Soundness is a property of interactive proof systems that requires that no prover can make the verifier accept for a wrong statement
y
∉
L
except with some small probability. The upper bound of this probability is referred to as the soundness error of a proof system.
More formally, for every prover
(
P
~
)
, and every
y
∉
L
:
Pr
[
(
⊥
,
(
accept
)
)
←
(
P
~
)
(
y
)
↔
(
V
)
(
y
)
]
<
2
−
80
.
The above definition uses the somewhat arbitrary soundness error 2^{−80}. As long as the soundness error is bounded by a polynomial fraction of the potential running time of the verifier (i.e.
≤
1
/
p
o
l
y
(

y

)
), it is always possible to amplify soundness until the soundness error becomes negligible relative to the running time of the verifier. This is achieved by repeating the proof and accepting only if all proofs verify. After
ℓ
repetitions, a soundness error
ϵ
will be reduced to
ϵ
ℓ
.