Relational Models for the Lambek Calculus with Intersection and Constants
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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V.
2023-12-01
|
| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/10120/pdf |
| _version_ | 1797268531459391488 |
|---|---|
| author | Stepan L. Kuznetsov |
| author_facet | Stepan L. Kuznetsov |
| author_sort | Stepan L. Kuznetsov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T01:33:58Z |
| format | Article |
| id | doaj.art-f03010db146e4c55bda65b89a83ea680 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:33:58Z |
| publishDate | 2023-12-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-f03010db146e4c55bda65b89a83ea6802024-03-08T10:43:59ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742023-12-01Volume 19, Issue 410.46298/lmcs-19(4:32)202310120Relational Models for the Lambek Calculus with Intersection and ConstantsStepan L. KuznetsovWe 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.https://lmcs.episciences.org/10120/pdfcomputer science - logic in computer sciencemathematics - logic03g15f.4.1 |
| spellingShingle | Stepan L. Kuznetsov Relational Models for the Lambek Calculus with Intersection and Constants Logical Methods in Computer Science computer science - logic in computer science mathematics - logic 03g15 f.4.1 |
| title | Relational Models for the Lambek Calculus with Intersection and Constants |
| title_full | Relational Models for the Lambek Calculus with Intersection and Constants |
| title_fullStr | Relational Models for the Lambek Calculus with Intersection and Constants |
| title_full_unstemmed | Relational Models for the Lambek Calculus with Intersection and Constants |
| title_short | Relational Models for the Lambek Calculus with Intersection and Constants |
| title_sort | relational models for the lambek calculus with intersection and constants |
| topic | computer science - logic in computer science mathematics - logic 03g15 f.4.1 |
| url | https://lmcs.episciences.org/10120/pdf |
| work_keys_str_mv | AT stepanlkuznetsov relationalmodelsforthelambekcalculuswithintersectionandconstants |