The Finite Model Property for Various Fragments of Linear Logic

To show that a formula A is not provable in propositional classical logic, it suuces to exhibit a nite boolean model which does not satisfy A. A similar property holds in the intuitionistic case, with Kripke models instead of boolean models (see for instance TvD88]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a nite model property. In particular, they are decidable: there is a semi-algorithm for provability (proof search) and a semi-algorithm for non provability (model search). For that reason, a logic which is undecidable, such as rst order logic, cannot satisfy a nite model property. The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces Gir87, Gir95], but it is undecidable LMSS92]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete LMSS92], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For aane logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable Kop95a]. Here, we show that the nite phase semantics is complete for MALL and for LLW, but not for MELL. In particular, this gives a new proof of the decidability of LLW. The noncommutative case is mentioned, but not handled in detail. 1. Syntax of linear logic Roman capitals A, B stand for formulas. The connectives of propositional linear logic are: the multiplicatives A (?A) ? = !A ? : One writes A ? B for A ? & B. Greek capitals ?, stand for sequents, which are multisets of formulas, so that exchange is implicit. Identity and cut are written as follows: