LSM is not generated by binary functions
  The material in this note is now superseded by arXiv:1108.5288v4.   Bulatov et al. [1] defined the operation of (efficient) pps_\omega-definability in order to study the computational complexity of certain approximate counting problems. They asked whether all log-supermodular functions can be defined by binary implication and unary functions in this sense. We give a negative answer to this question. 
Colin McQuillan
cs.CC