  We introduce the framework of Deep Weisfeiler Leman algorithms (DeepWL),
which allows the design of purely combinatorial graph isomorphism tests that
are more powerful than the well-known Weisfeiler-Leman algorithm.
  We prove that, as an abstract computational model, polynomial time
DeepWL-algorithms have exactly the same expressiveness as the logic Choiceless
Polynomial Time (with counting) introduced by Blass, Gurevich, and Shelah (Ann.
Pure Appl. Logic., 1999)
  It is a well-known open question whether the existence of a polynomial time
graph isomorphism test implies the existence of a polynomial time canonisation
algorithm. Our main technical result states that for each class of graphs
(satisfying some mild closure condition), if there is a polynomial time DeepWL
isomorphism test then there is a polynomial canonisation algorithm for this
class. This implies that there is also a logic capturing polynomial time on
this class.
