Bounded Depth Circuits with Weighted Symmetric Gates: Satisfiability, Lower Bounds and Compression

A Boolean function f:{0,1}^n -> {0,1} is weighted symmetric if there exist a function g: Z -> {0,1} and integers w_0, w_1, ..., w_n such that f(x_1, ...,x_n) = g(w_0+sum_{i=1}^n w_i x_i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2^n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. With an additional trick, we give an algorithm for the maximum satisfiability problem that runs in time poly(n^t)*2^{n-n^{1/O(t)}} for instances with n variables, O(n^t) clauses and arbitrary weights. To the best of our knowledge, this is the first moderately exponential time algorithm even for Max 2SAT instances with arbitrary weights. Through the analysis of our algorithms, we obtain average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits, where all the lower bounds are against the generalized Andreev function. Our average-case lower bounds might be of independent interest in the sense that previous ones for similar circuits with arbitrary symmetric gates rely on communication complexity lower bounds while ours are based on the restriction method

Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement

Language and Automata Theory and Applications

International audienc

Proceedings of the Third International Mobile Satellite Conference (IMSC 1993)

Satellite-based mobile communications systems provide voice and data communications to users over a vast geographic area. The users may communicate via mobile or hand-held terminals, which may also provide access to terrestrial cellular communications services. While the first and second International Mobile Satellite Conferences (IMSC) mostly concentrated on technical advances, this Third IMSC also focuses on the increasing worldwide commercial activities in Mobile Satellite Services. Because of the large service areas provided by such systems, it is important to consider political and regulatory issues in addition to technical and user requirements issues. Topics covered include: the direct broadcast of audio programming from satellites; spacecraft technology; regulatory and policy considerations; advanced system concepts and analysis; propagation; and user requirements and applications