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22
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.
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 27 (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 ...
Pseudorandom generators and structure of complete degrees
 In 17th Annual IEEE Conference on Computational Complexity
, 2002
"... It is shown that if there exist sets in E that requiresized circuits then sets that are hard for class P, and above, under 11 reductions are also hard under 11, sizeincreasing reductions. Under the assumption of the hardness of solving RSA or Discrete Log problem, it is shown that sets that are h ..."
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Cited by 23 (2 self)
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It is shown that if there exist sets in E that requiresized circuits then sets that are hard for class P, and above, under 11 reductions are also hard under 11, sizeincreasing reductions. Under the assumption of the hardness of solving RSA or Discrete Log problem, it is shown that sets that are hard for class NP, and above, under manyone reductions are also hard under (nonuniform) 11, and sizeincreasing reductions. 1
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completene ..."
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Cited by 14 (1 self)
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In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...
Amplifying lower bounds by means of selfreducibility
 In IEEE Conference on Computational Complexity
, 2008
"... We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the numb ..."
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Cited by 13 (4 self)
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We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 and has the selfreducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ɛd. If one were able to improve this lower bound to show that there is some constant ɛ>0 such that every TC 0 circuit family recognizing BFE has size n 1+ɛ, then it would follow that TC 0 ̸ = NC 1. We show that proving lower bounds of the form n 1+ɛ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0,TC 0 and NC 1 via existing “natural ” approaches to proving circuit lower bounds. We also show that problems with small uniform constantdepth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known timespace tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d. 1
Comparing reductions to NPcomplete sets
 Electronic Colloquium on Computational Complexity
, 2006
"... Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondetermin ..."
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Cited by 12 (4 self)
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Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNPcomplete for NP but not Turingcomplete. (3) Every problem that is manyone complete for NP is complete under lengthincreasing reductions that are computed by polynomialsize circuits. The first item solves one of Lutz and Mayordomo’s “Twelve Problems in ResourceBounded Measure ” (1999). We also show that every manyone complete problem for NE is complete under onetoone, lengthincreasing reductions that are computed by polynomialsize circuits. 1
A Lower Bound for Primality
, 1999
"... Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by s ..."
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Cited by 11 (5 self)
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Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC 0 [p] for any prime p. Similar lower bounds are presented for the set of squarefree numbers, and for the problem of computing the greatest common divisor of two numbers. 1 Introduction What is the computational complexity of the set of prime numbers? There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but  Supported in part by NSF grant CCR9734918. y Supported in part by NSF grant CCR9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS9...
On pseudorandom generators in NC 0
 In Proceedings of 26th Mathematical Foundations of Computer Science
, 2001
"... Abstract. In this paper we consider the question of whether NC 0 circuits can generate pseudorandom distributions. While we leave the general question unanswered, we show • Generators computed by NC 0 circuits where each output bit depends on at most 3 input bits (i.e, NC 0 3 circuits) and with stre ..."
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Cited by 10 (0 self)
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Abstract. In this paper we consider the question of whether NC 0 circuits can generate pseudorandom distributions. While we leave the general question unanswered, we show • Generators computed by NC 0 circuits where each output bit depends on at most 3 input bits (i.e, NC 0 3 circuits) and with stretch factor greater than 4 are not pseudorandom. • A large class of “nonproblematic ” NC 0 generators with superlinear stretch (including all NC 0 3 generators with superlinear stretch) are broken by a statistical test based on a linear dependency test combined with a pairwise independence test. • There is an NC 0 4 generator with a superlinear stretch that passes the linear dependency test as well as kwise independence tests, for any constant k. 1
The firstorder isomorphism theorem
 In Foundations of Software Technology and Theoretical Computer Science: 21st Conference
, 2001
"... Abstract. For any class C und closed under NC 1 reductions, it is shown that all sets complete for C under firstorder (equivalently, Dlogtimeuniform AC 0) reductions are isomorphic under firstorder computable isomorphisms. 1 ..."
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Cited by 8 (1 self)
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Abstract. For any class C und closed under NC 1 reductions, it is shown that all sets complete for C under firstorder (equivalently, Dlogtimeuniform AC 0) reductions are isomorphic under firstorder computable isomorphisms. 1
OneWay Functions and the Isomorphism Conjecture
, 2009
"... We study the Isomorphism Conjecture proposed by Berman and Hartmanis. It states that all sets complete for NP under polynomialtime manyone reductions are Pisomorphic to each other. From previous research it has been widely believed that all NPcomplete sets are reducible each other by onetoone ..."
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Cited by 3 (0 self)
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We study the Isomorphism Conjecture proposed by Berman and Hartmanis. It states that all sets complete for NP under polynomialtime manyone reductions are Pisomorphic to each other. From previous research it has been widely believed that all NPcomplete sets are reducible each other by onetoone and lengthincreasing polynomialtime reductions, but we may not hope for the full pisomorphism due to the existence of oneway functions. Here we showed two results on the relation between oneway functions and the Isomorphism Conjecture. Firstly, we imporve the result of Agrawal [Agrawal, CCC’02] to show that if regular oneway functions exist, then all NPcomplete sets are indeed reducible each other by onetoone, lengthincreasing and P/polyreductions. A consequence of this result is the complete description of the structure of manyone complete sets of NP relative to a random oracle: all NPcomplete sets are reducible each other by oneone and lengthincreasing polynomialtime reductions but (as already shown by [Kurtz etal, JACM 95]) they are not Pisomorphic. Neverthless, we also conjecture that (different from the random oracle world) all oneway functions should have some dense easy parts, which we call P/polyeasy cylinders, where they are P/polyinvertible. Then as our second result we show that if regular oneway functions exist and furthermore all oneone, lengthincreasing and P/polycomputable functions have P/polyeasy cylinders, then all manyone complete sets for NP are P/polyisomorphic.