Positive fragments of coalgebraic logics
Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunctio...
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| Language: | English |
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Logical Methods in Computer Science e.V.
2015-09-01
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| Online Access: | https://lmcs.episciences.org/1594/pdf |
| _version_ | 1797988647982596096 |
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| author | Adriana Balan Alexander Kurz Jiří Velebil |
| author_facet | Adriana Balan Alexander Kurz Jiří Velebil |
| author_sort | Adriana Balan |
| collection | DOAJ |
| description | Positive modal logic was introduced in an influential 1995 paper of Dunn as
the positive fragment of standard modal logic. His completeness result consists
of an axiomatization that derives all modal formulas that are valid on all
Kripke frames and are built only from atomic propositions, conjunction,
disjunction, box and diamond. In this paper, we provide a coalgebraic analysis
of this theorem, which not only gives a conceptual proof based on duality
theory, but also generalizes Dunn's result from Kripke frames to coalgebras for
weak-pullback preserving functors. To facilitate this analysis we prove a
number of category theoretic results on functors on the categories
$\mathsf{Set}$ of sets and $\mathsf{Pos}$ of posets: Every functor
$\mathsf{Set} \to \mathsf{Pos}$ has a $\mathsf{Pos}$-enriched left Kan
extension $\mathsf{Pos} \to \mathsf{Pos}$. Functors arising in this way are
said to have a presentation in discrete arities. In the case that $\mathsf{Set}
\to \mathsf{Pos}$ is actually $\mathsf{Set}$-valued, we call the corresponding
left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$ its posetification. A
$\mathsf{Set}$-functor preserves weak pullbacks if and only if its
posetification preserves exact squares. A $\mathsf{Pos}$-functor with a
presentation in discrete arities preserves surjections. The inclusion
$\mathsf{Set} \to \mathsf{Pos}$ is dense. A functor $\mathsf{Pos} \to
\mathsf{Pos}$ has a presentation in discrete arities if and only if it
preserves coinserters of `truncated nerves of posets'. A functor $\mathsf{Pos}
\to \mathsf{Pos}$ is a posetification if and only if it preserves coinserters
of truncated nerves of posets and discrete posets. A locally monotone
endofunctor of an ordered variety has a presentation by monotone operations and
equations if and only if it preserves $\mathsf{Pos}$-enriched sifted colimits. |
| first_indexed | 2024-04-11T08:06:29Z |
| format | Article |
| id | doaj.art-e5c24c20b0214f788c194cad7c12f7fa |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-11T08:06:29Z |
| publishDate | 2015-09-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-e5c24c20b0214f788c194cad7c12f7fa2022-12-22T04:35:32ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742015-09-01Volume 11, Issue 310.2168/LMCS-11(3:18)20151594Positive fragments of coalgebraic logicsAdriana Balanhttps://orcid.org/0000-0002-4494-6581Alexander KurzJiří VelebilPositive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories $\mathsf{Set}$ of sets and $\mathsf{Pos}$ of posets: Every functor $\mathsf{Set} \to \mathsf{Pos}$ has a $\mathsf{Pos}$-enriched left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$. Functors arising in this way are said to have a presentation in discrete arities. In the case that $\mathsf{Set} \to \mathsf{Pos}$ is actually $\mathsf{Set}$-valued, we call the corresponding left Kan extension $\mathsf{Pos} \to \mathsf{Pos}$ its posetification. A $\mathsf{Set}$-functor preserves weak pullbacks if and only if its posetification preserves exact squares. A $\mathsf{Pos}$-functor with a presentation in discrete arities preserves surjections. The inclusion $\mathsf{Set} \to \mathsf{Pos}$ is dense. A functor $\mathsf{Pos} \to \mathsf{Pos}$ has a presentation in discrete arities if and only if it preserves coinserters of `truncated nerves of posets'. A functor $\mathsf{Pos} \to \mathsf{Pos}$ is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets. A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves $\mathsf{Pos}$-enriched sifted colimits.https://lmcs.episciences.org/1594/pdfmathematics - category theorycomputer science - logic in computer science |
| spellingShingle | Adriana Balan Alexander Kurz Jiří Velebil Positive fragments of coalgebraic logics Logical Methods in Computer Science mathematics - category theory computer science - logic in computer science |
| title | Positive fragments of coalgebraic logics |
| title_full | Positive fragments of coalgebraic logics |
| title_fullStr | Positive fragments of coalgebraic logics |
| title_full_unstemmed | Positive fragments of coalgebraic logics |
| title_short | Positive fragments of coalgebraic logics |
| title_sort | positive fragments of coalgebraic logics |
| topic | mathematics - category theory computer science - logic in computer science |
| url | https://lmcs.episciences.org/1594/pdf |
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