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The finite model property for various fragments of linear logic

Published online by Cambridge University Press:  12 March 2014

Yves Lafont
Yves Lafont*
Affiliation:
Institut de Mathématiques de Luminy, UPR 9016 du CNRS, 163 Avenue de Luminy, Case 930, 13288 Marseille Cedex 9, France, E-mail: lafont@iml.univ-mrs.fr
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Extract

To show that a formula A is not provable in propositional classical logic, it suffices to exhibit a finite 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 [11]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a finite 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 first order logic, cannot satisfy a finite model property.

The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces [2, 3], but it is undecidable [9]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete [9], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For affine logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable [5].

Here, we show that the finite 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1202 - 1208
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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