V. Kabanets, C. Rackoff, and S. A. Cook. Efficiently approximable real-valued functions. Technical Report 00-034, Electronic Colloquium on Computational Complexity, April 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is promise AM (denoted prAM) complete. Indeed once it is known that M is an AM machine, a probabilistic nondeterministic Turing machine can simulate machine M on input x, thus deciding, with high probability, whether M accepts x or not. Another approach was introduced by V. Kabanets et al. in [KRC00]. They introduced a natural generalization of BPP, namely the class APP of real valued functions f : f0; 1g [0; 1] that can be approximated within any ffl 0, by a probabilistic Turing machine running in time polynomial in the input size and the precision 1 ffl. They showed that BPP is ....

V. Kabanets, C. Rackoff, and S. A. Cook. Efficiently approximable real-valued functions. Technical Report 00-034, Electronic Colloquium on Computational Complexity, April 2000.

....is indeed a BPP machine, is promise BPP (denoted prBPP) complete. Indeed once you know that M is a BPP machine, a probabilistic algorithm can simulate machine M on input x, thus deciding, with high probability, wether M accepts x or not; this puts L in prBPP. Another approach was introduced in [KRC00]. They introduced a natural generalization of BPP, namely the class APP of real valued functions f : f0; 1g [0; 1] that can be approximated within any ffl 0, by a probabilistic Turing machine running in time polynomial in the Address: Theorical Computer Science Department, University of ....

....construct a mapping from prBPP to APP, that maps every promise problem to a real valued function, and mapping complete promise problem to complete functions. We then prove that P APP = P prBPP . Finally we use our results to simplify proofs of important results about APP. Namely it is shown in [KRC00] that similarly to the case of BPP, the error probability for APP functions can be reduced exponentially. Their proof is rather technical and relies on a rather involved argument of repeated trials; on the other hand the idea of our proof is very simple: let f be any function in APP. We first map ....

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V. Kabanets, C. Rackoff, and S. A. Cook. Efficiently approximable real-valued functions. Technical Report 00-034, Electronic Colloquium on Computational Complexity, April 2000.

....but have the same general form. In particular, 1. We never use the acceptance probability guarantees for the algorithm on other inputs. Thus, we can derandomize algorithms even when acceptance seperations aren t guaranteed for all inputs, i.e. we can derandomize P romise Gamma BPP ( For01, KRC00] Intuitively, this means that randomized heuristics, that only perform well on some inputs, can also be simulated by a deterministic algorithm that performs well on the same inputs as the randomized algorithm. 2. The derandomization procedure only uses f x as an oracle. Although its correctness ....

....Approximation Recall that the Circuit Acceptance Probability Problem (CAPP) is the problem of computing the fraction of inputs accepted by a given Boolean circuit. This problem is easily solvable in probabilistic polynomial time, and, in a certain sense, is complete for promise BPP (see, e.g. KRC00, For01] We say that CAPP can be nontrivially approximated if, for every ffl 0, there is a nondeterministic time algorithm which, using advice of size n , approximates the acceptance probability of any given Boolean circuit of size n, to within an additive error 1=6, for infinitely many ....

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V. Kabanets, C. Rackoff, and S. Cook. Efficiently approximable real-valued functions. Electronic Colloquium on Computational Complexity, TR00-034, 2000.

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