Asymmetric Morita Baylis Hillman Reaction

Asymmetric Morita Baylis Hillman Reaction

Abstract

The main purpose of this paper is to discuss the asymmetric morita baylis Hillman reaction. It is interesting to note that we could introduce a notion of time by defining the “time” at which an operation occurs to be the maximum t such that the operation occurs after t system steps (or 0 if there is no such t). Axioms A5 and A6 of Part I imply that this maximum always exists. Axiom A5 and the assumption that there are no nonterminating elementary operation executions imply that “time” increases without bound—i.e., there are operations occurring at arbitrarily large “times”.

Asymmetric Morita Baylis Hillman Reaction

Introduction

Definition 1 A system step is an operation execution consisting of one normal elementary operation execution from every process. An operation execution A is said to occur after t system steps if there exist system steps S1, St such that

S1 -? · · · -? St -? A.

Since we only need the concept of eventuality, we will not consider this way of defining “time”. We can now define what it means for a property to hold “eventually”. Deadlock freedom and lockout freedom state that something eventually happens—for example, deadlock freedom states that so long as some process is executing it's trying operation, then some process eventually executes its critical section. Since “eventually X eventually happens” is equivalent to “X eventually happens”, requiring that these two properties eventually hold is the same as simply requiring that they hold. We say that the mutual exclusion and FCFS properties eventually hold if they can be violated only for a bounded “length of time”. Thus, the mutual exclusion property eventually holds if there is some t such that any two critical section executions CS[k] i and CS[m] j that both occur after t system steps are not concurrent. Similarly, the FCFS property holds eventually if it holds whenever both the doorway executions occur after t system steps. The value of t must be independent of the particular execution of the algorithm, but it may depend upon the number N of processes. If a property eventually holds under the above type of transient mal- function behavior, then we say that the algorithm is self-stabilizing for that property.

Remarks on “Forever”

In our definition of failure, we could not allow a malfunctioning process to fail again after it had resumed its normal behavior, since repeated malfunctioning and recovery can be indistinguishable from continuous malfunctioning. However, if an algorithm satisfies any of our properties under the assumption that a process may malfunction only once, then it will also satisfy the property under repeated malfunctioning and recovery—so long as the process waits long enough before malfunctioning again. The reason for this is that all our properties require that something either be true at all times (mutual exclusion, FCFS) or that something happens in the future (deadlock freedom, lockout freedom). If something remains true during a malfunction, then it will also remain true under repeated ...