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Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Al ..."
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Cited by 30 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem
 Journal of Computer and System Sciences
"... We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions. ..."
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Cited by 30 (12 self)
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We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions.
Reducing the Complexity of Reductions
 Computational Complexity
, 1997
"... We prove that the BermanHartmanis isomorphism conjecture is true under AC 0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC 0 computable manyone reductions. Isomorphism: The sets complete for C under AC 0 reductions are all i ..."
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Cited by 30 (13 self)
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We prove that the BermanHartmanis isomorphism conjecture is true under AC 0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC 0 computable manyone reductions. Isomorphism: The sets complete for C under AC 0 reductions are all isomorphic under isomorphisms computable and invertible by AC 0 circuits of depth three. Gap: The sets that are complete for C under AC 0 and NC 0 reducibility coincide. Stop Gap: The sets that are complete for C under AC 0 [mod 2] and AC 0 reducibility do not coincide. (These theorems hold both in the nonuniform and Puniform settings.) To prove the second theorem for Puniform settings, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.) 1 Introduction The notion of complete sets in complexity classes provides one of ...
DSPACE(n) ? = NSPACE(n): A degree theoretic characterization
 in Proc. 10th Structure in Complexity Theory Conference
, 1995
"... It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1 ..."
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Cited by 3 (3 self)
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It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1
News from the Isomorphism Front
"... this article. First, however, we will need to make a digression, while we discuss some recent progress on the isomorphism conjecture. ..."
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this article. First, however, we will need to make a digression, while we discuss some recent progress on the isomorphism conjecture.
A Small Span Theorem within P
"... The development of Small Span Theorems for various complexity classes and reducibilities plays a basic role in (resource bounded) measuretheoretic investigations of e cient reductions. A Small Span Theorem for a complexity class C and reducibility r is the assertion that, for all sets A in C, at le ..."
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The development of Small Span Theorems for various complexity classes and reducibilities plays a basic role in (resource bounded) measuretheoretic investigations of e cient reductions. A Small Span Theorem for a complexity class C and reducibility r is the assertion that, for all sets A in C, at least one of the cones below or above A is a negligible small class with respect to C, where the cones below orabove A refer to the sets fB: B r Ag and fB: A r Bg, respectively. That is, a Small Span Theorem rules out one of the four possibilities of the size of upper and lower cones for a set in C. Here we use the recent formulation of resourcebounded measure of Allender and Strauss which allows meaningful notions of measure on polynomialtime complexity classes. We showtwo Small Span Theorems for polynomialtime complexity classes and sublineartime reducibilities, namely a Small Span Theorem for P and Dlogtimeuniform NC0computable reductions, and for PNP and Dlogtimetransformations. Furthermore, we showthat, for every xed k, the hard set for P under Dlogtimeuniform AC 0reductions of depth k and size n k is a small class. In contrast, we show that every upper cone under Puniform NC 0reductions is not small. 1
The Isomorphism Conjecture for NP
, 2009
"... In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1 ..."
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In this article, we survey the arguments and known results for and against the Isomorphism Conjecture. 1