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*Dear all,*

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On Tuesday November 9th, Prakash Panangaden (McGill University, Canada) will give a talk, Quantitative Equational Logic, for our project colloquium from 10am (please note the unusual time). Further details can be found below.

If you would like to attend, please register through the following Google form:

https://forms.gle/6PoGNEfJVHLYDAdKA https://forms.gle/6PoGNEfJVHLYDAdKA

We later send you a zoom link by an email (using BCC).

For the latest information about ERATO colloquium / seminar, please see the webpage https://docs.google.com/document/d/1Qrg4c8XDkbO3tmns6tQwxn5lGHOrBON5LtHXXTpX... .

Jérémy Dubut (ERATO MMSD Colloquium Organizer)

Email: dubut@nii.ac.jp

-------*Tuesday November 9th, 10:00-11:30*

Speaker: *Prakash Panangaden (McGill University, Canada) *

Title: Quantitative Equational Logic

Abstract: Equations are at the heart of mathematical reasoning and reasoning with equations is the subject of equational logic. There are some landmark results in equational logic due to Birkhoff: the completeness theorem and the variety theorem. In the closely related subject of universal algebra there are results about algebraic structures defined equationally. Among these a major result is the existence of free algebras satisfying a universal property. In categorical terms, one can define monads on SET whose Eilenberg-Moore category gives the algebras and whose Kleisli category defines the free algebras. Together with Gordon Plotkin and Radu Mardare, we developed a theory of approximate equational reasoning by introducing the symbol =ε where ε is a (small) real number. One should think of s =ε t as meaning that s and t are within ε in some suitable sense. It turns out that one can define the notion of approximate equational reasoning and prove an analogue of the completeness theorem and the variety theorem. One can also define a quantitative algebra, which is an algebra equipped with a metric and give a construction of free algebras. This time the categorical description amounts to defining monads on EMET, the category of extended metric spaces and nonexpansive maps. More important than the theory is the existence of interesting examples that are very pertinent for probabilistic reasoning. There have been several interesting developments since the original paper in 2016. I will mention some of these but will not go into depth. Our work was aided by the contributions of Giorgio Bacci who played a major role in some of the later developments.

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