  Holant problems capture a class of Sum-of-Product computations such as
counting matchings. It is inspired by holographic algorithms and is equivalent
to tensor networks, with counting CSP being a special case. A classification
for Holant problems is more difficult to prove, not only because it implies a
classification for counting CSP, but also due to the deeper reason that there
exist more intricate polynomial time tractable problems in the broader
framework.
  We discover a new family of constraint functions $\mathscr{L}$ which define
polynomial time computable counting problems. These do not appear in counting
CSP, and no newly discovered tractable constraints can be symmetric. It has a
delicate support structure related to error-correcting codes. Local holographic
transformations is fundamental in its tractability. We prove a complexity
dichotomy theorem for all Holant problems defined by any real valued constraint
function set on Boolean variables and contains two 0-1 pinning functions.
Previously, dichotomy for the same framework was only known for symmetric
constraint functions. he set $\mathscr{L}$ supplies the last piece of
tractability. We also prove a dichotomy for a variant of counting CSP as a
technical component toward this Holant dichotomy.
