A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659-667, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....It was observed in [ARZ99] that the nonuniform collapse NL poly = UL poly of [RA00] holds also in the uniform case under a very plausible hypothesis. Namely, NL = UL if there is a set in DSPACE(n) that has exponential hardness in the sense of [NW94] More recently, it has been pointed out by [KvM02] that this same conclusion can be weakened to a worst case circuit lower bound. That is, NL = UL if there is a set in DSPACE(n) such as SAT, for example) that requires circuits (or even branching programs) of size 2 , for some # 0. So almost certainly it is the case that NL and UL are equal, ....

....= UL implies that all of the preceding conditions are equivalent. Let us mention one additional preliminary observation. Proposition 3.12 If KSA (n) O(logn) for every dense 1 L, then RSPACE(n) DSPACE(n) The hypothesis of Proposition 3. 12 is very likely to be true; as already mentioned, KvM02] presents a likely condition (that there is a set in DSPACE(n) A set is 1 sparse if it contains at most one string of any given length. A language is dense if, for each n, A contains at least half of the strings of length n or no strings of length n. that requires branching programs of size ....

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31:1501--1526, 2002.

.... two and depth b ### p#n# accepts a randomly chosen input of length p#n# (where the constant b and the polynomial p depend only on our language A,and do not depend on ) As in [3] we will make use of the hardness versusrandomness techniques of [24, 6] In particular, some of the results of [24, 6, 18] are summarized in [3] in the following form. Definition 29 For all large n, any # and any Boolean function f # ##; ## ###; ## there is a pseudorandom generator G # ##; ## ## ##; ## with the property that the function G ### is computable in space O#n given access to the ....

....n, any # and any Boolean function f # ##; ## ###; ## there is a pseudorandom generator G # ##; ## ## ##; ## with the property that the function G ### is computable in space O#n given access to the Boolean function f , and such that the following theorem holds. Theorem 30 ([6, 18]) There is a constant k depending on such that if T is a set such that # ## ### #### #r # T # # ## #### # #G ### #x# # T ## # #=#, then there exists an oracle circuit C of size n with oracle T that computes f and queries T non adaptively. Closer examination of the proof techniques that ....

[Article contains additional citation context not shown here]

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing, 2002. To appear; a preliminary version appeared in STOC '99.

.... Gamma Pr y2f0;1g n [C(y) 1]j 6 1=n: For the case of the empty oracle A, we will omit the mention of A and simply call the generator SIZE(n) pseudorandom. Finally, we call a generator G : f0; 1g quick if its output can be computed in deterministic time 2 O(l(n) Theorem 11 ( BFNW93, KM99] There is a polynomial time computable function F : f0; 1g ffi ffl and d 2 N such that and if r is the truth table of an n variable Boolean function of A oracle circuit complexity at least ffid (n) pseudorandom generator mapping f0; 1g . Such a circuit C may not ....

....large n 2 N, f n has circuit complexity greater than n . Then, for every ffl 0, MA ) 2. If the Boolean functions f n from Statement (1) above are such that, for every d 2 N and infinitely many n 2 N, f n has circuit complexity greater than n ) Klivans and van Melkebeek [KM99] show that a quick SIZE SAT (n) pseudorandom generator G : f0; 1g allows one to simulate every language in AM in nondeterministic time 2 for some k 2 N. Thus, if the truth tables of Boolean functions of superpolynomial SAT oracle circuit complexity can be generated nondeterministically ....

[Article contains additional citation context not shown here]

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659--667, 1999.

....and Wigderson [NW94] Arvind and Kobler [AK97] observe that the analysis of the Nisan Wigderson generator 7 extends to non deterministic circuits, which implies the existence of pseudorandom generators for non deterministic computations, under certain assumptions. Klivans and Van Melkebeek [KvM99] observe similar generalizations of the Impagliazzo Wigderson generator for arbitrary non uniform complexity measures having certain closure properties (which does not include non deterministic circuit size, but includes the related measure size of circuits having an NP oracle ) A novel aspect ....

