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Similar separations, as well as the fact that random oracles separate classes with probability 0 or 1 (as a consequence of the Kolmogorov's zero–one law), led to the creation of the Random Oracle Hypothesis, that two "acceptable" complexity classes C 1 and C 2 are equal if and only if they are equal (with probability 1) under a random oracle ...
For arbitrary groups, it is known that the hidden subgroup problem is solvable using a polynomial number of evaluations of the oracle. [3] However, the circuits that implement this may be exponential in log | G | {\displaystyle \log |G|} , making the algorithm not efficient overall; efficient algorithms must be polynomial in the number of ...
For a fixed E NP (relativized) complete problem, the oracle will give correct answers with high probability if queried with the problem instance followed by a random string of length kn (n is instance length; k is an appropriate small constant). Start with n=1.
As an example, consider a random sample of size 6. In that case, the sample median is usually defined as the midpoint of the interval delimited by the 3rd and 4th order statistics. However, we know from the preceding discussion that the probability that this interval actually contains the population median is [clarification needed]
Simon's problem considers access to a function : {,} {,}, as implemented by a black box or an oracle. This function is promised to be either a one-to-one function, or a two-to-one function; if is two-to-one, it is furthermore promised that two inputs and ′ evaluate to the same value if and only if and ′ differ in a fixed set of bits. I.e.,
In language, the status of an item (usually through what is known as "downranking" or "rank-shifting") in relation to the uppermost rank in a clause; for example, in the sentence "I want to eat the cake you made today", "eat" is on the uppermost rank, but "made" is downranked as part of the nominal group "the cake you made today"; this nominal ...
Practically, this means that a randomly sampled real tensor (from a continuous probability measure on the space of tensors) of size will be a rank-1 tensor with probability zero, a rank-2 tensor with positive probability, and rank-3 with positive probability. On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 ...
In statistics, ranking is the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted.. For example, if the numerical data 3.4, 5.1, 2.6, 7.3 are observed, the ranks of these data items would be 2, 3, 1 and 4 respectively.