A Second-Order Approach to Complex Event Recognition

TL;DR

This paper introduces SO-CEL, a second-order logic-based language for complex event recognition, unifying semantics and output, and demonstrating its expressive power and relation to automata models.

Contribution

It proposes SO-CEL, a novel second-order logic language for CER, compares its expressiveness with FO-CEL, and connects it to automata models, advancing formal understanding.

Findings

SO-CEL is equivalent to FO-CEL with unary predicates but incomparable in general.

Under binary predicates, SO-CEL is more expressive than FO-CEL.

SO-CEL captures a subclass of Unary Complex Event Automata satisfying the *-property.

Abstract

Complex Event Recognition (CER for short) refers to the activity of detecting patterns in streams of continuously arriving data. This field has been traditionally approached from a practical point of view, resulting in heterogeneous implementations with fundamentally different capabilities. The main reason behind this is that defining formal semantics for a CER language is not trivial: they usually combine first-order variables for joining and filtering events with regular operators like sequencing and Kleene closure. Moreover, their semantics usually focus only on the detection of complex events, leaving the concept of output mostly unattended. In this paper, we propose to unify the semantics and output of complex event recognition languages by using second order objects. Specifically, we introduce a CER language called Second Order Complex Event Logic (SO-CEL for short), that uses second order variables for managing and outputting sequences of events. This makes the definition of the main CER operators simple, allowing us to develop the first steps in understanding its expressive power. We start by comparing SO-CEL with a version that uses first-order variables called FO-CEL, showing that they are equivalent in expressive power when restricted to unary predicates but, surprisingly, incomparable in general. Nevertheless, we show that if we restrict to sets of binary predicates, then SO-CEL is strictly more expressive than FO-CEL. Then, we introduce a natural computational model called Unary Complex Event Automata (UCEA) that provides a better understanding of SO-CEL. We show that, under unary predicates, SO-CEL captures the subclass of UCEA that satisfy the so-called *-property. Finally, we identify the operations that SO-CEL is lacking to capture UCEA and introduce a natural extension of the language that captures the complete class of UCEA under unary predicates.