  Runtime verification is an effective automated method for specification-based
offline testing and analysis as well as online monitoring of complex systems.
The specification language is often a variant of regular expressions or a
popular temporal logic, such as LTL. This paper presents a novel and efficient
parallel algorithm for verifying a more expressive version of LTL
specifications that incorporates counting semantics, where nested quantifiers
can be subject to numerical constraints. Such constraints are useful in
evaluating thresholds (e.g., expected uptime of a web server). The significance
of this extension is that it enables us to reason about the correctness of a
large class of systems, such as web servers, OS kernels, and network behavior,
where properties are required to be instantiated for parameterized requests,
kernel objects, network nodes, etc. Our algorithm uses the popular {\em
MapReduce} architecture to split a program trace into variable-based clusters
at run time. Each cluster is then mapped to its respective monitor instances,
verified, and reduced collectively on a multi-core CPU or the GPU. Our
algorithm is fully implemented and we report very encouraging experimental
results, where the monitoring overhead is negligible on real-world data sets.
