## 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.

Original language | English |
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Title of host publication | LICS '16 : Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science |

Place of Publication | New York, NY. |

Pages | 700-709 |

Number of pages | 10 |

DOIs | |

Publication status | Published - 5 Jul 2016 |

Event | 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016 - New York, United States Duration: 5 Jul 2016 → 8 Jul 2016 |

### Conference

Conference | 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016 |
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Country/Territory | United States |

City | New York |

Period | 5/07/16 → 8/07/16 |

## Keywords

- algebraic reasoning
- quantitative algebra
- probabilistic programming
- semantics
- computer circuits
- reconfigurable hardware
- equality relations
- equational theory
- Kantorovich metric