In this lecture we continue our discussion of CFI constructions and complete the proof that these graphs give a query separating FPC from P. We also discuss the related notion of counting width.

We discuss relational machines formally and note that polynomial-time relational machines have the same expressive power as FPC. We turn to circuit complexity and discuss invariant circuits, i.e. those that take as input graphs and compute queries invariant under isomorphism. We introduce symmetric circuits, i.e. those where each permutation on the universe of the input graph extend to automorphisms of the circuit (which ensures invariance). We recount results due to Anderson and Dawar establishing an equivalence in expressive power between P-uniform families of symmetric circuits (with threshold gates) and FPC.

In the final part of this lecture we discuss some of the tools used to establish this circuit-based characterisation of FPC. In particular, we discuss what it means for a gate in a circuit to have a support and sketch the proof of a key result establishing bounds on the size of a support of a gate in terms of the size of its orbit. We sketch a translation from symmetric circuits to formulas. We also return to the bijection games used to characterise counting logics and show that they can be played directly on circuits, and so establish a direct game-based characterisation (i.e. one that does not go through the equivalence with counting logics) of the expressive power of symmetric circuits. This result establishes a close connection between the sizes of the supports of the gates in a circuit and the counting width of the query computed, a crucial result used for establishing lower bounds.

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.