Redundant Implementations of Algebraic Machines

Abstract

This chapter extends the algebraic approach of Chapter 3 in order to provide fault tolerance to group and semigroup machines. The discussion characterizes redundant implementations using algebraic homomorphisms and demonstrates that for a particular error-correcting scheme there exist many possible redundant implementations, each potentially offering different fault coverage [Had-jicostis, 1999]. The fault model assumes that the error detecting/correcting mechanism is fault-free and considers faults that cause the redundant machine to transition to an incorrect state. Explicit connections to hardware implementations and hardware faults are addressed in Chapter 5 for linear time-invariant dynamic systems (implemented using delay, adder and gain elements) and in Chapter 6 for linear finite-state machines (implemented using XOR gates and flip-flops). The issue of faults in the error corrector is studied in Chapter 7. Related work appeared in the context of providing fault tolerance to arbitrary finite-state machines via external monitoring mechanisms [Iyengar and Kinney, 1985; Leveugle and Saucier, 1990; Parekhji et al., 1991; Robinson and Shen, 1992; Leveugle et al., 1994; Parekhji et al., 1995]. This work, however, was not formulated in an algebraic setting and does not make use of algebraic properties and structure.