Complexity of Deciding Detectability in Discrete Event Systems

TL;DR

This paper investigates the computational complexity of detectability in discrete event systems, demonstrating that even simple systems have intractable weak detectability, while strong detectability can be efficiently verified.

Contribution

It proves that weak detectability remains intractable for simple DESs, but shows that strong detectability can be efficiently verified using parallel computation.

Findings

Weak detectability is PSpace-complete even for simple DESs.

Strong detectability can be verified efficiently with parallel algorithms.

Structural simplicity does not guarantee tractability of detectability problems.

Abstract

Detectability of discrete event systems (DESs) is a question whether the current and subsequent states can be determined based on observations. Shu and Lin designed a polynomial-time algorithm to check strong (periodic) detectability and an exponential-time (polynomial-space) algorithm to check weak (periodic) detectability. Zhang showed that checking weak (periodic) detectability is PSpace-complete. This intractable complexity opens a question whether there are structurally simpler DESs for which the problem is tractable. In this paper, we show that it is not the case by considering DESs represented as deterministic finite automata without non-trivial cycles, which are structurally the simplest deadlock-free DESs. We show that even for such very simple DESs, checking weak (periodic) detectability remains intractable. On the contrary, we show that strong (periodic) detectability of DESs can be efficiently verified on a parallel computer.