We show that metric temporal logic can be viewed as linear time-invariant filtering, by interpreting addition, multiplication, and their neutral elements, over the (max,min,0,1) idempotent dioid. Moreover, by interpreting these operators over the field of reals (+,*,0,1), one can associate various quantitative semantics to a metric-temporal-logic formula, depending on the filter's kernel used: square, rounded-square, Gaussian, low-pass, band-pass, or high-pass. This remarkable connection between filtering and metric temporal logic allows us to freely navigate between the two, and to regard signal-feature detection as logical inference. To the best of our knowledge, this connection has not been established before. We prove that our qualitative, filtering semantics is identical to the classical MTL semantics. We also provide a quantitative semantics for MTL, which measures the normalized, maximum number of times a formula is satisfied within its associated kernel, by a given signal. We show that this semantics is sound, in the sense that, if its measure is 0, then the formula is not satisfied, and it is satisfied otherwise. We have implemented both of our semantics in Matlab, and illustrate their properties on various formulas and signals, by plotting their computed measures.Comment: 10 page

Abstract. Motivated by the problem of verifying the correctness of arrhythmia-detection algorithms, we present a formalization of these algorithms in the language of Quantitative Regular Expressions. QREs are a flexible formal language for specifying complex numerical queries over data streams, with provable runtime and memory consumption guarantees. The medical-device algorithms of interest include peak detection (where a peak in a cardiac signal indicates a heartbeat) and various discriminators, each of which uses a feature of the cardiac signal to distinguish fatal from non-fatal arrhythmias. Expressing these algorithms' desired output in current temporal logics, and implementing them via monitor synthesis, is cumbersome, error-prone, computationally expensive, and sometimes infeasible. In contrast, we show that a range of peak detectors (in both the time and wavelet domains) and various discriminators at the heart of today's arrhythmia-detection devices are easily expressible in QREs. The fact that one formalism (QREs) is used to describe the desired end-to-end operation of an arrhythmia detector opens the way to formal analysis and rigorous testing of these detectors' correctness and performance. Such analysis could alleviate the regulatory burden on device developers when modifying their algorithms. The performance of the peak-detection QREs is demonstrated by running them on real patient data, on which they yield results on par with those provided by a cardiologist.

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The testing of Autonomous Vehicles (AVs) requires driving the AV billions of miles under varied scenarios in order to find bugs, accidents and otherwise inappropriate behavior. Because driving a real AV that many miles is too slow and costly, this motivates the use of sophisticated `world simulators', which present the AV's perception pipeline with realistic input scenes, and present the AV's control stack with realistic traffic and physics to which to react. Thus the simulator is a crucial piece of any CAD toolchain for AV testing. In this work, we build a test harness for driving an arbitrary AV's code in a simulated world. We demonstrate this harness by using the game Grand Theft Auto V (GTA) as world simulator for AV testing. Namely, our AV code, for both perception and control, interacts in real-time with the game engine to drive our AV in the GTA world, and we search for weather conditions and AV operating conditions that lead to dangerous situations. This goes beyond the current state-of-the-art where AVs are tested under ideal weather conditions, and lays the ground work for a more comprehensive testing effort. We also propose and demonstrate necessary analyzes to validate the simulation results relative to the real world. The results of such analyses allow the designers and verification engineers to weigh the results of simulation-based testing.