A Multiresolution Analysis of Temporal Logic

Abstract

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 paper 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. Then a conservative logical approximation of Y is also provided: namely, we show that Y is over approximated by the language of a formula − 1. By iterating the decomposition, we obtain a sequence of lower-resolution formulas − 1, − 2, − 3,... which thus constitute a multiresolution analysis of. 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. 1