Section: New Results

A completely symmetric approach to proof and refutation

Participants : Olivier Delande, Dale Miller, Alexis Saurin.

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 [11] 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 [34] 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).