45550157
Space Complexity in Propositional Calculus
We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a read-only input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two different space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.
10033815
Expanders In Group Algebras
Let G be a finite group and let p be a prime such that (p, |G|) = 1. We study conditions under which the Abelian group $$\Bbb F$$p[G] has a few G-orbits whose union generate it as an expander (equivalently, all the discrete Fourier coefficients (in absolute value) of this generating set are bounded away uniformly from one).We prove a (nearly sharp) bound on the distribution of dimensions of irreducible representations of G which implies the existence of such expanding orbits. We further show a class of groups for which such a bound follows from the expansion properties of G. Together, these lead to a new iterative construction of expanding Cayley graphs of nearly constant degree.
7565108
Space complexity in propositional calculus
We study space complexity in the framework of propositional proofs. We consider a natural model analogous to space bounded Turing machines with a read-only input tape, and such popular propositional proof systems as Resolution, Polynomial Calculus and Frege systems. We study two different space measures. The first, introduced by [5] for Resolution and extended here to other systems, is the structured measure which counts the number of clauses/monomials kept is memory simultaneously. The other is an unstructured °measure related to the number of b i t s describing the memory content. We develop lower bound techniques that enable proving tight linear lower bounds for the first measure, and tight quadratic lower bounds for the second, for large classes of tautologies (including familiar ones like the pigeonhole principle), in both Resolution and (extensions of) Polynomial Calculus. We also prove some structural results concerning the clause space for Resolution and Frege Systems.
14219969
Superpolynomial Lower Bounds for Monotone Span Programs
monotone span programs computing explicit functions. The best previous lower bound was by Beimel, Gál, Paterson [7]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. We prove an lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes.We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields.
5341041
Lower bounds on arithmetic circuits via partial derivatives
In this paper we describe a new technique for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials (that hold over fields of characteristic zero) and iterated matrix products (that hold for all fields).
6164458
Rubber bands, convex embeddings and graph connectivity
We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium.As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.