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Generic Weakest Precondition Semantics from Monads Enriched with Order
"... Abstract. We devise a generic framework where a weakest precondition semantics, in the form of indexed posets, is derived from a monad whose Kleisli category is enriched by posets. It is inspired by Jacobs’ recent identification of a categorical structure that is common in various predicate transfo ..."
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Abstract. We devise a generic framework where a weakest precondition semantics, in the form of indexed posets, is derived from a monad whose Kleisli category is enriched by posets. It is inspired by Jacobs’ recent identification of a categorical structure that is common in various predicate transformers, but adds generality in the following aspects: 1) different notions of modality (such as “may ” vs. “must”) are captured by EilenbergMoore algebras; 2) nested branching—like in games and in probabilistic systems with nondeterministic environments—is modularly modeled by a monad on the EilenbergMoore category of another. 1
After Example 3, p.2
, 2014
"... implies... ” Here, λ ∗. ηI(i) is the function of type 1 → T I mapping ∗ ∈ 1 to ηI(i). Proof of Lemma 1, p. 7 The proof in the paper only covers the case when I, ∅. We thus cover the case I = ∅ below. From x[⊑J]J∅y, we have!#T J(x) ⊑J!#T J(y); here!T J: ∅ → T J is the unique function. Define a fu ..."
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implies... ” Here, λ ∗. ηI(i) is the function of type 1 → T I mapping ∗ ∈ 1 to ηI(i). Proof of Lemma 1, p. 7 The proof in the paper only covers the case when I, ∅. We thus cover the case I = ∅ below. From x[⊑J]J∅y, we have!#T J(x) ⊑J!#T J(y); here!T J: ∅ → T J is the unique function. Define a function t: J → T ∅ by t = λ j ∈ J. x. From the substitutivity of ⊑, we have t#(!#T J(x)) ⊑ ∅ t#(!#T J(y)). Now t#◦!#T J = (t#◦!T J) # =!#T ∅ = η = idT∅. Therefore x ⊑ ∅ y. Equation (4) of Theorem 11, p.15