An Exponential Lower Bound for the Size of Monotone Real Circuits

We prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary real-valued functions (including AND and OR). Our proof is relatively simple and direct and uses the method of counting bottlenecks. The generalization was proved independently by Pudlák using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus.     

Upper and lower bounds for some depth-3 circuit classes

We investigate the complexity of depth-3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MOD _m gates at level one. We show that the fan-in of the AND gates can be reduced toO(logn) in the case wherem is unbounded, and to a constant in the case wherem is constant. We then use these upper bounds to derive exponential lower bounds for this class of circuits. In the unboundedm case, this yields a new proof of a lower bound of Grolmusz; in the constantm case, our result sharpens his lower bound. In addition, we prove an exponential lower bound if OR gates are also permitted on level two andm is a constant prime power.Dedicated to the memory of Roman Smolensky     

The Monotone Circuit Complexity of Quadratic Boolean Functions

Several results on the monotone circuit complexity and the conjunctive complexity, i.e., the minimal number of AND gates in monotone circuits, of quadratic Boolean functions are proved. We focus on the comparison between single level circuits, which have only one level of AND gates, and arbitrary monotone circuits, and show that there is an exponential gap between the conjunctive complexity of single level circuits and that of general monotone circuits for some explicit quadratic function. Nearly tight upper bounds on the largest gap between the single level conjunctive complexity and the general conjunctive complexity over all quadratic functions are also proved. Moreover, we describe the way of lower bounding the single level circuit complexity and give a set of quadratic functions whose monotone complexity is strictly smaller than its single level complexity.     

Lower bounds on monotone arithmetic circuits with restricted depths

We consider monotone arithmetic circuits with restricted depths to compute monotone multivariate polynomials such as the elementary symmetric functions, convolution of several vectors and raising a matrix to a power. We develop general lower- and upper-bound techniques that seem to generate almost-matching bounds for all the functions considered. These results imply exponential lower bounds for circuits of bounded depths which compute any of these functions. We also obtain several examples for which negation can reduce the size exponentially.     

Multilinear formulas,maximal-partition discrepancy and mixed-sources extractors

We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients' vector of a polynomial and the coefficients' vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits: monotone circuits, orthogonal formulas, non-canceling formulas, and noise-resistant formulas. One ingredient of our proof is an explicit map that has exponentially small discrepancy for every partition of the input variables into two sets of roughly the same size. We give two additional applications of this explicit map: to extractors construction and to communication complexity.     

Graph Complexity and Slice Functions

Abstract. A graph-theoretic approach to study the complexity of Boolean functions was initiated by Pudlák, Rödl, and Savický [PRS] by defining models of computation on graphs. These models generalize well-known models of Boolean complexity such as circuits, branching programs, and two-party communication complexity. A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of {2-slice functions} would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. We prove, for instance, that a lower bound of n ^1+Ω(1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 ^Ω(?) lower bound on the formula size over a complete basis of another explicit function g on l variables, where l=Θ( log n) . We also consider lower bound questions for depth-3 bipartite graph complexity. We prove a weak lower bound on this measure using algebraic methods. For instance, our result gives a lower bound of Ω(( log n) ³ / ( log log n) ⁵ ) for bipartite graphs arising from Hadamard matrices, such as the Paley-type bipartite graphs. Lower bounds for depth-3 bipartite graph complexity are motivated by two significant applications: (i) a lower bound of n ^Ω(1) on the depth-3 complexity of an explicit n -vertex bipartite graph would yield superlinear size lower bounds on log-depth Boolean circuits for an explicit function, and (ii) a lower bound of $\exp((\log \log n)^{\omega(1)})$ would give an explicit language outside the class Σ ₂ ^cc of the two-party communication complexity as defined by Babai, Frankl, and Simon [BFS]. Our lower bound proof is based on sign-representing polynomials for DNFs and lower bounds on ranks of ±1 matrices even after being subjected to sign-preserving changes to their entries. For the former, we use a result of Nisan and Szegedy [NS] and an idea from a recent result of Klivans and Servedio [KS]. For the latter, we use a recent remarkable lower bound due to Forster [F1].     

