Many Hard Examples in Exact Phase Transitions 1

This paper analyzes the resolution complexity of two random CSP models, i.e. Model RB/RD for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, this paper proves that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose a model with both many hard instances and exact phase transitions. Finally, conclusions and future work are presented, as well as a discussion of the main difference between Model RB/RD and some other models with exact phase transitions.