A neutral approach to proof and refutation in MALL

We propose a setting in which the search for a proof of B or a refutation of B (a proof of ¬ B) can be carried out simultaneously: this is in contrast to the usual approach in automated deduction where we need to commit to proving either B or ¬-B. Our neutral approach to proof and refutation is described as a two player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a winning counter-strategy translates to a refutation of the formula. The game is described for multiplicative and additive linear logic without atomic formulas, A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win (that is, neither B nor ¬-B might be provable).