Causality serves as an abstract notion of time for concurrent systems. A computation is causal, or simply valid, if each observation of a computation event is preceded by the observation of its causes. The present work establishes that this simple requirement is equally relevant when the occurrence of an event is invertible. We propose a conservative extension of causal models for concurrency that accommodates reversible computations. We first model reversible computations using a symmetric residuation operation in the general model of configuration structures. We show that stable configuration structures, which correspond to prime algebraic domains, remain stable under the action of this residuation. We then derive a semantics of reversible computations for prime event structures, which is shown to coincide with a switch operation that dualizes conflict and causality.

翻译：因果关系为并发系统提供了一种抽象的时间概念。若计算中每个事件的发生都以其因果关系的发生为前提，则该计算是因果的，或简称为有效的。本研究证明，当事件的发生可逆时，这一基本要求同样适用。我们提出了一种适用于可逆计算的并发因果模型的保守扩展。首先，我们在配置结构的一般模型中，利用对称剩余运算对可逆计算进行建模。研究表明，对应于素代数域的稳定配置结构在此剩余运算作用下仍保持稳定。随后，我们推导出素事件结构中可逆计算的语义，并证明其与一种对偶化冲突和因果关系的切换操作相一致。