Quantitative Algebraic Reasoning - Mardare, Radu; Panangaden, Prakash; Plotkin, Gordon (Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, 2016)
We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We define an equality relation indexed by rationals: a = ε b which we think of as saying that "a is approximately equal to b up to an error of ε ". We have 4 interesting examples where we have a quantitative equational theory whose free algebras correspond to well known structures. In each case we have finitary and continuous versions. The four cases are: Hausdorff metrics from quantitive semilattices; p-Wasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed barycentric algebras and the total variation metric from a variant of barycentric algebras.