Tape diagrams provide a graphical notation for categories equipped with two monoidal products, $\otimes$ and $\oplus$, where $\oplus$ is a biproduct. Recently, they have been generalised to handle Kleisli categories of arbitrary monoidal monads. In this work, we show that for the subdistribution monad, tapes are isomorphic to stochastic matrices of subdistributions of string diagrams. We then exploit this result to provide a complete axiomatisation of probabilistic Boolean circuits.

翻译：带图为一类配备两个幺半积（$\otimes$与$\oplus$）的范畴提供了一种图形化表示方法，其中$\oplus$为双积。近期，该方法被推广至处理任意幺半单子的Kleisli范畴。本文证明，对于子分布单子而言，带图同构于弦图子分布的随机矩阵。我们进而利用这一结果为概率布尔电路提供了完整的公理化体系。