scholarly journals On structuring proof search for first order linear logic

2006 ◽  
Vol 360 (1-3) ◽  
pp. 42-76 ◽  
Author(s):  
Paola Bruscoli ◽  
Alessio Guglielmi
Keyword(s):  
Linear Logic ◽  
Order Linear ◽  
First Order ◽  
Proof Search

The finite model property for various fragments of linear logic

1997 ◽  
Vol 62 (4) ◽  
pp. 1202-1208 ◽  
Author(s):  
Yves Lafont
Keyword(s):  
Classical Logic ◽  
Linear Logic ◽  
Boolean Model ◽  
Finite Model ◽  
Model Property ◽  
Boolean Models ◽  
Finite Model Property ◽  
First Order ◽  
Proof Search ◽  
Model Search

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.

scholarly journals First-order linear logic without modalities is NEXPTIME-hard

1994 ◽  
Vol 135 (1) ◽  
pp. 139-153 ◽  
Author(s):  
P. Lincoln ◽  
A. Scedrov
Keyword(s):  
Linear Logic ◽  
Order Linear ◽  
First Order
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