In computational complexity theory, the Sipser–Lautemann theorem or Sipser–Gács–Lautemann theorem states that bounded-error probabilistic polynomial (BPP) time is contained in the polynomial time hierarchy, and more specifically Σ2 ∩ Π2.
In 1983, Michael Sipser showed that BPP is contained in the polynomial time hierarchy.
[1] Péter Gács showed that BPP is actually contained in Σ2 ∩ Π2.
Clemens Lautemann contributed by giving a simple proof of BPP’s membership in Σ2 ∩ Π2, also in 1983.
[2] It is conjectured that in fact BPP=P, which is a much stronger statement than the Sipser–Lautemann theorem.
[2] Without loss of generality, a machine M ∈ BPP with error ≤ 2−|x| can be chosen.
(All BPP problems can be amplified to reduce the error probability exponentially.)
The basic idea of the proof is to define a Σ2 sentence that is equivalent to stating that x is in the language, L, defined by M by using a set of transforms of the random variable inputs.
Since the output of M depends on random input, as well as the input x, it is useful to define which random strings produce the correct output as A(x) = {r | M(x,r) accepts}.
The key to the proof is to note that when x ∈ L, A(x) is very large and when x ∉ L, A(x) is very small.
By using bitwise parity, ⊕, a set of transforms can be defined as A(x) ⊕ t={r ⊕ t | r ∈ A(x)}.
The first main lemma of the proof shows that the union of a small finite number of these transforms will contain the entire space of random input strings.
The general idea of lemma one is to prove that if A(x) covers a large part of the random space
then there exists a small set of translations that will cover the entire random space.
So, for all r in R, The probability that there will exist at least one element in R not in S is Therefore Thus there is a selection for each
such that The previous lemma shows that A(x) can cover every possible point in the space using a small set of translations.
Complementary to this, for x ∉ L only a small fraction of the space is covered by