.... generator for arbitrary non uniform complexity measures having certain closure properties (which does not include non deterministic circuit size, but includes the related measure size of circuits having an NP oracle ) A novel aspect in our view (that is not explicit in [IW97, AK97, KvM99] is to see the Impagliazzo Wigderson construction as an algorithm that takes two inputs: the truth table of a predicate and a seed. The Impagliazzo Wigderson analysis says something meaningful even when the predicate is not fixed and hard (for an appropriate complexity measure) but rather ....

A. Klivans and D. van Milkebeek. Graph non-isomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the 31st ACM Symposium on Theory of Computing, 1999.

....one needs only a uniform hardness assumption on the predicate (rather than a circuit complexity assumption) Their conclusion is also weaker, obtaining only an average case deterministic simulation of BPP for infinitely many input lengths. Arvind and Kobler [AK97] and Klivans and van Melkebeek [KvM99] show that if the predicate is hard on average for nondeterministic circuits, then the output of the generator is indistinguishable from uniform for nondeterministic adversaries. Therefore it is possible to derandomize classes involving randomness and nondeterminism, such as AM. Trevisan [Tre99] ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on the Theory of Computing, pages 659--667, Atlanta, Georgia, 1--4 May 1999.

.... Impagliazzo and Wigderson [IW98] show that if the predicate has certain additional properties (such as downward self reducibility ) then one needs only a uniform hardness assumption on the predicate (rather circuit complexity assumption) Arvind and Kobler [AK97] and Klivans and van Melkebeek [KvM98] show that if the predicate is hard on average for nondeterministic circuits, then the output of the generator is indistinguishable from uniform for nondeterministic adversaries. Therefore it is possible to derandomize classes involving randomness and nondeterminism, such as AM. Trevisan [Tre98] ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomialtime hierarchy collapses. Technical Report TR-98-12, University of Chicago, Department of Computer Science, December 1998. Extended abstract in these proceedings.

....require exponential size circuits. It also seems likely that E contains problems that are significantly harder than this hypothesis assumes. For instance, it seems likely that E requires circuits of this size, even if the circuits are allowed to have oracle gates for SAT [AK01] It is shown in [KvM99] that this stronger hypothesis about the complexity of problems in E implies that AM = MA = NP. In [KvM99] the same conclusion AM=NP is shown to follow from a weaker assumption: namely that the hard set A is in NE#coNE. This hypothesis was subsequently weakened further by [MV99] If one is ....

....harder than this hypothesis assumes. For instance, it seems likely that E requires circuits of this size, even if the circuits are allowed to have oracle gates for SAT [AK01] It is shown in [KvM99] that this stronger hypothesis about the complexity of problems in E implies that AM = MA = NP. In [KvM99] the same conclusion AM=NP is shown to follow from a weaker assumption: namely that the hard set A is in NE#coNE. This hypothesis was subsequently weakened further by [MV99] If one is willing to settle for a weaker conclusion than BPP=P, then it su#ces to start with a much weaker assumption. ....

[Article contains additional citation context not shown here]

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In ACM Symposium on Theory of Computing (STOC), pages 659--667, 1999.

....how the nondeterminism versus determinism problem and the time versus space problem are related to the problem of derandomization. In particular, we show two ways of derandomizing the complexity class AM under uniform assumptions, which was only known previously under non uniform assumptions [13, 14]. First, we prove that either AM = NP or it appears to any nondeterministic polynomial time adversary that NP is contained in deterministic subexponential time in nitely often. This implies that to any nondeterministic polynomial time adversary, the graph non isomorphism problem appears to have ....

....then we can use them to construct pseudo random generators suitable for derandomization purposes. Nisan and Wigderson [5] weakened the hardness assumption and derived a whole range of tradeo s between hardness and randomness, and their methodology formed the basis of a series of later work [3, 9, 10, 18, 13, 14, 7, 8, 11]. Since then, the assumption for derandomizing BPP has been relaxed from average case hardness to worst case hardness, culminating in the result of Impagliazzo and Wigderson [9] stating that P = BPP if E requires 1 exponential size circuits. Sudan, Trevisan, and Vadhan [18] are able to achieve ....

