We have so far seen that polynomial-time relational machines, P-uniform families of symmetric circuits, and FPC all have the same expressive power. We also have a means of characterising the expressive power of these models via bijection games. This suggests a robust theory of symmetric computation. In this lecture we discuss a natural symmetric model of computation that arose for independent reasons in linear programming.

We describe what it means for a system of linear equations to decide a language (roughly, it defines a polytope whose projection includes all members in the language and excludes those not in the language). We discuss a recent result establishing a close connection between polynomial-size families of symmetric linear programs, bounded-variable polynomial-size families of formulas of counting logics, and polynomial-size families of symmetric circuits (with threshold gates). In order to sketch a proof of this result we first discuss a theorem establishing that linear program feasibility is in FPC, which itself is proved by showing that the ellipsoid method can be implemented in the logic. We also discuss how this can be used to prove that the perfect matching problem is definable in FPC. We finally return to the characterisation of symmetric linear programs and sketch the proof. We note that we use similar tools in our analysis of symmetric linear programs as for the characterisation of symmetric circuits (e.g. supports).

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.