Equivalence of Decentralized Observation, Diagnosis, and Control Problems in Discrete-event Systems

TL;DR

This paper proves that decentralized observation, diagnosis, and control problems in discrete-event systems are fundamentally equivalent in computational complexity, revealing their shared undecidability and enabling problem decomposition.

Contribution

It establishes a formal Turing equivalence among these problems, allowing their decomposition and transfer of undecidability results across problem types.

Findings

Control problems are undecidable due to their equivalence with observation problems.

Diagnosis problems are also undecidable, consistent with previous results.

Observation problems are shown to be Turing equivalent to control and diagnosis problems.

Abstract

This paper demonstrates an equivalence between observation problems, control problems (with partial observation), and diagnosis problems of decentralized discrete-event systems, namely, the three classes of problems are Turing equivalent, as one class Turing reduces to another. The equivalence allows decomposition of a control problem into a collection of simpler control sub\-/problems, which are each equivalent to an observation problem; and similarly allows converting a diagnosis problem to a formally simpler observation problem. Since observation problems in their most general formulation have been shown to be undecidable in previous work, the equivalence produced here demonstrates that control problems are also undecidable; whereas the undecidability of diagnosis problems is a known result.