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17
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.
TimeSpace Tradeoffs in the Counting Hierarchy
, 2001
"... We extend the lower bound techniques of [14], to the unboundederror probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform ..."
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Cited by 19 (4 self)
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We extend the lower bound techniques of [14], to the unboundederror probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform TC 0 circuits for iterated multiplication [9]. Here is an
On TC^0, AC^0, and Arithmetic Circuits
 Journal of Computer and System Sciences
, 2000
"... Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constantdepth polynomialsize ..."
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Cited by 18 (6 self)
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Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constantdepth polynomialsize arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding # Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship.
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
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Cited by 17 (3 self)
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Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynom ..."
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Cited by 16 (4 self)
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisf ..."
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Cited by 11 (5 self)
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We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODpSat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.
On Proofs About Threshold Circuits and Counting Hierarcies (Extended Abstract)
, 1998
"... ) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth t ..."
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Cited by 9 (2 self)
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) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth threshold circuits of polynomial and quasipolynomial size. Then we dene certain secondorder theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the socalled RSUV isomorphism. 1 Introduction A phenomenon that is commonly observed in Complexity Theory is that proofs of results about counting complexity classes (#P , Mod p P etc.) can often be scaled down to yield results about small depth circuit classes with the corresponding counting gates. For example, Toda's result [17] that every problem in the Polynomial Hierarchy can be solved in polynomial time with an oracle for #P correspond...
Uniform Characterizations of Complexity Classes
 Complexity Theory Column 23, ACMSIGACT News
, 1999
"... In the past few years, generalized operators (a. k. a. leaf languages) in the context of polynomial time machines, and gates computing arbitrary groupoidal functions in the context of Boolean circuits have raised some interest. We survey results from both areas, point out connections between them, ..."
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Cited by 7 (3 self)
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In the past few years, generalized operators (a. k. a. leaf languages) in the context of polynomial time machines, and gates computing arbitrary groupoidal functions in the context of Boolean circuits have raised some interest. We survey results from both areas, point out connections between them, and present relations to a generalized quantifier concept known from finite model theory. 1 Introduction There is an "amusing and instructive way of looking at [...] diverse complexity classes" [Pap94a, p. 504] that are of current focal interest in computational complexity theory. This way makes instrumental use of characterizations of classes in terms of conditions on computations trees of nondeterministic polynomialtime Turing machines. As an example, let us look at the class NP. By definition, a language A 2 NP is given by a nondeterministic polynomialtime machine (NPTM) M such that for all inputs x, we have that x belongs to A if and only if in the computation tree that M produces wh...
Monoids and computations
 International Journal on Algebra and Computation
, 2004
"... Abstract. This contribution wishes to argue in favour of increased interaction between experts on finite monoids and specialists of theory of computation. Developing the algebraic approach to formal computations as well as the computational point of view on monoids will prove to be beneficial to bot ..."
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Cited by 6 (1 self)
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Abstract. This contribution wishes to argue in favour of increased interaction between experts on finite monoids and specialists of theory of computation. Developing the algebraic approach to formal computations as well as the computational point of view on monoids will prove to be beneficial to both communities. We give examples of this twoway relationship coming from temporal logic, communication complexity and boolean circuits. Although mostly expository in nature, our paper proves some new results along the way. 1 Introduction This paper is an invitation to members of the semigroup community to increasetheir interaction with colleagues working in the theory of computation. We are convinced that computational theorists generate ideas that can be extremelyprofitable in the study of semigroups and that, in turn, semigroup theorists can adapt their techniques to problems in theoretical informatics. Increased collaboration will be beneficial to both groups. Our first point is that the notion of formal computation is fascinating bothfrom a strictly mathematical point of view and also from an epistemological one. Investigations of computational issues give rise to questions that satisfy the mostdemanding criteria of mathematical "beauty " and "elegance". The bonus is that it is not so farfetched to view the human brain as a computational device sothat our whole understanding of the universe is mediated through some sort of "computation".
On the b 1 bitcomprehension rule
 Logic Colloquium 98
, 2000
"... Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counte ..."
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Cited by 5 (0 self)
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Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counterexample computations using TC 0functions. The latter is used to show that a collapse of stronger theories to � b 1CR implies that NP is contained in nonuniform TC 0. 1