AbstractInteractive proofs of proximity allow a sublinear-time verifier to
check that a given input is close to the language, using a
small amount of communication with a powerful (but untrusted)
prover. In this work, we consider two natural
minimally interactive variants of such
proofs systems, in which the prover only sends a single message,
referred to as the proof.
The first variant, known as -proofs of
Proximity (), is fully non-interactive, meaning that the
proof is a function of the input only. The second variant,
known as -proofs of Proximity (), allows the proof
to additionally depend on the verifier's (entire) random string. The
complexity of both s and s is the total number of bits
that the verifier observes—namely, the sum of the proof length
and query complexity.
Our main result is an exponential separation between the power of
s and s. Specifically, we exhibit an explicit and
natural property $$\Pi$$
Π
that admits an with complexity
$$O(\log n)$$
O
(
log
n
)
, whereas any for $$\Pi$$
Π
has complexity
$$\tilde{\Omega}(n^{1/4})$$
Ω
~
(
n
1
/
4
)
, where n denotes the length of the input
in bits. Our lower bound also yields an alternate proof,
which is more general and arguably much simpler, for a
recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown
a $$\Omega(n^{1/6})$$
Ω
(
n
1
/
6
)
lower bound for the same property $$\Pi$$
Π
.Lastly, we also consider the notion of oblivious proofs of proximity, in which
the verifier's queries are oblivious to the proof.
In this setting, we show
that s can only be quadratically stronger than s. As an
application of this result, we show an exponential separation
between the power of public and private coin for oblivious
interactive proofs of proximity.
The average number of block interchanges needed to sort a permutation and a recent result of Stanley