Geodesic
Proof nets for the Lambek calculus with one division and a~negative-polarity modality for weakening
Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 247-257.

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In this paper, we introduce a variant of the Lambek calculus allowing empty antecedents. This variant uses two connectives: the left division and a unary modality that occurs only with negative polarity and allows weakening in antecedents of sequents. We define the notion of a proof net for this calculus, which is similar to those for the ordinary Lambek calculus and multiplicative linear logic. We prove that a sequent is derivable in the calculus under consideration if and only if there exists a proof net for it. Thus, we establish a derivability criterion for this calculus in terms of the existence of a graph with certain properties. The size of the graph is bounded by the length of the sequent. 
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A. E. Pentus; M. R. Pentus. Proof nets for the Lambek calculus with one division and a~negative-polarity modality for weakening. Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 247-257. http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a13/

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