A new lower bound on the monotone network complexity of Boolean sums

Summary Neciporuk [3], Lamagna/Savage [1] and Tarjan [6] determined the monotone network complexity of a set of Boolean sums if each two sums have at most one variable in common. By this result they could define explicitely a set of n Boolean sums which depend on n variables and whose monotone complexity is of order n ^3/2. In the main theorem of this paper we prove a more general lower bound on the monotone network complexity of Boolean sums. Our lower bound is for many Boolean sums the first nontrivial lower bound. On the other side we can prove that the best lower bound which the main theorem yields is the n ^3/2-bound cited above. For the proof we use the technical trick of assuming that certain functions are given for free.     

Upper bounds for reversible circuits based on Young subgroups

We present tighter upper bounds on the number of Toffoli gates needed in reversible circuits. Both multiple controlled Toffoli gates and mixed polarity Toffoli gates have been considered for this purpose. The calculation of the bounds is based on a synthesis approach based on Young subgroups that results in circuits using a more generalized gate library. Starting from an upper bound for this library we derive new bounds which improve the existing bound by around 77%.     

A note on monotone complexity and the rank of matrices

We shall give simpler proofs of some lower bounds on monotone computations. We describe a simple condition on combinatorial structures, such that the rank of the matrix associated with these structures gives lower bounds on monotone span program size and monotone formula size. We also prove an upper bound on the rank of the corresponding matrices, and show that such structures can be constructed from self-avoiding families. As a corollary, we obtain an upper bound on the size of self-avoiding families, which solves a problem posed by Babai and Gál [Combinatorica 19 (3) (1999) 301-319].     

A characterization of span program size and improved lower bounds for monotone span programs

We give a characterization of span program size by a combinatorial-algebraic measure. The measure we consider is a generalization of a measure on covers which has been used to prove lower bounds on formula size and has also been studied with respect to communication complexity.?In the monotone case our new methods yield lower bounds for the monotone span program complexity of explicit Boolean functions in n variables over arbitrary fields, improving the previous lower bounds on monotone span program size. Our characterization of span program size implies that any matrix with superpolynomial separation between its rank and cover number can be used to obtain superpolynomial lower bounds on monotone span program size. We also identify a property of bipartite graphs that is suficient for constructing Boolean functions with large monotone span program complexity. Received: September 30, 2000.     

Graph Complexity and Slice Functions

   Abstract. A graph-theoretic approach to study the complexity of Boolean functions was initiated by Pudlák, R?dl, and Savicky [PRS] by defining models of computation on graphs. These models generalize well-known models of Boolean complexity such as circuits, branching programs, and two-party communication complexity. A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of {2-slice functions} would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. We prove, for instance, that a lower bound of n ^1+Ω(1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 ^Ω(ℓ) lower bound on the formula size over a complete basis of another explicit function g on l variables, where l=Θ( log n) . We also consider lower bound questions for depth-3 bipartite graph complexity. We prove a weak lower bound on this measure using algebraic methods. For instance, our result gives a lower bound of Ω(( log n) ³ / ( log log n) ⁵ ) for bipartite graphs arising from Hadamard matrices, such as the Paley-type bipartite graphs. Lower bounds for depth-3 bipartite graph complexity are motivated by two significant applications: (i) a lower bound of n ^Ω(1) on the depth-3 complexity of an explicit n -vertex bipartite graph would yield superlinear size lower bounds on log-depth Boolean circuits for an explicit function, and (ii) a lower bound of

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

In this paper, we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds, we obtain lower bounds for these models.For depth-3 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/2}))}\) for depth-3 multilinear formulas, for some explicit polynomial.For depth-4 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/4}))}\) for depth-4 multilinear formulas, for some explicit polynomial.A regular formula consists of alternating layers of \({+,\times}\) gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp\({\left(n^{1- \delta}\right)}\), for regular depth-d multilinear formulas with formal degree at most n and size exp\({(n^\delta)}\), where \({\delta = O(1/{\sqrt{5}^d})}\). This result implies a lower bound of roughly exp\({(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}\) for such formulas.We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas, we go straight to read-once algebraic branching programs).     