[Article contains additional citation context not shown here]

A. Klivans and D. van Melkebeek, Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses, Proceedings of the 31th Annual ACM Symposium on Theory of Computing, pages 659-667, 1999.

....Nisan and Wigderson [NW94] Arvind and K obler [AK97] observe that the analysis of the Nisan Wigderson generator extends to non deterministic circuits, which implies the existence of pseudorandom generators for non deterministic computations, under certain assumptions. Klivans and Van Melkebeek [KvM99] observe similar generalizations of the Impagliazzo Wigderson generator for arbitrary non uniform complexity measures having certain closure properties (which does not include non deterministic circuit size, but includes the related measure size of circuits having an NP oracle ) The ....

....the observation that the results of [BFNW93] can be stated in a general form where the hard predicate is given as an oracle, and the proof of security can also be seen as the existence of an oracle machine with certain properties. A novel aspect in our view (that is not explicit in [IW97, AK97, KvM99, IW98] is to see the Impagliazzo Wigderson construction as an algorithm that takes two inputs: the truth table of a predicate and a seed. The Impagliazzo Wigderson analysis says something meaningful even when the predicate is not xed and hard (for an appropriate complexity measure) but rather ....

A. Klivans and D. van Milkebeek. Graph non-isomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the 31st ACM Symposium on Theory of Computing, pages 659-667, 1999.

....So it appears that a somewhat nonstandard hardness assumption is required. Still, it is of interest to weaken the assumption, as we do next. Using Worst case Hardness. Our next goal is to start with a reasonable worst case complexity assumption, such as the one used by Klivans and van Melkebeek [KvM99] that E = DTIME(2 O(n) contains a problem that is not solvable by NP circuits of size 2 o(n) We would like to show that such an assumption implies the existence of polynomial time computable predicates with strong average case hardness against NP circuits; by the previous results, such ....

....is needed, which involves the use of approximate counting algorithms with an NP oracle [Sip83, Sto85, JVV86] ministic extractors. This looks like the standard problem of worst case to average case reduction, as solved in [BFNW93, Imp95, IW97, STV99] and observed to extend to NP circuits in [KvM99] However, in all such results, one gets predicates that are hard to predict with an advantage that is at least inversely proportional to the size of the adversary (and, for a stronger reason, on the time needed to compute the predicate) It then follows that an extractor computable in time t(n) ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proceedings of 31st ACM Symposium on Theory of Computing, pages 659--667, 1999.

....one needs only a uniform hardness assumption on the predicate (rather than a circuit complexity assumption) Their conclusion is also weaker, obtaining only an average case deterministic simulation of BPP for in nitely many input lengths. Arvind and K obler [AK97] and Klivans and van Melkebeek [KvM99] show that if the predicate is hard on average for nondeterministic circuits, then the output of the generator is indistinguishable from uniform for nondeterministic adversaries. Therefore it is possible to derandomize classes involving randomness and nondeterminism, such as AM. Trevisan [Tre99] ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on the Theory of Computing, pages 659-667, Atlanta, Georgia, 1-4 May 1999.

.... m;n with hardness H(G m;n ) n allows one to approximate, to within 1=n, the acceptance probability of any Boolean circuit of size at most n, in deterministic time poly(2 m ; n) and hence, to simulate every n time BPP algorithm in deterministic time poly(2 m ; n) Klivans and van Melkebeek [KM99] noticed that Theorems 11 and 12 relativize. That is, there is an ecient algorithm for converting the truth tables of Boolean functions that have high relativized circuit complexity with respect to an oracle A into an ecient generator that is pseudorandom for A oracle circuits. For instance, the ....

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659-667, 1999.

