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 lowcomplexity proofs of the required arithmetic properties.

Tropically Convex Constraint Satisfaction

“…Such relations have been called tropically convex in the literature [12]. 2 This class is a non trivial extension of the max-atoms problem (for instance it contains relations such as x (y + z)/2), and it is not covered by the known reduction to mean payoff games [28,1,3]. Indeed, it is open whether the CSP for tropically convex semilinear relations can be reduced to mean payoff games (in fact, Zwick and Paterson [32] believe that mean payoff games are "strictly easier" than simple stochastic games, which reduce to our problem via the results presented in Section 4).…”

Automatizability and Simple Stochastic Games

“…IGOP states informally that in any finite undirected graph where all nodes are labelled by integers, there exists a vertex whose value is at least as large as its neighbors. This principle is expressible as a CNF formula, and we actually prove that if depth-2 Frege, augmented with IGOP, is weakly automatizable, then SSGs are in P. This raises the very interesting question as to the exact proof theoretic strength required to prove IGOP: If it has a polynomial size depth-2 Frege proof, then our result subsumes that of [4]. On the other hand, if not, then we have found a natural CNF formula separating depth-2 from depth-3 Frege, and furthermore, expose an essential difference between mean payoff games and simple stochastic games.…”

“…Results and Related Work. In a recent paper, Atserias and Maneva made an important new link between automatizability and game theory by proving that solving mean payoff games (MPGs) is reducible to the weak automatizability of depth-2 Frege systems and to feasible interpolation of depth-3 Frege systems [4]. In this paper, we prove that if depth-3 Frege systems are weakly automatizable, then simple stochastic games are solvable in polynomial time, thus establishing a link between SSGs and an important open problem in proof complexity.…”

“…Our proof builds on [4] in that we use and further develop very low depth circuits that they invent for performing addition of a constant number of integers. However, at a high level our proofs are very different.…”

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

“…The exact complexity of this problem is unknown, but is known to lie in NP ∩ co-NP (see [43]). The connection with low-depth proof systems was found by Atserias and Maneva who showed that every MPG can be converted in polynomial time into a set of clauses that is either satisfiable, in which case the first player has a winning strategy, or has a polynomial-size DNF-refutation, in which case the second player has a winning strategy [4]. In particular, this shows that if DNFresolution were automatizable or even weakly automatizable in polynomial time, then MPGs would be solvable in polynomial time.…”