Abstract
In this letter, in discrete-event systems modeled by labeled finite-state automata (LFSAs), we show new thinking on the tools of detector and concurrent composition and derive two new algorithms for verifying strong periodic detectability (SPD) without any assumption that run in NL; we also reconsider the tool of observer and derive a new algorithm for verifying strong periodic Ddetectability (SPDD) without any assumption that runs in PSPACE. These results strengthen the NL upper bound on verifying SPD and the PSPACE upper bound on verifying SPDD for deadlock-free and divergence-free LFSAs in the literature. In our algorithms, the two assumptions are removed by verifying the negations of these properties.