Logic Colloquium: Model theory of invariant probabilistic constructions

Colloquium | September 8 | 4:10-5 p.m. | 60 Evans Hall

 Cameron Freer, Founder, Borelian Corporation and Chief Scientist, Remine LLC

 Department of Mathematics

There is a long history of studying the logic obtained by assigning probabilities, instead of truth values, to first-order formulas. In a 1964 paper, Gaifman studied probability distributions on countable structures that are invariant under renaming of the underlying set – which he called "symmetric measure-models", and which are essentially equivalent to what today are known as $S_\infty $-invariant measures. In this paper, he asked the question of which first-order theories admit invariant measures concentrated on the models of the theory.

We answer this question of Gaifman, a key first step towards understanding the model theory of these measures, which can be thought of as "probabilistic structures". In this talk, we will also discuss related questions, such as how many probabilistic structures are models of a given theory, and when probabilistic structures are almost surely isomorphic to a single classical model.

Joint work with Nathanael Ackerman and Rehana Patel.