On the computational power of winner-take-all

This article initiates a rigorous theoretical analysis of the computational power of circuits that employ modules for computing winner-take-all. Computational models that involve competitive stages have so far been neglected in computational complexity theory, although they are widely used in computational brain models, artificial neural networks, and analog VLSI. Our theoretical analysis shows that winner-take-all is a surprisingly powerful computational module in comparison with threshold gates (also referred to as McCulloch-Pitts neurons) and sigmoidal gates. We prove an optimal quadratic lower bound for computing winner-take-all in any feedforward circuit consisting of threshold gates. In addition we show that arbitrary continuous functions can be approximated by circuits employing a single soft winner-take-all gate as their only nonlinear operation. Our theoretical analysis also provides answers to two basic questions raised by neurophysiologists in view of the well-known asymmetry between excitatory and inhibitory connections in cortical circuits: how much computational power of neural networks is lost if only positive weights are employed in weighted sums and how much adaptive capability is lost if only the positive weights are subject to plasticity.     

Lower bounds for monotone span programs

Span programs provide a linear algebraic model of computation. Lower bounds for span programs imply lower bounds for formula size, symmetric branching programs, and contact schemes. Monotone span programs correspond also to linear secret-sharing schemes. We present a new technique for proving lower bounds for monotone span programs. We prove a lower bound of (m ^2.5) for the 6-clique function. Our results improve on the previously known bounds for explicit functions.     

Property Testing Lower Bounds via Communication Complexity

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of ??(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.     

Enclosure of solutions of weakly nonlinear elliptic boundary value problems and their computation

An algorithm is presented to compute approximations as well as continuous bounds for solutions of weakly nonlinear elliptic boundary value problems. The given problem is majorized in some sense and the obtained new problem is solved by a finite element method. The finite element solution is computed by a monotone iteration process and at last transformed to a continuous (lower) bound for a solution. Convergence is proved and mesh refinement effects are discussed. Two numerical examples are given.     

Exponential Sums and Circuits with a Single Threshold Gate and Mod-Gates

Consider circuits consisting of a threshold gate at the top, Mod _m gates at the next level (for a fixed m ), and polylog fan-in AND gates at the lowest level. It is a difficult and important open problem to obtain exponential lower bounds for such circuits. Here we prove exponential lower bounds for restricted versions of this model, in which each Mod _m -of-AND subcircuit is a symmetric function of the inputs to that subcircuit. We show that if q is a prime not dividing m , the Mod _q function requires exponential-size circuits of this type. This generalizes recent results and techniques of Cai et al. [CGT] (which held only for q=2 ) and Goldmann (which held only for depth two threshold over Mod _m circuits). As a further generalization of the [CGT] result, the symmetry condition on the Mod _m subcircuits can be relaxed somewhat, still resulting in an exponential lower bound. The basis of the proof is to reduce the problem to estimating an exponential sum, which generalizes the notion of ``correlation" studied by [CGT]. This identifies the type of exponential sum that will be instrumental in proving the general case. Along the way we substantially simplify previous proofs. Received June 1997, and in final form October 1998.     

Lower Bounds for Randomized Exclusive Write PRAMs

In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and we present randomized lower bounds which match the best deterministic lower bounds known. (For the CREW PRAM model, the lower bound is asymptotically optimal.) These are the first nontrivial randomized lower bounds known for the compaction problem on these models. We show that our lower bounds also apply to the problem of approximate compaction. Next we examine the problem of computing boolean functions on the CREW PRAM model, and we present a randomized lower bound which improves on the previous best randomized lower bound for many boolean functions, including the OR function. (The previous lower bounds for these functions were asymptotically optimal, but we improve the constant multiplicative factor.) We also give an alternate proof for the randomized lower bound on PARITY, which was already optimal to within a constant additive factor. Lastly, we give a randomized lower bound for integer merging on an EREW PRAM which matches the best deterministic lower bound known. In all our proofs, we use the Random Adversary method, which has previously only been used for proving lower bounds on models with Concurrent Write capabilities. Thus this paper also serves to illustrate the power and generality of this method for proving parallel randomized lower bounds. Received October 2, 1995, and in final form January 29, 1997.