Mean-payoff games and propositional proofs

Abstract

We associate a CNF-formula to every instance of the mean-payoff game problem in such a way that if the value of the game is non-negative the formula is satisfiable, and if the value of the game is negative the formula has a polynomial-size refutation in Σ 2 -Frege (i.e. DNF-resolution). This reduces mean-payoff games to the weak automatizability of Σ 2 -Frege, and to the interpolation problem for Σ 2 , 2 -Frege. Since the interpolation problem for Σ 1 -Frege (i.e. resolution) is solvable in polynomial time, our result is close to optimal up to the computational complexity of solving mean-payoff games. The proof of the main result requires building low-depth formulas that compute the bits of the sum of a constant number of integers in binary notation, and low-complexity proofs of the required arithmetic properties.