Metastability-Containing Turing Machines

2026-04-19

AI Summary

This study investigates the computational power of Turing machines operating under metastable (uncertain) inputs. It introduces, for the first time, a metastability-tolerant mechanism into a universal computational model, integrating complexity theory with explicit Turing machine constructions to analyze closure properties under uncertainty. The main contributions include proving that the metastable closure of general Turing machines is uncomputable; demonstrating that for EXPTIME problems, even a single uncertain input bit renders them EXPTIME-complete; and showing that while P problems remain solvable in polynomial time under logarithmically bounded uncertainty, they become coNP-complete under arbitrarily many uncertain bits. Furthermore, the work presents a universal metastability-tolerant Turing machine that incurs at most exponential time overhead and is amenable to hardware implementation.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Calibration & Uncertainty Quantification
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• metastability
• Turing Machine
• uncertain input
• computability
• computational complexity

Innovation

• metastability-containing
• Turing Machine
• computational complexity
• uncertain inputs
• universal computation