Giuseppe Greco
580 California St., Suite 400
San Francisco, CA, 94104
Neighbourhood Semantics for Graded Modal Logic
2021, Bulletin of the Section of Logic
https://doi.org/10.18778/0138-0680.2021.12
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Abstract
We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation.
References (23)
- Lemma 5.5. Let F = (W, {ν n } n∈N ) be a neighbourhood frame satisfying ( ). Then for w ∈ W , 1. If ν 1 (w) = ∅, then ν n (w) = ∅ for n > 1.
- If ν 1 (w) = ∅, then ν n (w) =↑ P ≥n (A) for n > 1, where A is the maximum atomic set in ν 1 (w).
- Proof: For item 1, we prove by contradiction. Assume that ν 1 (w) = ∅ and for some n > 1, X ∈ ν n (w). By ( 3), X = ∅. By ( 4) and ( 5), there exists X ⊆ X such that X is atomic in ν 1 (w). By ( 3), X = ∅. By atomicity of X , ν 1 (w) = ∅, contradiction . Now we prove item 2 and assume that X ∈ ν n (w). By ( 4), there exists a minimal element of ν n (w) such that Y ⊆ X. By ( 5), |Y | ≥ n and Y is atomic in ν 1 (w). Since A is the maximum atomic set of ν 1 (w), Y ⊆ A. Since |Y | ≥ n, Y ∈ P ≥n (A). Since Y ⊆ X, X ∈↑P ≥n (A). Assume that X ∈↑P ≥n (A). Then there exists Y ∈ P ≥n (A) such that
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- ∈ R [w ] with uZu . Let X = {u }. It follows that X = 1 and that X and X form a Z-pair. Consider the case that i > 1. We may assume that X = {u} ∪ Y , where Y ⊆ R[w] and u ∈ Y . It follows that |Y | = i -1 ≥ 1. By induction hypothesis, there exists an Y ⊆ R [w ] such that Y = i -1 and that Y and Y forms a Z-pair. Since u ∈ R[w], by Lemma 7.7, there exists u ∈ R [w ] with uZu . If u ∈ Y , let X = Y ∪ {u }. Then X = i and X and X forms a Z-pair. If u ∈ Y , there are two subcases: ∃y ∈ Y ∃v ∈ R [w ]\Y : yZv and for all y ∈ Y and v ∈ R [w ]\Y , not yZv . Consider the case that ∃y ∈ Y ∃v ∈ R [w ]\Y : yZv . Let X = Y ∪ {v }. Then X = i. Since Y and Y form a Z-pair, uZu and yZv , X and X form a Z-pair. Consider the case that for all y ∈ Y and v ∈ R [w ] \ Y , not yZv .
- Since X ∈↑ P ≥i (R[w]), by (Forth), there exists B ∈↑ P ≥i (R [w ]) such that ∀b ∈ B ∃x ∈ X : xZb . Since B ∈↑P ≥i (R [w ]), there exists B ⊆ B such that B ⊆ R [w ] and B ≥ i. Since Y = i -1, there exists b ∈ B such that b ∈ R [w ]\Y . Since for all y ∈ Y and v ∈ R [w ]\Y , not yZv , we have for all y ∈ Y , not yZb . Since ∀b ∈ B ∃x ∈ X : xZb and X = {u} ∪ Y , we have uZb . Let X = Y ∪ {b }. Then X = i. Since Y and Y form a Z-pair and uZb , X and X form a Z-pair. Claim (2) can be proved in a similar way by using (Back).
- Proposition 7.9. Let M = (W, R, V ) and M = (W , R , V ) be Kripke models and Z ⊆ W × W a non-empty relation such that Z : M g M . Define a tuple of relations Z = (Z 1 , Z 2 , . . .) as: Z 1 = {({w}, {w }) | wZw }, and Z n = {(X, X ) | |X| = X = n, X and X form a Z-pair}, for n > 1. Then Z : M gt M . Proof: Since Z is non-empty, Z 1 is non-empty. So item (1) in Definition 7.5 is satisfied. Items (2), (3) and (4) are satisfied by the definition of Z. Items (5) and (6) are satisfied by Lemma 7.8. Item (7) is satisfied by the definition of Z i and the definition of Z-pairs. In summary, we showed how to construct a graded bisimulation out of a graded tuple bisimulation (Prop. 7.6), and vice versa (Prop. 7.9). Hence, graded bisimulation (Def. 7.3) and graded tuple bisimulation (Def. 7.5) are equivalent. Another notion of bisimulation called resource bisimulation was proposed in [1], which is very similar to the notion later proposed in References
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