We develop a fixed-point extension of quantitative equational logic and give semantics in one-bounded complete quantitative algebras. Unlike previous related work about fixed-points in metric spaces, we are working with the notion of approximate equality rather than exact equality. The result is a novel theory of fixed points which can not only provide solutions to the traditional fixed-point equations but we can also define the rate of convergence to the fixed point. We show that such a theory is the quantitative analogue of a Conway theory and also of an iteration theory; and it reflects the metric coinduction principle. We study the Bellman equation for a Markov decision process as an illustrative example.
翻译:我们开发了定量等式逻辑的固定点延伸,并在一个限制的完整定量代数中给出了语义。与以往关于指标空间中固定点的相关工作不同,我们正在研究近似平等而不是确切平等的概念。结果是一个关于固定点的新理论,它不仅能够为传统的固定点方程提供解决办法,而且我们可以确定与固定点的趋同率。我们表明,这种理论是Conway理论和迭代理论的定量类比;它反映了标准硬体化原则。我们研究Markov决策过程的Bellman等式作为示例。