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We investigate online monitoring algorithms over dense-time and continuous-time signals for properties written in metric temporal logic (MTL). We consider an abstract algebraic semantics based on complete lattices. This semantics includes as special cases the standard Boolean (qualitative) semantics and the widely-used real-valued robustness (quantitative) semantics. Our semantics also extends to truth values that are partially ordered and allows the modeling of uncertainty in satisfaction. We propose a compositional approach for the construction of online monitors that transform exact representations of piecewise constant (dense-time and continuous-time) signals. These monitors are based on a class of infinite-state deterministic signal transducers that (1) are allowed to produce the output signal with some bounded delay relative to the input signal, and (2) do not introduce unbounded variability in the output signal. A key ingredient of our monitoring framework is an efficient algorithm for sliding-window aggregation over dense-time signals. We have implemented and experimentally evaluated our monitoring framework by comparing it to the recently proposed online monitoring tools Reelay and RTAMT. 

Regular expressions (regexes) are ubiquitous in modern software. There is a variety of implementation techniques for regex matching, which can be roughly categorized as (1) relying on backtracking search, or (2) being based on finite-state automata. The implementations that use backtracking are often chosen due to their ability to support advanced pattern-matching constructs. Unfortunately, they are known to suffer from severe performance problems. For some regular expressions, the running time for matching can be exponential in the size of the input text. In order to provide stronger guarantees of matching efficiency, automata-based regex matching is the preferred choice. However, even these regex engines may exhibit severe performance degradation for some patterns. The main reason for this is that regexes used in practice are not exclusively built from the classical regular constructs, i.e., concatenation, nondeterministic choice and Kleene's star. They involve additional constructs that provide succinctness and convenience of expression. The most common such construct is bounded repetition (also called counting), which describes the repetition of the pattern a fixed number of times. In this paper, we propose a new algorithm for the efficient matching of regular expressions that involve bounded repetition. Our algorithms are based on a new model of automata, which we call nondeterministic bit vector automata (NBVA). This model is chosen to be expressively equivalent to nondeterministic counter automata with bounded counters, a very natural model for expressing patterns with bounded repetition. We show that there is a class of regular expressions with bounded repetition that can be matched in time that is independent from the repetition bounds. Our algorithms are general enough to cover the vast majority of challenging bounded repetitions that arise in practice. We provide an implementation of our approach in a regex engine, which we call BVA-Scan. We compare BVA-Scan against state-of-the-art regex engines on several real datasets.