## 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).