M. Agrawal, E. Allender and S. Rudich, Reductions in circuit complexity: An isomorphism theorem and a gap theorem,J.Comp.Sys. Sci. 57 (1998), 127--143.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....distribution as well as the circuit itself. We say a sequence of circuits Cn n : 0, 1 n # 0, 1 #(n) is in NC 0 c if Cn is in NC 0 c for all n. The function #(n) is known as the stretch function of the sequence and the stretch factor is #(n) n. Definition 2. A function f : N # [0, 1] is said to be negligible if for every c 0, there is some constant n c such that f(n) n c for every n # n c . It is said to be overwhelming if 1 f is negligible. Definition 3 (from [10] A function G : 0, 1 n # 0, 1 #(n) is a pseudorandom generator if every non uniform ....

....Avg j (t j ) 2 # (Avg j t j ) 2 . Also, Pr[y 1 = 1 x = 1] Avg j # j and Pr[y 1 = 1 x = 0] Avg j # j . Therefore Pr[y 1 y 2 = 11] # 1 2 (Pr[y 1 = 1 x = 1] 2 Pr[y 1 = 1 x = 0] 2 ) 1 2 ( 1 2 #) 2 (1 2 #) 2 ) 1 2 (1 2 2# 2 ) Now consider the calculation of F 1,2 [1, 1] in Step (ii) The expected value of F 1,2 [1, 1] is at least 1 4 1 2 2c . Therefore, if the deviation of F 1,2 [1, 1] from its expectation is bounded by 1 2 2c 1 , we have F 1,2 [1, 1] 0.25 # 1 2 2c 1 . Using Cherno# Bounds (see McDiarmid [15] the probability that F 1,2 [1, ....

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M. Agrawal, E. Allender and S. Rudich, "Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem"; Journal of Computer and System Sciences, Vol 57(2): pages 127-143, 1998.

....behave the same as the classical complete degrees. All are different except for the many one and 1 tt, which coincide. See Theorem 17) In connection with the isomorphism problem (See Theorem 13) degrees defined by reducibilities even stronger than p m have been studied. In particular in [AAR96] the AC 0 degree is shown to collapse to the NC 0 degree for every complexity classe C that is closed under NC 1 reductions. On the class NP all relations of polynomial time complete degrees necessarily remain open questions (without a proof that P 6= NP) Even from the assumption P 6= NP it ....

....of circuits of polynomial size and constant depth with bounded fan in. The class AC 0 is the class of circuits of polynomial size and constant depth with unbounded fan in. There has recently be some good progress with respect to AC 0 reductions by Agrawal, Allender and Rudich. Theorem 13 ([AAR96]) All AC 0 complete sets for NP are AC 0 isomorphic The proof of the above theorem relies heavily on the nonuniformity of the reduction. It is open whether an isomorphism theorem is true for NP when uniform reductions are used. On the other hand it is true that the existence of one way ....

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M. Agrawal, E. Allender, and S. Rudich. Reductions in circuit complexity: An isomorphism theorem and a gap theorem. Submitted for publication to JCSS Special Issue on the Eleventh Annual IEEE Conference on Computational Complexity. A preliminary version appeared in the proceedings of that conference, 1996, pp. 2-11., 1996.

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M. Agrawal, E. Allender and S. Rudich, Reductions in circuit complexity: An isomorphism theorem and a gap theorem,J.Comp.Sys. Sci. 57 (1998), 127--143.

....Some exciting developments (such as [FFK96, JPY94, Ro95] are too recent to be mentioned in the surveys. Q: What does this have to do with factorization A: I m getting to that. In spite of the general feeling that the original BermanHartmanis conjecture is probably false, it was shown in [AAR96] that the BermanHartmanis conjecture for AC 0 reductions is actually true. Q: Please state this more clearly. A: For any reasonable complexity class, all of the sets that are complete under AC 0 reductions are isomorphic under bijections computable and invertible by AC 0 circuits. In ....

....objects of study in complexity theory, because the natural computational problems 3 Actually [ABI93] proves something much stronger, but that would require another digression. that we are interested in tend to cluster into classes of complete sets for complexity classes. The result of [AAR96] shows that, not only are the complete sets interreducible, but in fact they are isomorphic in a very restrictive sense. A: Remember, this is only true for sets that are complete under AC 0 reducibility. Q: Yes, but you said that all of the sets that are complete under other reducibilities are ....

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M. Agrawal, E. Allender and S. Rudich, Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem, to appear in J. Comp. Sys. Sci. Some of the results in this paper were originally announced in [AA96] and [AAIPR97].

....1 question can be phrased this way) it is worth noting that a set A in NP is presented in [AAIPR97] such that there is no AC 0 m reduction from PARITY to A. If Primes were complete for NP (or for any other reasonable complexity class) under AC 0 m reductions, the isomorphism theorems of [AAR98, AAIPR97] show that Primes would be isomorphic to all of the other complete sets for that class, under isomorphisms computable and invertible by P uniform depth three AC 0 circuits. In particular, there would be an isomorphism of this sort between Primes and Primes Theta f0; 1g . Among other things, ....

M. Agrawal, E. Allender and S. Rudich, Reductions in circuit complexity: An isomorphism theorem and a gap theorem, J. Comp. Sys. Sci. 57 (1998), 127--143.

....1 question can be phrased this way) it is worth noting that a set A in NP is presented in [AAIPR97] such that there is no AC 0 m reduction from PARITY to A. If Primes were complete for NP (or for any other reasonable complexity class) under AC 0 m reductions, the isomorphism theorems of [AAR98, AAIPR97] show that Primes would be isomorphic to all of the other complete sets for that class, under isomorphisms computable and invertible by P uniform depth three AC 0 circuits. In particular, there would be an isomorphism of this sort between Primes and Primes Theta f0; 1g . Among other things, ....

M. Agrawal, E. Allender and S. Rudich, Reductions in circuit complexity: An isomorphism theorem and a gap theorem, J. Comp. Sys. Sci. 57 (1998), 127--143.

....1 question can be phrased this way) it is worth noting that a set A in NP is presented in [AAIPR97] such that there is no AC 0 m reduction from PARITY to A. If Primes were complete for NP (or for any other reasonable complexity class) under AC 0 m reductions, the isomorphism theorems of [AAR96, AAIPR97] show that Primes would be isomorphic to all of the other complete sets for that class, under isomorphisms computable and invertible by P uniform depth three AC 0 circuits. In particular, there would be an isomorphism of this sort between Primes and Primes Theta f0; 1g . Among other things, ....

M. Agrawal, E. Allender and S. Rudich, Reductions in circuit complexity: An isomorphism theorem and a gap theorem, J. Comp. Sys. Sci., to appear.

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M. Agrawal, E. Allender, and S. Rudich. Reductions in circuit complexity: An isomorphism theorem and a gap theorem. Submitted for publication to JCSS Special Issue on the Eleventh Annual IEEE Conference on Computational Complexity. A preliminary version appeared in the proceedings of that conference, 1996, pp. 2-11., 1996.

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