R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203-208, New York, 1998. ACM.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....instances of USAT in polynomial time, we would have a probabilistic polynomial time algorithm that solves almost all instances of all problems from the class NP. Exact definition are somewhat complicated so, due to the lack of space, we refer the interested reader to [7] and [26] Note that in [2], arguments are given that this problem may not be NP hard. 3 New Results The following modification of the above results provides the desired complete explanation: For two global maxima, we cannot pinpoint one of them, but, as we will show, we can compute the next best thing: namely, we can ....

R. Beigel, H. Buhrman, and L. Fortnow, "NP might not be as easy as detecting unique solutions", In: Proceedings of the 30th ACM Symposium on the Theory of Computing, 1998, pp. 203--208.

....relative to this oracle was intricate and technical, relying on the notions of genericity and relativized Kolmogorov complexity. One of the open questions they posed was whether there exists a simpler oracle that would make the IC true. This was settled in 1997 when Beigel, Buhrman, and Fortnow [BBF98] used a straightforward and entirely computable construction to create an oracle relative to which the IC holds. To understand their result, we must first refer to three earlier theorems: 1. Grollmann and Selman [GS88] showed that one way functions exist iff P 6= UP. 2. Berman [Ber77] showed ....

....Theorem 2.1 There is an oracle C such that: 1. NP C = EXP C , so the complete m degree of NP C is a 1 li degree; 2. P C 6= UP C , so one way functions exist; 3. the IC C is true. Proof. Let C be the oracle A Phi G = f0x : x 2 Ag [ f1x : x 2 Gg, where A is the oracle constructed in [BBF98] and G is a UP A coUP A generic oracle. Because NP A = EXP A , NP C = EXP C . To see this, let L be a language accepted by an NP A machine M and let x be a string in L. Let N be a machine that first guesses the unique string in G that can affect the computation. If a path s guess ....

[Article contains additional citation context not shown here]

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203--208. ACM, New York, 1998. 3

....accepting computation for each input. Obviously, P # UP # NP, but it is not known whether any of these inclusions are proper. The results of Valiant and Vazirani [53] provide some evidence for the hypothesis P # UP = NP. On the other hand, there is an oracle A with P A = UP A # NP A [14]. Finally, we note that separating P from UP would also have consequences for the area of cryptography: it is known that there are polynomial time one way functions if and only if P # UP [23] For the setting of nonuniform, logarithmically space bounded computations, Allender and Reinhardt [6] ....

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proc. of the 30th Ann. ACM Symp. on Theory of Computing (STOC), 203-- 208, 1998.

....secure public key cryptosystem exists (i.e. one which cannot be cracked in polynomial time) only if p separators exist. Homer and Selman [HS92] constructed an oracle relative to which all # p 2 complete sets are polynomial time isomorphic, and no p separators exist. Beigel, Buhrman, and Fortnow [BBF98] showed that there is an oracle relative to which the isomorphism conjecture holds, and p separators exist. Relative to any oracle for which P = PSPACE the isomorphism conjecture is true, and p separators do no exist. 5 Disjunctive reductions and m reductions This section is mainly intended for ....

....[FFK96] and NP simple sets exist (as we will show presently) Hence the assumption that NP simple sets exist will not be enough to show the isomorphism conjecture true or false with relativizing techniques. In the other direction we can use an oracle constructed by Beigel, Buhrman, and Fortnow [BBF98] which makes the isomorphism conjecture true, while NP simple sets do not exist (since NP = coNP) Furthermore the isomorphism conjecture will be false, and NP simple sets do not exist relative to any oracle making P = PSPACE. Hence the isomorphism conjecture will not prove the existence, or ....

Richard Beigel, , Harry Burhman, and Lance Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of the 30th ACM Symposium on the Theory of Computing (STOC-98), pages 203--208, 1998. 26

No context found.

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203-208, New York, 1998. ACM.

No context found.

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203-208, New York, 1998. ACM.

No context found.

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Thirtieth Annual ACM Symposium on Theory of Computing (STOC), 1998. To appear.

....we construct an oracle world where our rst result can not be improved to deterministic reductions. We show that there is an oracle such that BPP 6 P R CD t for any polynomial t. The construction of the oracle is an extension of the techniques developed by Beigel, Buhrman and Fortnow [BBF98] 2 De nitions and Notations We assume the reader familiar with standard notions in complexity theory as can be found e.g. in [BDG88] Strings are elements of , where = f0; 1g. 3 For a string s and integers n; m jsj we use the notation s[n: m] for the string consisting of the nth ....

