About 25 years ago, it came to light that a single combinatorial property determines both an important dividing line in model theory and machine learning. The following years saw a fruitful exchange of ideas between PAC-learning and the model theory of NIP structures. In this article, we point out a new and similar connection between model theory and machine learning, this time developing a correspondence between stability and learnability in various settings of online learning. In particular, this gives many new (...) examples of mathematically interesting classes which are learnable in the online setting. (shrink)

We set up a general context in which one can prove Sauer-Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer-Shelah type lemma in the context of op-rank, a notion of Guingona and Hill [4].

We study and develop a notion of isogeny for superstable groups inspired by the notion in algebraic groups and differential algebraic notions developed by Cassidy and Singer. We prove several fundamental properties of the notion. Then we use it to formulate and prove a uniqueness results for a decomposition theorem about superstable groups similar to one proved by Baudisch. Connections to existing model theoretic notions and existing differential algebraic notions are explained.

In this paper, it is shown that if [Formula: see text] is a complete type of Lascar rank at least 2, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations [Formula: see text], [Formula: see text] such that p has a nonalgebraic forking extension over [Formula: see text]. Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over [Formula: see text]. The results are (...) also formulated in a more general setting. (shrink)