Graduate Thesis Or Dissertation
 

A Multiresolution Analysis of Temporal Logic

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/w0892k099

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  • Is it possible to determine whether a signal violates a formula in Signal Temporal Logic (STL), if the monitor only has access to a low-resolution version of the signal? We answer this question affirmatively by demonstrating that temporal logic has a multiresolution structure, which parallels the multiresolution structure of signals. A formula in discrete-time Signal Temporal Logic (STL) is equivalently defined via the set of signals that satisfy it, known as its language. If a wavelet decomposition x = y + d is performed on each signal x in the language, we end up with two signal sets Y and D, where Y contains the low-resolution approximation signals y, and D contains the detail signals d needed to reconstruct the x’s. This thesis provides a complete computational characterization of both Y and D using a novel constraint set encoding of STL, s.t. x satisfies a formula if and only if its decomposition signals satisfy their respective encoding constraints. We obtain a sequence of lower-resolution formulas φ−1, φ−2, φ−3, ... which thus constitute a multiresolution analysis of φ. This work lays the foundation for multiresolution monitoring in distributed systems. One potential application of these results is a multiresolution monitor that can detect specification violation early by simply observing a low-resolution version of the signal to be monitored. We illustrate these results with experiments on synthetic signals.
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