Common use of Monads and Modalities for Probabilistic Branching Clause in Contracts

Monads and Modalities for Probabilistic Branching. are subject to: (x y) z x (y z), x 0 x and x y y x, where is the Kleene equality. A generalized effect algebra is a PCM (M, , 0) that is positive (x y = 0 x = y = 0) and cancellative (x y = x z y = z). ❽ ❽ ❽❽ ⇒ ❽ ❽ ❽ ❽ ⇒ ' A generalized effect module is a generalized effect algebra M with a scalar multiplication · : [0, 1] × M → M that satisfies (r s) · x ' (r · x) (s · x), r · (x y) ' (r · x) (r · y), 1 · x = x and r · (s · x) = (r · s) · x. Here for r, s ∈ [0, 1] the partial ❽ sum r s = r + s is defined when r + s ≤ 1. P One of the following monads replaces in Section 2. We impose the restriction of countable supports.