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This is achieved by constructing random constraints such that, with non-negligible probability, exactly one solution satisfies these additional constraints if the solution space is not empty. Isolation lemmas have important applications in computer science, such as the Valiant–Vazirani theorem and Toda's theorem in computational complexity ...
The Valiant–Vazirani theorem is a theorem in computational complexity theory stating that if there is a polynomial time algorithm for Unambiguous-SAT, then NP = RP.It was proven by Leslie Valiant and Vijay Vazirani in their paper titled NP is as easy as detecting unique solutions published in 1986.
The PCP theorem states that NP = PCP[O(log n), O(1)],. where PCP[r(n), q(n)] is the class of problems for which a probabilistically checkable proof of a solution can be given, such that the proof can be checked in polynomial time using r(n) bits of randomness and by reading q(n) bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1/2.
Given a claimed solution x with length n, which might be false, the prover produces a proof π which states x solves L (x ∈ L, the proof is a string ∈ Σ ∗). And the verifier is a randomized oracle Turing Machine V (the verifier) that checks the proof π for the statement that x solves L (or x ∈ L) and decides whether to accept the ...
One concept is the set of all patterns of bits in = {,} that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, { ( a , b ) ∣ 0 ≤ a ≤ π / 2 , π ≤ b ≤ 13 } {\displaystyle \{(a,b)\mid 0\leq a\leq \pi /2,\pi \leq b\leq {\sqrt {13}}\}} , each of which contains only the positive ...
The averaging argument is the following claim: if we have a circuit such that (,) = with probability at least , where is chosen at random and is chosen independently from some distribution over {,} (which might not even be efficiently sampleable) then there exists a single string {,} such that [(,) = ()].
The conditional probability at any interior node is the average of the conditional probabilities of its children. The latter property is important because it implies that any interior node whose conditional probability is less than 1 has at least one child whose conditional probability is less than 1.
In the simplest case, if one allocates balls into bins (with =) sequentially one by one, and for each ball one chooses random bins at each step and then allocates the ball into the least loaded of the selected bins (ties broken arbitrarily), then with high probability the maximum load is: [8]