13/11/2008 Dale Miller (INRIA-LIX, Palaiseau)

Proof and refutation in MALL as a game

We present a setting in which the search for a proof of B or a refutation of B (ie, a proof of (not B)) can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving (not B) as two, possibly unrelated, activities. Our 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 counter-winning strategy translates to a refutation of the formula. The game is described for multiplicative and additive linear logic (MALL). 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 (not B) might be provable). [This talk is based on papers in GaLoP05 and LICS08 written with Olivier Delande and Alexis Saurin.]