Clock Systems in Stochastic System Theory

Stochastic Moore machines (low-brow version)

Definition.An interface for an open stochastic system consists of a pair

(A, B)

of measurable spaces, which we write as

(B A)

Definition.An open stochastic system on an interface

(B A)

consists of a measurable space

X

, an update function

φ : A \times X \to Δ (X)

, and an output function

r : X \to B

Definition.A morphism of open stochastic systems from

(A_{1}, B_{1}, X_{1}, ϕ_{1}, r_{1})

(A_{2}, B_{2}, X_{2}, ϕ_{2}, r_{2})

consists

x b a : X_{1} \to X_{2} : B_{1} \to B_{2} : A_{1} \times B_{1} \to A_{2}