Section: New Results
A completely symmetric approach to proof and refutation
A couple of years ago, Miller and Saurin proposed a neutral approach to proof and refutation. The goal was to describe an entirely neutral setting where a step in a “proof search” could be seen as a step in either building a proof of a formula or a proof of its negation. The early work was limited to essentially a simple generalization to additive logic. Delande was able to generalize that work to capture multiplicative connectives as well. His thesis  contains two game semantics for multiplicative additive linear logic (MALL): the first is sequential and the second is concurrent. The concurrent game was used to capture full completeness results between MALL (focused) sequent calculus proofs and winning strategies.
The paper  provides a detailed description of the sequent approach to both additive and multiplicative linear logic. In that paper, the search for a proof of B or a refutation of B (i.e. , a proof of ¬B ) can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Their 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 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).