Robert Beals

Las Vegas algorithms for matrix groups

We consider algorithms in finite groups, given by a list of generators. We give polynomial time Las Vegas algorithms (randomized, with guaranteed correct output) for basic problems for finite matrix groups over the rationals (and over algebraic number fields): testing membership, determining the order, finding a presentation (generators and relations), and finding basic building blocks: center, composition factors, and Sylow subgroups. These results extend previous work on permutation groups into the potentially more significant domain of matrix groups. Such an extension has until recently been considered intractable. In case of matrix groups G of characteristic p, there are two basic types of obstacles to polynomial-time computation: number theoretic (factoring, discrete log) and large Lie-type simple groups of the same characteristic p involved in the group. The number theoretic obstacles are inherent and appear already in handling abelian groups. They can be handled by moderately efficient (subexponential) algorithms. We are able to locate all the nonabelian obstacles in a normal subgroup N and solve all problems listed above for G/N.<<ETX>>

A black-box group algorithm for recognizing finite symmetric and alternating groups, I

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree n of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of n is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of S n : the conditional probability that a random element σ ∈ S n is an n-cycle, given that σ n = 1, is at least 1/10.

On the Complexity of Negation-Limited Boolean Networks

A theorem of Markov precisely determines the number r of NEGATION gates necessary and sufficient to compute a system of boolean functions F. For a system of boolean functions on n variables,

Polynomial-time theory of matrix groups

r\leq b(n)=\lceil\log_2(n+1)\rceil

Deciding finiteness of matrix groups in deterministic polynomial time

. We call a circuit using b(n) NEGATION gates negation-limited. We continue recent investigations into negation-limited circuit complexity, giving both upper and lower bounds. A circuit with inputs x1,..., xn and outputs

Efficient distributed quantum computing

\neg x_1, \ldots, \neg x_n

Symmetry and complexity

is called an inverter, for which

Permutations with Restricted Cycle Structure and an Algorithmic Application

r=\lceil\log_2(n+1)\rceil

More on the complexity of negation-limited circuits

. Fischer has constructed negation-limited inverters of size O(n2 log n) and depth O(log n). Recently, Tanaka and Nishino have reduced the circuit size to O(n log2 n) at the expense of increasing the depth to log2 n. We construct negation-limited inverters of size O(n log n), with depth only O(log n), and we conjecture that this is optimal. We also improve a technique of Valiant for constructing monotone circuits for slice functions (introduced by Berkowitz). Next, we introduce some lower bound techniques for negation-limited circuits. We provide a 5n+3 log(n+1)-c lower bound for the size of a negation-limited inverter. In addition, we show that for two different restricted classes of circuit, negation-limited inverters require superlinear size.

Algorithms for matrix groups and the Tits alternative

We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles. Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log. One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity). As a by-product, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N, does the group generated by A have a semisimple quotient of order > N? We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.