....on all possible outputs of the generator. The resulting algorithm is deterministic, so has not only one sided error, but zero error. Under stronger (but still plausible) assumptions, analogous pseudorandom generators can be made for constant round public coin interactive proof systems [AK97, KvM99, BV99] These generators can be used to derandomize such a proof system by replacing the verifier s messages (which consists of random coin flips) with all possible outputs of the generator. This preserves the prover s complexity and the result is a deterministic proof system (i.e. an NP proof ....

Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proceedings of the Thirty-first Annual ACM Symposium on Theory of Computing, pages 659--667, Atlanta, 1--4 May 1999.

....in [NW88] They showed that every dicult problem (in E) could be used to construct a pseudo random generator. The quality of this NWgenerator (i.e. its seed length) was shown to relate to the diculty of the given function. Their work has been quantitatively improved and qualitatively extended ([BFNW93, Imp95, IW97, IW98, STV99, ISW99, KvM99, CNS99]) but their construction remains central to work in this area. While the best hardness vs. randomness trade o to be expected from such a construction should yield a seed whose size is linear in the input size of the given hard function, this was not achieved yet, and the best construction so ....

....of hard functions . Speci cally one assumes that there exists a function f = ff l g which is computable in time 2 O(l) yet every circuit of size k(l) where k is an integer function which measures the hardness of f ) cannot compute f . Using this assumption a large number of papers [BFNW93, Imp95, IW97, IW98, STV99, KvM99, CNS99] construct Pseudorandom generators. Most of these results use the Nisan Wigderson generator exactly as stated in [NW88] and improvements are gained by a pre processing stage of hardness ampli cation . In some sense, which is made exact in [ISW99] the best possible pseudo random generator ....

Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, 1999.

....proofs where the verifier sends to the prover a random string, the prover answers, and then the verifier decides whether to accept or to reject, deterministically. As will be stated later, AM contains interesting problems not known to be contained in NP. It is also conceivable that NP = AM [KvM99] 2.2 Main Results By definition NP # IP(k) for every k. The papers introducing the notion of interactive proofs had some examples of problems having interactive proofs and not known to be in NP: quadratic non residuosity [GMR89] and some group theoretic problems [Bab85] Goldreich et al. ....

A. Klivans and D. van Milkebeek. Graph non-isomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the 31st ACM Symposium on Theory of Computing, pages 659--667, 1999.

.... Impagliazzo and Wigderson [IW98] show that if the predicate has certain additional properties (such as downward self reducibility ) then one needs only a uniform hardness assumption on the predicate (rather circuit complexity assumption) Arvind and Kobler [AK97] and Klivans and van Melkebeek [KvM98] show that if the predicate is hard on average for nondeterministic circuits, then the output of the generator is indistinguishable from uniform for nondeterministic adversaries. Therefore it is possible to derandomize classes involving randomness and nondeterminism, such as AM. Trevisan [Tre98] ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. Technical Report TR-98-12, University of Chicago, Department of Computer Science, December 1998. Extended abstract in these proceedings.

.... Impagliazzo and Wigderson [IW98] show that if the predicate has certain additional properties (such as downward self reducibility ) then one needs only a uniform hardness assumption on the predicate (rather circuit complexity assumption) Arvind and Kobler [AK97] and Klivans and van Melkebeek [KvM98] show that if the predicate is hard on average for nondeterministic circuits, then the output of the generator is indistinguishable from uniform for nondeterministic adversaries. Therefore it is possible to derandomize classes involving randomness and nondeterminism, such as AM. Trevisan [Tre98] ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomialtime hierarchy collapses. Technical Report TR-98-12, University of Chicago, Department of Computer Science, December 1998. Extended abstract in these proceedings.

.... Impagliazzo and Wigderson [IW98] show that if the predicate has certain additional properties (such as downward self reducibility ) then one needs only a uniform hardness assumption on the predicate (rather circuit complexity assumption) Arvind and Kobler [AK97] and Klivans and van Melkebeek [KvM98] show that if the predicate is hard on average for nondeterministic circuits, then the output of the generator is indistinguishable from uniform for nondeterministic adversaries. Therefore it is possible to derandomize classes involving randomness and nondeterminism, such as AM. Trevisan [Tre98] ....

