(\log n)$ bits proof size for the low space case and $polylog(n)$ many rounds and proof size for NC.

* We also demonstrate the power of our compilers for problems not captured by the above families. We show that for Graph Non-Isomorphism, one of the striking demonstrations of the power of interaction, there is a 4-round protocol with $O(\log n)$ proof size, improving upon the $O(n \log n)$ proof size of Kol et al.

* For many problems we show how to reduce proof size below the naturally seeming barrier of $\log n$. By employing our RAM compiler, we get a 5-round protocols with proof size $O(\log \log n)$ for a family of problems including Fixed Automorphism, Clique and Leader Election (for the later two problems we actually get $O(1)$ proof size).

* Finally we discuss how to make these proofs non-interactive arguments via random oracles.

Our compilers capture many natural problems and demonstrates the difficultly in showing lower bounds in these regimes.