Sipser-Lautemann theorem
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In computational complexity theory, the Sipser-Lautemann theorem or Sipser-Gács-Lautemann theorem states that BPP (Bounded-error Probablistic Polynomial) 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. 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.
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[edit] Proof
Michael Sipser's version of the proof works as follows. 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 ∩ Π2 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. Using this fact, a Σ2 sentence and a Π2 sentence can be generated that is true if and only if x∈L (see corollary).
[edit] Lemma 1
The general idea of lemma one is to prove that if A(x) covers a large part of the random space R = {1,0} | r | then there exists a small set of translations that will cover the entire random space. In more mathematical language:
If , then , where such that
Proof. Randomly pick t1,t2,...,t|r|. Let S= ∪i A(x) ⊕ ti (the union of all transforms of A(x)).
So, :
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
- .
[edit] Lemma 2
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 A(x). Therefore the set of random strings causing M(x,r) to accept cannot be generated by a small set of vectors ti.
R is the set of all accepting random strings, exclusive-or'd with vectors ti.
[edit] Corollary
An important corollary of the lemmas shows that the result of the proof can be expressed as a Σ2 expression, as follows.
That is, x is in language L if and only if there exist |r| binary vectors, where for all random bit vectors r, TM M accepts at least one random vector ⊕ ti.
The above expression is in Σ2 in that it is first existentially then universally quantified. Therefore BPP ∈ Σ2. Because BPP is closed under complement, this proves BPP ∈ Σ2∩Π2
[edit] Lautemann's Proof
Here we present the proof (due to Lautemann) that BPP ∈ Σ2. See Trevisan's notes for more information.
[edit] Lemma 3
Based on the definition of BPP we define the following:
If L is in BPP then there is an algorithm A such that for every x,
where m is the number of random bits | r | = m = | x | O(1) and A runs in time | x | O(1)
Proof: Let A' be a BPP algorithm for L. For every x, . A' uses m'(n) random bits where n = |x|.
Do k(n) repetitions of A' and accept if and only if at least executions of A' accept. Define this new algorithm as A. So A uses k(n)m'(n) random bits and . We can then find k(n) with k(n) = θ(logm'(n)) such that
[edit] Theorem 1
Proof: Let L be in BPP and A as in Lemma 3. We want to show
where m is the number of random bits used by A on input x.
Given , then
< 1.
So
So (y1,...,ym) exists.
Conversely, suppose . Then
.
So
So there is a z such that for all
[edit] References
- M. Sipser. A complexity theoretic approach to randomness In Proceedings of the 15th ACM Symposium on Theory of Computing, 330--335. ACM Press, 1983
- C. Lautemann, BPP and the polynomial hierarchy Inf. Proc. Lett. 14 215-217, 1983
- Luca Trevisan's Lecture Notes, University of California, Berkeley, http://www.cs.berkeley.edu/~luca/notes/