Metastability-Containing Turing Machines

TL;DR

This paper explores the computational limits of Turing Machines under uncertain inputs, focusing on metastability, and introduces a hardware-realizable universal machine for metastable closure computation.

Contribution

It establishes the non-computability of metastable closure in general, analyzes specific cases for EXPTIME and polynomial time problems, and presents a hardware-realizable universal Turing Machine.

Findings

Metastable closure of a Turing Machine is generally non-computable.

Resolving a single uncertain bit in EXPTIME problems is EXPTIME-complete.

A universal Turing Machine can compute metastable closure with exponential time overhead.

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.