Mean-payoff games and propositional proofs.

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 @S"2-Frege (i.e. DNF-resolution). This reduces mean-payoff games to the weak automatizability of @S"2-Frege, and to the interpolation problem for @S"2","2-Frege. Since the interpolation problem for @S"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.