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On the Computational Power of Depth 2 Circuits with Threshold and Modulo Gates
, 2000
"... We investigate the computational power of depth two circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, thresholdMOD r circuits) and circuits with two levels of MOD gates (MOD p MOD q circuits). In particular, we will show ..."
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Cited by 55 (4 self)
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We investigate the computational power of depth two circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, thresholdMOD r circuits) and circuits with two levels of MOD gates (MOD p MOD q circuits). In particular, we will show the following results. (i) For all prime numbers p and integers q; r, it holds that if p divides r but not q then all thresholdMOD q circuits for MOD r have exponentially many nodes. (ii) For all integers r, all problems computable by depth two fAND;OR;NOTg circuits of polynomial size have thresholdMOD r circuits with polynomially many edges. (iii) There is a problem computable by depth 3 fAND;OR;NOTgcircuits of linear size and constant bottom fanin which for all r needs thresholdMOD r circuits with exponentially many nodes. (iv) For p; r different primes, and q 2; k positive integers, where r does not divide q; every MOD p k MOD q circuit for MOD r has e...
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 quasipoly ..."
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Cited by 36 (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.
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 bound ..."
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Cited by 33 (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.
Lower Bounds for (MOD p  MOD m) Circuits
 Proc. 39th IEEE FOCS
, 1998
"... Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circui ..."
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Cited by 1 (0 self)
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Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien (1990), Krause and Pudlak (1994), and others, characterizing symmetric functions computable by small (MOD p ; AND t ; MODm ) circuits. Applying a degreedecreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien (1990), and other related lower bounds for certain (MOD p ; MODm ; AND) circuits. Most of the previous lower bounds on circuits with modular gates used special denitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MODm gates. Our methods are applicable, and our lower bounds are valid, for the most general modular gates as well. 1
A DegreeDecreasing Lemma for (MOD q  MOD p) Circuits
, 2001
"... plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexit ..."
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plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexity theory context as well as in the theory of parallel computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([13], [22], [9], [14], [15], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [9] generalized this result
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"... Lower bounds for circuits with MODm gates Let CC o(n) [m] be the class of circuits that have size o(n) and in which all gates are MODm gates. • We show that CC[m] circuits cannot compute MODq in sublinear size when m, q> 1 are coprime integers. No nontrivial lower bounds were known before on t ..."
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Lower bounds for circuits with MODm gates Let CC o(n) [m] be the class of circuits that have size o(n) and in which all gates are MODm gates. • We show that CC[m] circuits cannot compute MODq in sublinear size when m, q> 1 are coprime integers. No nontrivial lower bounds were known before on the size of CC[m] circuits of constant depth for computing MODq. On the other hand, our results show circuits of type MAJ ◦ CC o(n)[m] need exponential size to compute MODq. Using Bourgain’s recent breakthrough result on estimates of exponential sums, we extend our bound to the case where small fanin AND gates are allowed at the bottom of such circuits i.e. circuits of type MAJ ◦ CC[m] ◦ ANDɛ log n, where ɛ> 0 is a sufficiently small constant. • CC[m] circuits of constant depth need superlinear number of wires to compute both the AND and MODq functions. To prove this, we show that any circuit computing such functions has a certain connectivity property that is similar to that of superconcentration. We show a superlinear lower bound on the number of edges of such graphs extending results on superconcentrators. 1
Some Properties of MODm Circuits Computing
"... We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit w ..."
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We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii) We show some properties of such depth two circuits computing symmetric functions. As a consequence we might think of the strategy for deriving lower bounds on modular circuits: Suppose that a polynomial size constant depth modulo 2p circuit C computes a symmetric function. If we can show that the circuit obtained by applying the procedure given in (i) to the circuit C cannot satisfy the properties described in (ii), then we have a superpolynomial lower bound on the size of a constant depth modulo 2p circuit computing a certain symmetric function.
Some Properties of Modulo m Circuit Computing Simple Functions
"... In this note, we investigate the complexity of circuits with modulo gates only in two ways. (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii ..."
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In this note, we investigate the complexity of circuits with modulo gates only in two ways. (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii) We show some properties of such depth two circuits computing symmetric functions. Thus if we can show that, for any linear size constant depth modulo 6 circuit C, a circuit obtained from C by the procedure described in (i) can not satisfy the properties described in (ii), then we could have a superlinear lower bound on the size of a constant depth modulo 6 circuit computing some symmetric functions. Key words circuit complexity, modulo gate, constant depth circuit, symmetric function, lower bound 1 Introduction To derive a strong lower bound on the size complexity in various circuit models is a big challenge in theoretical computer science. In particular, constant depth circuits are wide...