An Experimental Survey of Extended Resolution Effects for SAT Solvers on the Pigeonhole Principle

It has been proven that extended resolution (ER) has more powerful reasoning than general resolution for the pigeonhole principle in Cook’s paper. This fact indicates the possibility that a solver based on extended resolution can exceed Boolean satisfiability problem solvers (SAT solvers for short) based on general resolution. However, few studies have provided practical evidence of this assumption. This paper explores how extended resolution can improve SAT solvers by using the pigeonhole principle, which was the first problem solved by ER in polynomial steps. In fact, although Cook’s paper introduced how to add auxiliary variables, there is no evidence that these variables are really useful for practical solvers. We try to answer the question: If the SAT solver can add appropriate auxiliary variables as proposed in Cook’s paper, can the solver enhance its performance by utilizing these variables? Experimental results show that if the solver properly prioritizes the extended variables in the search, the solver can end the search in a shorter time.