  We present a general method for converting any family of unsatisfiable CNF
formulas that is hard for one of the simplest proof systems, tree resolution,
into formulas that require large rank in any proof system that manipulates
polynomials or polynomial threshold functions of degree at most k (known as
Th(k) proofs). Such systems include Lovasz-Schrijver and Cutting Planes proof
systems as well as their high degree analogues.
  These are based on analyzing two new proof systems, denoted by T^cc(k) and
R^cc(k). The proof lines of T^cc(k) are arbitrary Boolean functions, each of
which can be evaluated by an efficient k-party randomized communication
protocol. They include Th{k-1} proofs as a special case. R^cc(k) proofs are
stronger and only require that each inference be locally checkable by an
efficient k-party randomized communication protocol.
  Our main results are the following:
  (1) When k is O(loglogn), for any unsatisfiable CNF formula F requiring
resolution rank r, there is a related CNF formula G=Lift_k(F) requiring
refutation rank r^Omega(1/k) log^O(1) n in all R^cc(k) systems.
  (2) There are strict hierarchies for T^cc(k) and R^cc(k) systems with respect
to k when k is O(loglogn in that there are unsatisfiable CNF formulas requiring
large rank R^cc(k) refutations but having log^O(1) n rank Th(k) refutations.
  (3) When k is O(loglogn) there are 2^(n^Omega(1/k)) lower bounds on the size
of tree-like T^cc(k) refutations for large classes of lifted CNF formulas.
  (4) A general method for producing integrality gaps for low rank R^cc(2)
inference (and hence Cutting Planes and Th(1) inference) based on related gaps
for low rank resolution. These gaps are optimal for MAX-2t-SAT.
