A Dichotomy for Real Boolean Holant Problems

We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions. It is proved that for every set F of real-valued constraint functions, Holant(F) is either P-time computable or #P-hard. The classification has an explicit criterion. This is a culmination of much research on this problem, and it uses many previous results and techniques. Dealing with some concrete functions plays an important role in this proof. In particular, two functions, called f6 and f8, and their associated families exhibit intriguing and extraordinary closure properties related to Bell states in quantum information theory.