The logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: γ, xn → y / γ, xm → y. It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the finite model property with respect to its algebraic semantics and hence that the logic is decidable.
"The finite model property for knotted extensions of propositional linear logic." J. Symbolic Logic 70 (1) 84 - 98, March 2005. https://doi.org/10.2178/jsl/1107298511