Proof and refutation in MALL as a game

Abstract

We present a setting in which 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. 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 ¬B might be provable).

Least and Greatest Fixed Points in Linear Logic

“…Focusing organizes proofs in stripes of asynchronous and synchronous rules, removing irrelevant interleavings and inducing a reading of the logic based on macroconnectives aggregating stripes of usual connectives. Focusing is useful to justify game theoretic semantics [Delande and Miller 2008;Delande et al 2010;Miller and Saurin 2006] and has been central to the design of Ludics [Girard 2001]. From the viewpoint of proof search, focusing plays the essential role of reducing the space of the search for a cut-free proof, by identifying situations when backtracking is unnecessary.…”

A Proof Theory for Model Checking

“…The formula (∀n.lt n 10 ⊃ lt n 10) has a proof that involves generating all numbers less than 10 and then showing that they are, in fact, all less than 10. Similarly, a proof of the formula ∀n∀m∀p(lt n 10 ⊃ lt m 10 ⊃ plus n m p ⊃ plus m n p) exists and consists of enumerating all pairs of numbers n, m with n and m less than 10 and checking that the result of adding n + m yields the same value as adding m + n.…”

A framework for proof certificates in finite state exploration

“…Keeping sequents mostly asynchronous allows the asynchronous phase to deal with most of the context: that way the synchronous phase is left with a single, meaningful formula. (The structure of focused proofs based on switchable formulas is similar to the structure of simple games in the game-theoretic analysis of focused proofs in [7,Section 4].) While the restriction to switchable formulas provides a match to the model checking problems we develop here, that restriction is not needed for using clerks and experts (the examples in [5] involve non-switchable formulas).…”