Quantitative algebraic reasoning

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Mardare, Radu and Panangaden, Prakash and Plotkin, Gordon; (2016) Quantitative algebraic reasoning. In: LICS '16 : Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science. ACM, USA, pp. 700-709. ISBN 9781450343916 (https://doi.org/10.1145/2933575.2934518)

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Abstract

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.

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