Computer science theory provides many different measures of complexity of a system including Kolmogorov complexity, logical depth, computational depth, and Levin complexity. However, these measures are all defined only for deterministic Turing machines, i.e., deterministic dynamics of the underlying generative process whose output we are interested in. Therefore, by construction they cannot capture complexity of the output of stochastic processes - like those in the real world. Motivated by this observation, we combine probabilistic Turing machines with a prior over the inputs to the Turing machine to define a complete stochastic process of Turing machines. We call this a stochastic process Turing machine. We use stochastic process Turing machines to define a set of new generative complexity measures based on Turing machines, which we call stochastic depth. As we discuss, stochastic depth is related to other such measures including Kolmogorov complexity and Levin complexity. However, as we elaborate, it has many desirable properties that those others measures lack. In addition, stochastic depth is closely related to various thermodynamic properties of computational systems. Stochastic process Turing machines and stochastic depth allow us to study complex, stochastic systems like the human brain, societies, and evolution all from within the framework of formal computation.

翻译：计算机科学理论提供了多种衡量系统复杂性的度量标准，包括柯尔莫哥洛夫复杂度、逻辑深度、计算深度和莱文复杂度。然而，这些度量均仅针对确定性图灵机定义，即仅适用于我们关注其输出的底层生成过程的确定性动力学。因此，这些度量在构造上无法刻画随机过程输出的复杂性——例如现实世界中的诸多过程。受此启发，我们将概率图灵机与图灵机输入的先验分布相结合，定义了完整的图灵机随机过程，称之为随机过程图灵机。基于随机过程图灵机，我们提出了一类新的基于图灵机的生成性复杂度度量，称为随机深度。如文中所述，随机深度与柯尔莫哥洛夫复杂度、莱文复杂度等其他度量存在理论关联，但同时也具备许多其他度量所缺乏的优良性质。此外，随机深度与计算系统的多种热力学特性密切相关。随机过程图灵机与随机深度的提出，使得我们能够在形式化计算框架内研究人类大脑、社会系统、演化过程等复杂的随机系统。