Adam Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. Technical Report TR-98-12, University of Chicago, Department of Computer Science, December 1998.

....tradeoffs by Babai et al. 5] and by Impagliazzo and Wigderson [17] can be cast as an application with = KT, a time bounded Kolmogorov measure introduced in [3] which essentially measures the circuit complexity of the Boolean function defined by the string described. The observation from [21] that the construction of [17] relativizes with respect to any oracle A can be viewed in terms of a Kolmogorov measure which we denote as KT . This interpretation plays a crucial role in Trevisan s recent construction of extractors out of pseudorandom generators [36] Other connections are ....

....in space O(n ) given access to the Boolean function f on inputs of size at most . Moreover, if f is PSPACE robust, then there is a constant c independent of #, such that each bit of G with oracle access to f . The following hardness versus randomness tradeoff holds. Theorem 8 ([5, 21]) Let f be a Boolean function, # 0, and G be the pseudorandom generator described above. Let T be a set and p(n) a polynomial. If 1 p(n) for all large n, then there exists a polynomial size oracle circuit family Cn n#IN with oracle T that computes f and queries T ....

[Article contains additional citation context not shown here]

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In ACM Symposium on Theory of Computing (STOC), pages 659--667, 1999.

....by Babai et al. BFNW93] and by Impagliazzo and Wigderson [IW97] can be cast as an application with = KT, a time bounded Kolmogorov measure introduced in [All01] which essentially measures the circuit complexity of the Boolean function defined by the string described. The observation from [KvM99] that the construction of [IW97] relativizes with respect to any oracle A can be viewed in terms of a Kolmogorov measure which we denote as KT . This interpretation plays a crucial role in Trevisan s recent construction of extractors out of pseudo random generators [Tre01] Other connections ....

....for any x of size n ,the function G (x) is computable in space O(n ) given access to f for inputs of size at most . Moreover, for some constant c independent of #,iff is PSPACE robust, then each bit of (x) is computable in time n with oracle access to f . Theorem 17 ( BFNW93, KvM99] Let f be a function, # 0,andG be the pseudo random generator described above. Let T be a set and p(n) a polynomial. If for all large n T ] #1 p(n) then there exists a polynomial size oracle circuit family C n#IN with oracle T that computes f and queries T ....

[Article contains additional citation context not shown here]

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In ACM Symposium on Theory of Computing (STOC), pages 659--667, 1999.

....requires exponential size branching programs then space bounded randomized computations can be simulated deterministically with only a constant factor overhead in space. A preliminary version of this paper appeared in the Proceedings of the 31st ACM Symposium on the Theory of Computing [KvM99] y Supported in part by NSF grant CCR 9701304. z Supported in part by the NSF through grants CCR 92 53582 and CCR 97 32922 while at the University of Chicago, by the European Union through TMR grant ERB 4001 GT 96 0783 while visiting CWI and the University of Amsterdam, and by the Fields ....

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the 31st ACM Symposium on the Theory of Computing, pages 659-667. ACM, 1999.

No context found.

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659-667, 1999.

No context found.

A. R. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-- 1526, 2002.

No context found.

Klivans, A.R. and van Melkebeek, D. 2002. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput. 31(5):1501--1526.

No context found.

A. R. Klivans, D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proc. 31st ACM Symp. on Theory of Computing, 1999, 1999.

No context found.

A. R. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659--667, 1999.

No context found.

A. Klivans and D. van Melkebeek. Graph Nonisomorphism has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses. In 31st STOC, pages 659--667, 1998. To appear in SICOMP.

No context found.

A. R. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In 31st Annual ACM Symposium on Theory of Computing, pages 659--667, 1999.

No context found.

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31:1501--1526, 2002.

No context found.

A. Klivans and D. van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659-667, 1999.

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