The rise of automated systems and sensor networksin virtually all areas of industry and social life means many technologies produce streams of events rich with information. These technologies demand algorithms for evaluating queries on streams, coordinating systems that communicate with events, and monitoring streams with respect to specified constraints. In monitoring, constraints that define correct behavior, e.g., business goals, legal requirements, resource limitations, or safety and security concerns are specified in a formal language; then, an event stream is analyzed at runtime to determine if the constraints are satisfied or violated. To make monitoring effective, it is important to detect constraint violations at the earliest possible time, which we call the early violation detection problem.

We study early violation detection for a class of constraints called rulesthat restrict time gaps between events and compare events' data contents. We show that (1) the general problem of early violation detection for an arbitrary set of rules is unsolvable and (2) early violation detection is possible for various subclasses of rules. For (1), we show early violation detection is closely related to the problem of finite satisfiability (whether or not a given set of rules can be satisfied by a finite event stream) and prove that finite satisfiability for a set of rules is undecidable with a reduction from the empty-tape Turing machine halting problem, which implies that early violation detection is unsolvable in general. For (2), we study restricted classes of rules. A recent proof of Kamp's Theorem provides a translation algorithm for ``dataless'' rules through translation to linear temporal logic. yielding formulas hyper-exponential in the size of the input rule. We present translation algorithms for two subclasses of dataless rules, improving the output size from hyper-exponential to single- and double-exponential, respectively. For rules with data, we first present a technique that calculates deadlines from time gaps between events, then use deadlines for early violation detection for individual rules. We extend these algorithms to monitor an acyclic set of rules by applying a chase process and satisfiability testing. We also report the performance of an implementation of these algorithms. Finally, we consider acyclic sets of rules with aggregation functions over time windows, combining the chase and satisfiability techniques with an encoding of aggregation functions in Presburger arithmetic.