Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 247-257
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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/
@article{FPM_2020_23_2_a13,
author = {A. E. Pentus and M. R. Pentus},
title = {Proof nets for the {Lambek} calculus with one division and a~negative-polarity modality for weakening},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {247--257},
year = {2020},
volume = {23},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a13/}
}
TY - JOUR
AU - A. E. Pentus
AU - M. R. Pentus
TI - Proof nets for the Lambek calculus with one division and a negative-polarity modality for weakening
JO - Fundamentalʹnaâ i prikladnaâ matematika
PY - 2020
SP - 247
EP - 257
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UR - http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a13/
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%A A. E. Pentus
%A M. R. Pentus
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%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2020
%P 247-257
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%N 2
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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.
[1] Lambek I., “Matematicheskoe issledovanie struktury predlozheniya”, Matematicheskaya lingvistika, Sbornik perevodov, eds. Yu. A. Shreider i dr., Mir, M., 1964, 47–68
[2] Pentus A. E., Pentus M. R., “Atomarnaya teoriya levogo deleniya dvustoronnikh idealov polukolets s edinitsei”, Fundament. i prikl. matem., 17:5 (2011/2012), 129–146
