ADEPT: A Heuristic Program for Proving Theorems of Group Theory

A computer program, named ADEPT (A Distinctly Empirical Prover of Theorems), has been written which proves theorems taken from the abstract theory of groups. Its operation is basically heuristic, incorporating many of the techniques of the human mathematician in a "natural" way. This program has proved almost 100 theorems, as well as serving as a vehicle for testing and evaluating special-purpose heuristics. A detailed description of the program is supplemented by accounts of its performance on a number of theorems, thus providing many insights into the particular problems inherent in the design of a procedure capable of proving a variety of theorems from this domain. Suggestions have been formulated for further efforts along these lines, and comparisons with related work previously reported in the literature have been made.

An Algorithm for Group Formation and Maximal Independent Set in an Amorphous Computer

Amorphous computing is the study of programming ultra-scale computing environments of smart sensors and actuators cite{white-paper}. The individual elements are identical, asynchronous, randomly placed, embedded and communicate locally via wireless broadcast. Aggregating the processors into groups is a useful paradigm for programming an amorphous computer because groups can be used for specialization, increased robustness, and efficient resource allocation. This paper presents a new algorithm, called the clubs algorithm, for efficiently aggregating processors into groups in an amorphous computer, in time proportional to the local density of processors. The clubs algorithm is well-suited to the unique characteristics of an amorphous computer. In addition, the algorithm derives two properties from the physical embedding of the amorphous computer: an upper bound on the number of groups formed and a constant upper bound on the density of groups. The clubs algorithm can also be extended to find the maximal independent set (MIS) and $Delta + 1$ vertex coloring in an amorphous computer in $O(log N)$ rounds, where $N$ is the total number of elements and $Delta$ is the maximum degree.