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. One of the following monads replaces in Section 2. We impose the restriction of countable supports. 5.1 (the (sub)distribution monad D=1, D≤1). The distri- bution monad D=1 on Set is such that: D=1X = {p : X → [0, 1] | The category of general effect modules (with a straightforward notion of their morphism, see [4]) is denoted by GEMod. An example of a generalized effect module is the set D≤1X of sub- distributions over X. Here p ❽ q ∈ D≤1X is given by (p ❽ q)(x) = Σ ∈ 1. The set ≤1X comes with an obvious scalar multiplica- tion, too. The unit interval [0, 1] is another example; so is its prod- ucts [0, 1]X . See [4] for details. For our purpose of healthiness conditions, we have to study the monad map induced by the ≤1-algebra τtotal. It shall also be denoted by τtotal. The following is easy. Σ