The probabilistic bisimilarity distance provides a quantitative measure of behavioural difference for labelled Markov chains, but it may be discontinuous under perturbations of the transition probabilities. This lack of continuity undermines its applicability to empirically derived models, where transition probabilities are often approximations. Recently, we introduced robust probabilistic bisimilarity as a sufficient condition for continuity at distance zero. In this paper, we show that it is also a necessary condition, that is, two states are robustly probabilistic bisimilar if and only if their probabilistic bisimilarity distance is small for any small enough perturbation of the transition probabilities. We further extend robustness to non-bisimilar state pairs to establish a complete characterization for continuity of the probabilistic bisimilarity distance. Based on this characterization, we develop a polynomial time algorithm to decide continuity. Finally, we complement our theoretical contributions with an experimental evaluation demonstrating the proposed approach in practice. Our results show that the extra step of deciding continuity requires minimal additional cost when compared to computing the probabilistic bisimilarity distance.

Abstract Despite its prevalence, probabilistic bisimilarity suffers from a lack of robustness under minuscule perturbations of the transition probabilities. This can lead to discontinuities in the probabilistic bisimilarity distance function, undermining its reliability in practical applications where transition probabilities are often approximations derived from experimental data. Motivated by this limitation, we introduce the notion of robust probabilistic bisimilarity for labelled Markov chains, which ensures the continuity of the probabilistic bisimilarity distance function. We also propose an efficient algorithm for computing robust probabilistic bisimilarity and show that it performs well in practice, as evidenced by our experimental results.

An 8-channel wearable wireless device for surface EEG monitoring is presented. The entire multi-channel recording, quantization, and motion artifact removal is implemented on a 4-layer polyimide flexible substrate. The recording electrodes and active shielding are also integrated on the same substrate, yielding the smallest form factor compared to the state of the art. Thanks to the dry non-contact electrodes, the system is quickly mountable with minimal assistance required, making it an ideal frontal and temporal-lobe EEG monitoring device in emergency departments. The flexible main board is connected to a rechargeable battery on one end and to a 13×17mm2rigid board on the other end. The mini rigid board hosts a low-power programmable FPGA and a BLE 5.0 transceiver, which add diagnostic capability and wireless operation features to the device, respectively. The device performance in terms of voltage gain (260 V/V), bandwidth (DC-700 Hz), input-referred noise, motion artifact removal, and wireless communication throughput (up to 1Mbps) is experimentally validated and the overall power consumption is measured to be 27mW. The entire wearable solution with the battery weight 9.2 grams.