St̊almarck’s Method versus Resolution: A Comparative Theoretical Study

This Master’s thesis presents a comparative analysis of St̊almarck’s proof method and resolution from a theoretical perspective. We give (to our knowledge) the first complete explicit formal description of the dilemma proof system underlying St̊almarck’s method. Based on this description we prove a number of simulation and separation results between different subsystems of dilemma (defined by restrictions on possible branching assumptions and rules for merging the results derived in distinct branches). The key result of the thesis is that dilemma depth translates into resolution width. More precisely, a dilemma refutation in depth d and length L of a k-CNF formula F can be transformed to a resolution refutation of F in width O (kd) and length ( Lk ) . From this depth-width relation it follows that for k-CNF formulas with k fixed, resolution p-simulates dilemma restricted to minimum-depth proofs. Furthermore, the running time of the minimum-width proof search algorithm suggested by Ben-Sasson and Wigderson is shown to be polynomial in the running time of St̊almarck’s method. Finally, using results by Beame et al. we show that if F is a uniformly random 3-CNF formula with ∆n clauses on n variables, then with high probability it holds for the dilemma hardness HD (F ) that Ω ( n/∆ ) ≤ HD (F ) ≤ O(n/∆) (where is an arbitrarily small but positive constant).