....2 coNP via an obvious predicate. As argued above, this gives us our contradiction. 2 Now we proceed to construct the oracle. Theorem 4.3 There exists an oracle A such that EXP NP A NP A =poly P A = P A Proof . The proof parallels the construction from Beigel, Buhrman and Fortnow [BBF98] who construct an oracle such that P A = P A and NEXP A = NP A . We will use a similar setup. Let M A be a nondeterministic linear time Turing machine such that the language L A de ned by w 2 L A , #M A (w) mod 2 = 1 is P A complete for every A. For every oracle A, let K ....

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Thirtieth Annual ACM Symposium on Theory of Computing (STOC), 1998. To appear.

....in a relativized world by setting P = UP and C = EXP. Homer and Selman [HS92] were able to do this for C = p 2 , and our present oracle works for C = P NP . An oracle for C = NP con rms the isomorphism conjecture [BH77] Oracles making the conjecture true are known since [FFK96] However, [BBF98] gives an oracle relative to which P = UP and NP = EXP. 3 2 Preliminaries We assume the reader familiar with standard notions in structural complexity theory, as are de ned e.g. in [BDG88] Nonetheless, we will in this section, recall some notions that we feel are not common knowledge, and x ....

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203-208, New York, 1998. ACM.

....as the many one complete degree. We cannot extend theorem 4.1 to get NP 6= coNP and P NP[1] EXP since Homer, Kurtz and Royer [HKR93] give a relativizable proof that the 1 tt complete degree for EXP is the same as the many one complete degree. In a later paper, Beigel, Buhrman and Fortnow [BBF98] give a relativized world where the 1 tt complete degree for NP is not the same as the many one complete degree. 5 Consequences of P NP[1] P NP[2] In this section we examine collapses that occur if P NP[1] P NP[2] Kadin [Kad88] showed that the polynomial time hierarchy collapse under ....

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203--208. ACM, New York, 1998.

....that in general circuit lower bounds imply oracle separations. We show that in general exponential circuit lower bounds imply in fact a pair of collapses, which in turn imply Furst et al. s separation. Our approach is based on combining oracle construction techniques of Beigel, Buhrman, and Fortnow [BBF98] with bounded query techniques of Amir, Beigel, and Gasarch [ABG90] 2 Preliminaries We assume that the reader is familiar with the basic notions of complexity theory. See [Pap94] for example. In this section, we recall some definitions that will be used in this article. For any number r, a ....

R. Beigel, H. Buhrman, and L. Fortnow, NP might not be as easy as detecting unique solution,, Proceedings of the 30th ACM Symposium on Theory of Computing, 1998.

....as the many one complete degree. We cannot extend theorem 4.1 to get NP 6= coNP and P NP[1] EXP since Homer, Kurtz and Royer [HKR93] give a relativizable proof that the 1 tt complete degree for EXP is the same as the many one complete degree. In a later paper, Beigel, Buhrman and Fortnow [BBF98] give a relativized world where the 1 tt complete degree for NP is not the same as the many one complete degree. 5 Consequences of P NP[1] P NP[2] In this section we examine collapses that occur if P NP[1] P NP[2] Kadin [Kad88] showed that the polynomial time hierarchy collapse under ....

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of the 30th ACM Symposium on the Theory of Computing, pages 203--208. ACM, New York, 1998.

....Sha92] for QBF . Last we construct an oracle world where our first result can not be improved to deterministic reductions. We show that there is an oracle such that BPP 6 P R CD t for any polynomial t. The construction of the oracle is an extension of the techniques developed in Beigel et al. [BBF98]. 2 Definitions and Notations We assume the reader familiar with standard notions in complexity theory as can be found e.g. in [BDG88] Strings are elements of Sigma , where Sigma = f0; 1g. For a string s and integers n; m jsj we use the notation s[n: m] for the string consisting of the ....

....As all parts of this proof relativize, we get the result for any oracle. 2 Now we proceed to construct the oracle. Theorem 4.2 There exists an oracle A such that EXP NP A ae NP A =poly PhiP A = P A Proof . The proof of the construction parallels the one from Beigel, Buhrman and Fortnow [BBF98], who construct an oracle such that P A = PhiP A and NEXP A = NP A . We will use a similar setup. Let M A be a nondeterministic linear time Turing machine such that the language L A defined by w 2 L A , #M A (w) mod 2 = 1 is PhiP A complete for every A. For every oracle A, ....

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Thirtieth Annual ACM Symposium on Theory of Computing (STOC), 1998. To appear.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC