Quantitative equational reasoning

Prakash Panangaden

Reasoning with equations is a central part of mathematics. Typically we think of solving equations but another role they play is to define algebraic structures like groups or vector spaces. Equational logic was formalized and developed by Birkhoff in the 1930s and led to a subject called universal algebra. Universal algebra was used in formalizing concepts of data types in computer science. In this talk I will present a quantitative analogue of equational logic: we write expressions like s =_\epsilon t with the intended interpretation "s is within \epsilon of t". It turns out that the metatheory of equational logic can be redeveloped in this setting. Perhaps this seems like sterile theory but what makes it come alive is some striking examples. A notion of distance between probability distributions called the Kantorovich metric (frequently called the Wasserstein metric) has become important in the theory of probabilistic systems and in parts of machine learning. It turns out that this metric emerges naturally as the "free algebra" of some simple equational axioms in our extended sense.