We use the computer constructions of the Monster to determine words in the generators for an element in each conjugacy class, up to algebraic conjugacy. We show how to distinguish the conjugacy classes. You can(not) access the Monster elements database on-line.
to be precise.
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(Preprint and program, tar.gz, 124k.)
LMS J. Comput. Math. 13 (2010) 82–89We use the technique of Fischer matrices to write a program to produce the character table of a group of shape
from the character tables of
G, G:2, 2.G and 2.G:2. We also supply a
simplified proof of a result frequently used in the Fischer matrices
method.
We define a simple, intuitive trajectory-based equivalence relation which accurately captures the relationship between programs and the slices produced by the most commonly used and well-known slicing algorithms, Weiser’s and the PDG algorithm, even for non-terminating programs. Our new equivalence, unlike other approaches, is based on standard semantics. It deals with non-termination by stipulating that a program and its slice must ‘agree’ in all terminating ‘approximations’ based on finite loop unfoldings. We prove that this equivalence is preserved by Weiser’s algorithm and all slicing algorithms based on data and control dependence. We also demonstrate that this new equivalence more accurately captures the behaviour of slicing algorithms than previous work.
There are several similar, but not identical, definitions of control dependence in the literature. These definitions are given in terms of control flow graphs which have had extra restrictions imposed (for example, end-reachability). We define two new generalisations of non-termination insensitive and non-termination sensitive control dependence called weak and strong control-closure. These are defined for all finite directed graphs, not just control flow graphs and are hence allow control dependence to be applied to a wider class of program structures than before. We investigate all previous forms of control dependence in the literature and prove that, for the restricted graphs for which each is defined, vertex sets are closed under each if and only if they are either weakly or strongly control-closed. Low polynomial-time algorithms for producing minimal weakly and strongly control-closed sets over generalised control flow graphs are given. This paper is the first to define an underlying semantics for control dependence: we define two relations between graphs called weak and strong projections, and prove that the graph induced by a set of vertices is a weak/strong projection of the original if and only if the set is weakly/strongly control-closed. Thus, all previous forms of control dependence also satisfy our semantics. Weak and strong projections, therefore, precisely capture the essence of controldependence in our generalisations and all the previous, more restricted forms. More fundamentally, these semantics can be thought of as correctness criteria for future definitions of control dependence.
