Quantitative Equational Reasoning

Giorgio Bacci, Radu Mardare, Prakash Panangaden, Gordon D. Plotkin

Research output: Contribution to book/anthology/report/conference proceedingBook chapterResearchpeer-review


Equational logic has been a central theme in mathematical reasoning and in reasoning about programs. We introduce a quantitative analogue of equational reasoning that allows one to reason about approximate equality. The equality symbol is annotated with a real number that describes how far apart two terms can be. We develop the counterparts of standard results of equational logic, in particular, a completeness theorem. We define quantitative algebras and free quantitative algebras which yield monads on categories of metric spaces. We show that key examples of probability metrics, in particular, the Kantorovich metric and the Wasserstein p-metrics, arise from simple quantitative theories. Finally we develop a quantitative version of the theory of effects in programming languages.

Original languageEnglish
Title of host publicationFoundations of Probabilistic Programming
EditorsGilles Barthe, Joost-Pieter Katoen, Alexandra Silva
PublisherCambridge University Press
Publication dateDec 2020
ISBN (Print)9781108488518
ISBN (Electronic)9781108770750
Publication statusPublished - Dec 2020


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