Anuj Dawar (University of Cambridge)
Gregory Wilsenach (University of Cambridge)
Abstract: The notion of a symmetric algorithm, i.e. one with an explicit combinatorial property that guarantees isomorphism-invariant computation, arises naturally in the context of database theory, finite model theory, circuit complexity, the theory of relational machines, and theory of linear programming. The apparently very different models of symmetric computation developed in these disparate fields have recently been shown to be closely related both in terms of expressive power and underlying theory. The set of ideas that emerge in all of these cases have coalesced into a coherent and robust theory of symmetric computation.
This course will introduce this exciting and emerging new theory. In particular, we will present the various symmetric models introduced in each of these fields and establish their close relationship. We will develop the common theory and the methods for proving upper and lower bounds on expressive power. These rest on the Weisfeiler-Leman equivalences, the related notion of counting width, and logical reductions. Lastly, we will discuss extensions of these models and a number of open questions.