Metastability is a spurious mode of operation in digital signals, where an electrical signal fails to settle into a stable state within a specified time, leading to uncertainty and potentially failing downstream hardware. A system that computes the closure over all possibilities, given an uncertain input, is called a Metastability-containing system. While prior work has addressed metastability-containing systems in the context of combinational and clocked circuits, state machines, and logic formulas, its implications for general-purpose computation remain largely unexplored. In this work, we study the metastability-containing systems within an abstract computational model: The Turing Machine. This approach allows us to investigate the computational limits and capabilities of Turing Machines operating under uncertain inputs. Specifically, we prove that in general the metastable closure of a Turing Machine is non-computable. Then we discuss cases where the meta-stable closure is computable: For EXPTIME problems, we prove that resolving even a single uncertain bit is EXPTIME-complete. In contrast, we prove that for polynomial time problems, the meta-stable closure is polynomial time computable for a logarithmic number of uncertain bits, but coNP-complete, when the number of undefined inputs is arbitrary. Finally, we describe a hardware-realizable Universal Turning Machine that computes the metastable closure of any given bounded-time Turing Machine with at most an exponential blowup in time.

翻译：亚稳态是数字信号中的一种虚假工作模式，指电信号未能在规定时间内稳定到确定状态，导致不确定性并可能使后续硬件失效。能够根据不确定输入计算所有可能结果闭包的系统称为亚稳态包含系统。尽管先前的研究已在组合电路、时序电路、状态机和逻辑公式的背景下探讨了亚稳态包含系统，但其对通用计算的影响仍鲜有涉及。本文在抽象计算模型——图灵机的框架下研究亚稳态包含系统，从而探究不确定输入下图灵机的计算极限与能力。具体而言，我们证明一般情况下图灵机的亚稳态闭包不可计算。随后讨论亚稳态闭包可计算的情形：对于EXPTIME问题，我们证明即便仅解析单个不确定比特也是EXPTIME完全的；相比之下，对于多项式时间问题，当不确定比特数为对数级时，亚稳态闭包可在多项式时间内计算，但当未定义输入数量任意时，则属于coNP完全问题。最后，我们描述了一种可硬件实现的通用图灵机，它能以最多指数级时间膨胀计算任意有界时间图灵机的亚稳态闭包。