[Show abstract][Hide abstract] ABSTRACT: A syntactic read-k-times branching program has the restriction that no variable occurs more than k times on any path (whether or not consistent) of the branching program. We rst extend the result in [30], to show that the =2 clique only function", which is easily seen to be computable by deterministic polynomial size read-twice programs, cannot be computed by nondeterministic polynomial size read-once programs, although its complement can be so computed. We then exhibit an explicit Boolean function f such that every nondeterministic syntactic read-k-times branching program for computing f has size exp .
[Show abstract][Hide abstract] ABSTRACT: We show that there is a set of points p 1 ; p 2 ; : : : ; p n such that any arithmetic circuit of depth d for polynomial evaluation (or interpolation) at these points has size Omega ` n log n log(2 + d= log n) ' : Moreover, for circuits of sub-logarithmic depth, we obtain a lower bound of OmegaGamma dn 1+1=d ) on its size. 1 Introduction To prove a superlinear lower bound for a natural problem is one of the greatest challenges of theoretical computer science. Algebraic complexity theory is the study of a restricted class of algorithms that can perform arithmetic operations on data (i.e., add, subtract, multiply and divide), but that do not care how the data is represented. This is a reasonable class of algorithms to consider when solving algebraic problems. We shall use arithmetic circuits as our model of computation. Two very natural measures of the complexity of such a circuit are its size (number of gates and wires) and depth (length of longest path from input to output...
[Show abstract][Hide abstract] ABSTRACT: In the first part of the paper we show that a subset S of a boolean cube B<sub>n</sub> embedded in the projective space P<sup>n</sup> can be approximated by a subset of B<sub>n</sub> defined by nonzeroes of a low-degree polynomial only if the values of the Hilbert function of S are sufficiently small relative to the size of S. The use of this property provides a simple and direct technique for proving lower bounds on the size of ACC[p<sup>r</sup>] circuits. In the second part we look at the problem of computing many-output function by ACC[p<sup>r</sup>] circuit and give an example when such a circuit can be correct only at exponentially small fraction of assignments
[Show abstract][Hide abstract] ABSTRACT: A syntactic read-k-times branching program has the restriction that no variable occurs more than k times on any path (whether or not consistent) of the branching program. We first extend the result in [31], to show that the "n/2 clique only function", which is easily seen to be computable by deterministic polynomial size read-twice programs, cannot be computed by nondeterministic polynomial size read-once programs, although its complement can be so computed. We then exhibit an explicit Boolean function f such that every nondeterministic syntactic read-k-times branching program for computing f has size exp {Mathematical expression}
[Show abstract][Hide abstract] ABSTRACT: We examine the class of polynomial threshold functions using harmonic analysis and applies the results to derive lower bounds related to AC 0 functions. A Boolean function is polynomial threshold if it can be represented as the sign of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that the class of polynomial threshold functions can be characterized using their spectral representation. It is proved that an n-variable Boolean function whose L 1 spectral norm is bounded by a polynomial in n is a polynomial threshold function, while a Boolean function whose L ∞ -1 spectral norm is not bounded by a polynomial in n is not a polynomial threshold function J. Bruck [SIAM J. Discrete Math. 3, 168-177 (1990; Zbl 0695.94022)]. The motivation is that the characterization of polynomial threshold functions can be applied to obtain upper and lower bounds on the complexity of computing with networks of linear threshold elements. Results related to the complexity of computing AC 0 functions are presented. More applications of the characterization theorem are presented in J. Bruck [loc. cit.] and K. Y. Siu and J. Bruck [SIAM J. Discrete Math. 4, 423-435 (1991; Zbl 0737.68042)].
[Show abstract][Hide abstract] ABSTRACT: It is shown that there is a set of points p <sub>1</sub>,
p <sub>2</sub>,. . ., p <sub>n</sub> such that any
algebraic program of depth d for polynomial evaluation (or
interpolation) at these points has size Ω( n log
n /log d ). Moreover, if d is a constant, then
a lower bound of Ω( n <sup>1+1/d</sup>) is obtained
[Show abstract][Hide abstract] ABSTRACT: The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC<sup>0</sup> functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L <sub>1</sub> spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L <sub>∞</sub><sup>-1</sup> spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC<sup>0</sup> functions are derived
[Show abstract][Hide abstract] ABSTRACT: The author investigates the question of whether or not a specific Boolean function in n variables can be interpolated by an analytic function in the same variables whose partial derivatives of all orders span a subspace of low dimension in the space of analytic functions. The upper and lower bounds for this dimension yield some weak circuit lower bounds. For a particular function, an Ω( n /log n )-size lower bound is obtained for its computation by a circuit whose gates are symmetric. For the same function an Ω( n ) lower bound is obtained for the circuit with mod<sub>k</sub> gates