R. Chang and P. Rohatgi. Random reductions in the Boolean hierarchy are not robust. Technical Report 90-1154, Department of Computer Science, Cornell University, October 1990.  Home/Search   Document Not in Database   Summary   Related Articles

....case holds. Thus, Prob i2f0;1g [ iF 2 Gammai 2 SAT Phi SAT ] 0: To improve the probability beyond 1=2, simply observe that if F 2 2 SAT, then there is a small probability of guessing a satisfying assignment. This fact can be used to improve the proability to 1=2 2 Gamman 2 . See [CR90] for details. 2 Note that the probability bound for the reduction described above is much better than the one for reducing SATSAT to USAT. Since USAT behaves so much like the D P P m complete sets, one would expect SAT Phi SAT to behave like co D P P m complete sets. In fact, SAT Phi ....

....SAT does not enjoy. However, the behavior of USAT we mentioned above is actually a property of any vv m complete set for D P , not just of USAT. Still, this asymmetry seems very unnatural and we believe that it is an artifact of the definition of vv m reductions. The following theorem [CR90] illustrates our point. Theorem 3. If SATSAT rp m SATSAT with probability 1=2 1=p(n) for some polynomial bound p, the Polynomial Hierarchy collapses. Now, consider a set G 2 co D P such that SATSAT rp m G with probability 1=2 1=poly. Then, the theorem shows that G cannot be in D P ....

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R. Chang and P. Rohatgi. Random reductions in the Boolean hierarchy are not robust. Technical Report 90-1154, Department of Computer Science, Cornell University, October 1990.

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