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By Lipsitz S.R., Ibrahim J.G.

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Both ∼ and ≡ are symmetric relations and thus, equivalence relations; 3. ∼ is a preorder (and not an ordering). 4. G ≡ G if and only if G∼G and G ∼G. 2 Bisimulation Semantics of G-Log 19 Proof. Follows immediately from the definition for 1 and 2. For 3, consider the graphs: A A ✒ ❅ ❘ ❅ A ✛ A ■ ❅ ❅ ✠ A ✲ A G1 G2 It holds that G1 ∼G2 and vice versa. However, the two graphs are not the same graph. 1), the two functions ensuring that G∼G and G ∼G are onto. This means that |N | = |N | and, thus, both functions are bijections.

We prove first the claim when ξ is ∼. G ∼-satisfies G when for all G1 G such that RRS ∼G1 there is a G2 G1 such that R∼G2 . But this is exactly the meaning of the formula Φ∼ R. To conclude the proof, notice that the way to compute ΨRξ when ξ is ∼ or ≡ ensures that the result holds. 3 (Rule application). Let G be a G-Log concrete graph and R a rule. Then, for ξ ∈ {∼ ∼, ≡}, G ∈ [[ R ]]ξ (G) → (MG |= ΨRξ ∧ G = G )∨ (MG |= ΨRξ ∧ MG |= ΦξR ∧ G = G )∨ (MG |= ΨRξ ∧ MG |= ΦξR ∧ G = G ∧ MG |= ΦR ) Proof. The proof is by case analysis.

Then, for ξ in {∼, ∼, ≡}, G, G ∈ [[ S ]]ξ if 22 2. Semantics Based on Bisimulation 1. G G , G |=ξ Ri , for i = 1, . . , n, and 2. G is minimal with respect to property (1). Let P be a program S1 , . . , Sn . For ξ in {∼, ∼, ≡}, G0 , Gn ∈ [[ P ]]ξ if and only if there are G1 , . . , Gn−1 such that Gi , Gi+1 ∈ [[ Si+1 ]]ξ , for i = 0, . . , n − 1. 8. Let R be a rule and G be a concrete graph such that G |= R. For ξ in {∼, ∼, ≡}, the ξ-view of G using R, denoted by G|R is the union of all the graphs G G such that R ξ G .

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Estimation with correlated censored survival data with missing covariates by Lipsitz S.R., Ibrahim J.G.

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