Fixed-Points for Quantitative Equational Logics

TL;DR

This paper introduces a novel fixed-point theory for quantitative equational logic using approximate equality, enabling solutions to fixed-point equations and convergence rate analysis within a metric framework.

Contribution

It develops a fixed-point extension of quantitative equational logic with semantics in complete quantitative algebras, incorporating approximate equality and convergence rates.

Findings

Provides a quantitative analogue of Conway and iteration theories.

Reflects the metric coinduction principle.

Applies to Bellman equations in Markov decision processes.

Abstract

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.