一句话：研究带交与零/单位常元的 Lambek 演算关系语义，厘清其弱/强完全性的成立边界。
本文研究带交运算以及零元/单位元常元的 Lambek 演算的关系语义（R-model）。已有工作（Andreka & Mikulas, 1994）对不含常元且禁止空前件的变体证明了强完全性。作者表明：一旦去掉禁止空前件的限制，强完全性不再成立；但在常元的非标准解释下可证明弱完全性，而标准解释下连弱完全性也失败。该弱完全性结果还可扩展到无穷情形，即所谓迭代除法（除法下的 Kleene 星）。此外，文章对无积（product-free）片段证明了若干强完全性结果。方法上采用关系模型并对常元解释与前提限制进行细致区分，从而厘清 Lambek 演算在关系语义下的完全性边界。
We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.
@article{paperbot2163,
title = {Relational Models for the Lambek Calculus with Intersection and Constants},
author = {Stepan L. Kuznetsov},
journal = {Logical Methods in Computer Science},
volume = {Volume 19, Issue 4},
year = {2023},
doi = {10.46298/lmcs-19(4:32)2023}
}