Hennessy-Milner Type Theorems for Fuzzy Multimodal Logics Over Heyting Algebras
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arXiv
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| Format: | Preprint |
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2025
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| author | Stanković, Marko Ćirić, Miroslav Ignjatović, Jelena |
| author_facet | Stanković, Marko Ćirić, Miroslav Ignjatović, Jelena |
| contents | In a recent paper, we have introduced two types of fuzzy simulations (forward and backward) and five types of fuzzy bisimulations (forward, backward, forward-backward, backward-forward and regular) between Kripke models for the fuzzy multimodal logics over a complete linearly ordered Heyting algebra. In this paper, for a given non-empty set $Ψ$ of modal formulae, we introduce the concept of a weak bisimulation between Kripke models. This concept can be used to express the degree of equality of fuzzy sets of formulae from $Ψ$ that are valid in two worlds $w$ and $w'$, that is, to express the degree of modal equivalence between worlds $w$ and $w'$ with respect to the formulae from $Ψ$. We prove several Hennessy-Milner type theorems. The first theorem determines that the greatest weak bisimulation for the set of plus-formulae between image-finite Kripke models coincides with the greatest forward bisimulation. The second theorem determines that the greatest weak bisimulation for the set of minus-formulae between domain-finite Kripke models coincides with the greatest backward bisimulation. Finally, the third theorem determines that the greatest weak bisimulation for the set of all modal formulae between the degree-finite Kripke models coincides with the greatest regular bisimulation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_10126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hennessy-Milner Type Theorems for Fuzzy Multimodal Logics Over Heyting Algebras Stanković, Marko Ćirić, Miroslav Ignjatović, Jelena Logic 03B45, 03B50, 03B52, 03E72 F.4.1 In a recent paper, we have introduced two types of fuzzy simulations (forward and backward) and five types of fuzzy bisimulations (forward, backward, forward-backward, backward-forward and regular) between Kripke models for the fuzzy multimodal logics over a complete linearly ordered Heyting algebra. In this paper, for a given non-empty set $Ψ$ of modal formulae, we introduce the concept of a weak bisimulation between Kripke models. This concept can be used to express the degree of equality of fuzzy sets of formulae from $Ψ$ that are valid in two worlds $w$ and $w'$, that is, to express the degree of modal equivalence between worlds $w$ and $w'$ with respect to the formulae from $Ψ$. We prove several Hennessy-Milner type theorems. The first theorem determines that the greatest weak bisimulation for the set of plus-formulae between image-finite Kripke models coincides with the greatest forward bisimulation. The second theorem determines that the greatest weak bisimulation for the set of minus-formulae between domain-finite Kripke models coincides with the greatest backward bisimulation. Finally, the third theorem determines that the greatest weak bisimulation for the set of all modal formulae between the degree-finite Kripke models coincides with the greatest regular bisimulation. |
| title | Hennessy-Milner Type Theorems for Fuzzy Multimodal Logics Over Heyting Algebras |
| topic | Logic 03B45, 03B50, 03B52, 03E72 F.4.1 |
| url | https://arxiv.org/abs/2502.10126 |