Fault-Tolerant Computation in Semigroups and Semirings

The traditional approach to fault-tolerant computation has been via modular redundancy. Although universal and simple, modular redundancy is inherently expensive and inefficient. By exploiting particular structural features of a computation or algorithm, recently developed Algorithm-Based Fault Tolerance (ABFT) techniques manage to offer more efficient fault coverage at the cost of narrower applicability and harder design. In the special case of arithmetic codes, previous work has shown that a variety of useful results and constructive procedures can be obtained when the computations take place in an abelian group. In this thesis, we develop a systematic algebraic approach for computations occurring in an abelian semigroup, thereby extending to a much more general setting many of the results obtained earlier for the group case. Examples of the application of these results to representative semigroups and higher semigroup-based algebraic structures, such as a semiring, are also included. Thesis Supervisor: George C. Verghese Title: Professor of Electrical Engineering